{"id":2400,"date":"2019-06-11T11:35:32","date_gmt":"2019-06-11T15:35:32","guid":{"rendered":"https:\/\/opentextbc.ca\/introalgebra\/chapter\/use-properties-of-angles-triangles-and-the-pythagorean-theorem\/"},"modified":"2025-09-22T19:26:34","modified_gmt":"2025-09-22T23:26:34","slug":"use-properties-of-angles-triangles-and-the-pythagorean-theorem","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/introalgebra\/chapter\/use-properties-of-angles-triangles-and-the-pythagorean-theorem\/","title":{"raw":"9.1 Use Properties of Angles, Triangles, and the Pythagorean Theorem","rendered":"9.1 Use Properties of Angles, Triangles, and the Pythagorean Theorem"},"content":{"raw":"[latexpage]\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li>Use the properties of angles<\/li>\r\n \t<li>Use the properties of triangles<\/li>\r\n \t<li>Use the Pythagorean Theorem<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<h1 data-type=\"title\">Use the Properties of Angles<\/h1>\r\n<p id=\"fs-id1172574714298\">Are you familiar with the phrase \u2018do a \\(180\\text{\u2019?}\\) It means to make a full turn so that you face the opposite direction. It comes from the fact that the measure of an angle that makes a straight line is \\(180\\) degrees. See <a class=\"autogenerated-content\" href=\"#CNX_BMath_Figure_09_03_046\">(Figure 1)<\/a>.<\/p>\r\n\r\n<div id=\"CNX_BMath_Figure_09_03_046\" class=\"bc-figure figure\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"152\"]<img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2020\/06\/CNX_BMath_Figure_09_03_046_img.jpg\" alt=\"The image is a straight line with an arrow on each end. There is a dot in the center. There is an arrow pointing from one side of the dot to the other, and the angle is marked as 180 degrees.\" width=\"152\" height=\"47\" data-media-type=\"image\/jpeg\" \/> Figure 1[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1172591978152\">An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle and the common endpoint is called the vertex. An angle is named by its vertex. In <a class=\"autogenerated-content\" href=\"#CNX_BMath_Figure_09_03_001\">(Figure 2)<\/a>, \\(\\angle A\\) is the angle with vertex at point \\(A\\). The measure of \\(\\angle A\\) is written \\(m\\angle A\\).<\/p>\r\n\r\n<div id=\"CNX_BMath_Figure_09_03_001\" class=\"bc-figure figure\">\r\n<div class=\"bc-figcaption figcaption\">\\(\\angle A\\) is the angle with vertex at \\(\\text{point}\\phantom{\\rule{0.2em}{0ex}}A\\).<\/div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"201\"]<img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_001.jpg\" alt=\"The image is an angle made up of two rays. The angle is labeled with letter A.\" width=\"201\" height=\"73\" data-media-type=\"image\/jpeg\" \/> Figure 2[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1172584183371\">We measure angles in degrees, and use the symbol \u00b0\u00a0 to represent degrees. We use the abbreviation \\(m\\) to for the <em data-effect=\"italics\">measure<\/em> of an angle. So if \\(\\angle A\\) is 27\u00b0, we would write \\(m\\angle A=27\\).<\/p>\r\n<p id=\"fs-id1411514\">If the sum of the measures of two angles is \\(\\text{180}\\)\u00b0, then they are called supplementary angles. In <a class=\"autogenerated-content\" href=\"#CNX_BMath_Figure_09_03_003\">(Figure 3)<\/a>, each pair of angles is supplementary because their measures add to \\(\\text{180}\\)\u00b0. Each angle is the <em data-effect=\"italics\">supplement<\/em> of the other.<\/p>\r\n\r\n<div id=\"CNX_BMath_Figure_09_03_003\" class=\"bc-figure figure\">\r\n<div class=\"bc-figcaption figcaption\">The sum of the measures of supplementary angles is \\(\\text{180}\\)\u00b0.<\/div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"574\"]<img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_003.jpg\" alt=\"Part a shows a 120 degree angle next to a 60 degree angle. Together, the angles form a straight line. Below the image, it reads 120 degrees plus 60 degrees equals 180 degrees. Part b shows a 45 degree angle attached to a 135 degree angle. Together, the angles form a straight line. Below the image, it reads 45 degrees plus 135 degrees equals 180 degrees.\" width=\"574\" height=\"177\" data-media-type=\"image\/jpeg\" \/> Figure 3[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1277381\">If the sum of the measures of two angles is \\(\\text{90}\\)\u00b0, then the angles are complementary angles. In <a class=\"autogenerated-content\" href=\"#CNX_BMath_Figure_09_03_004\">(Figure 4)<\/a>, each pair of angles is complementary, because their measures add to \\(\\text{90}\\)\u00b0. Each angle is the <em data-effect=\"italics\">complement<\/em> of the other.<\/p>\r\n\r\n<div id=\"CNX_BMath_Figure_09_03_004\" class=\"bc-figure figure\">\r\n<div class=\"bc-figcaption figcaption\">The sum of the measures of complementary angles is \\(\\text{90}\\)\u00b0.<\/div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"286\"]<img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_004.jpg\" alt=\"Part a shows a 50 degree angle next to a 40 degree angle. Together, the angles form a right angle. Below the image, it reads 50 degrees plus 40 degrees equals 90 degrees. Part b shows a 60 degree angle attached to a 30 degree angle. Together, the angles form a right angle. Below the image, it reads 60 degrees plus 30 degrees equals 90 degrees.\" width=\"286\" height=\"185\" data-media-type=\"image\/jpeg\" \/> Figure 4[\/caption]\r\n\r\n<\/div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Supplementary and Complementary Angles<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1136622\">If the sum of the measures of two angles is \\(\\text{180}\\)\u00b0, then the angles are supplementary.<\/p>\r\n<p id=\"eip-173\">If \\(\\angle A\\) and \\(\\angle B\\) are supplementary, then \\(m\\angle A+m\\angle B=\\text{180}\\)\u00b0.<\/p>\r\n<p id=\"eip-id1165148725040\">If the sum of the measures of two angles is \\(\\text{90}\\)\u00b0, then the angles are complementary.<\/p>\r\n<p id=\"eip-id1165144076222\">If \\(\\angle A\\) and \\(\\angle B\\) are complementary, then \\(m\\angle A+m\\angle B=\\text{90}\\)\u00b0.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1610226\">In this section and the next, you will be introduced to some common geometry formulas. We will adapt our Problem Solving Strategy for Geometry Applications. The geometry formula will name the variables and give us the equation to solve.<\/p>\r\n<p id=\"fs-id1172596651337\">In addition, since these applications will all involve geometric shapes, it will be helpful to draw a figure and then label it with the information from the problem. We will include this step in the Problem Solving Strategy for Geometry Applications.<\/p>\r\n\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">HOW TO: Use a Problem Solving Strategy for Geometry Applications<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol id=\"eip-id1170325410595\" class=\"stepwise\" type=\"1\">\r\n \t<li><strong data-effect=\"bold\">Read<\/strong> the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.<\/li>\r\n \t<li><strong>Identify<\/strong> what you are looking for.<\/li>\r\n \t<li><strong>Name<\/strong> what you are looking for and choose a variable to represent it.<\/li>\r\n \t<li><strong data-effect=\"bold\">Translate<\/strong> into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.<\/li>\r\n \t<li><strong data-effect=\"bold\">Solve<\/strong> the equation using good algebra techniques.<\/li>\r\n \t<li><strong data-effect=\"bold\">Check<\/strong> the answer in the problem and make sure it makes sense.<\/li>\r\n \t<li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n\r\nThe next example will show how you can use the Problem Solving Strategy for Geometry Applications to answer questions about supplementary and complementary angles.\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1172599768030\" data-type=\"problem\">\r\n<p id=\"fs-id1468569\">An angle measures \\(\\text{40}\\)\u00b0. Find a) its supplement, and b) its complement.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1222879\" data-type=\"solution\">\r\n<div data-type=\"title\">Solution<\/div>\r\n<table id=\"eip-id1168469720851\" style=\"width: 100%;\" summary=\"Step 1 says, \u201cRead the problem. Draw the figure and label it with the given information.\u201d The word \u201cread\u201d is in bold. Beside this is a picture of two angles that together form a right angle. One is labeled as c degrees, the other as 40 degrees. Step 2 says, \u201cIdentify what you are looking for.\u201d The word \u201cidentify\u201d is in bold. Beside this it says, \u201cthe complement of a 40 degree angle.\u201d Step 3 says, \u201cName. Choose a variable to represent it.\u201d The word \u201cname\u201d is in bold. Beside this it says, \u201clet c equal the measure of the complement.\u201d Step 4 says, \u201cTranslate. Write the appropriate formula for the situation and substitute in the given information.\u201d The word \u201ctranslate\u201d is in bold. Beside this is the equation c plus 40 equals 90. Step 5 says, \u201cSolve the equation.\u201d The word \u201csolve\u201d is in bold. Beside this is c equals 50. Step 6 says \u201cCheck\u201d in bold. Beside this is 50 plus 40 followed by an equal sign with a question mark, then 90. The next line says 90 equals 90 with a check mark. Step 7 says, \u201cAnswer the question.\u201d The word \u201canswer\u201d is in bold. Beside this is the answer, \u201cThe complement of the 40 degree angle is 50 degrees.\u201d\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>a)<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 1. <strong data-effect=\"bold\">Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n<td><span id=\"eip-id1168466323010\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_047_img-01.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong data-effect=\"bold\">Identify<\/strong> what you are looking for.<\/td>\r\n<td><span id=\"eip-id1168469768902\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_047_img-02.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong data-effect=\"bold\">Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td><span id=\"eip-id1168469696175\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_047_img-03.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong data-effect=\"bold\">Translate.<\/strong>\r\nWrite the appropriate formula for the situation and substitute in the given information.<\/td>\r\n<td><span id=\"eip-id1168466314136\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_047_img-04.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168469631749\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_047_img-05.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong data-effect=\"bold\">Solve<\/strong> the equation.<\/td>\r\n<td><span id=\"eip-id1168469420724\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_047_img-06.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong data-effect=\"bold\">Check:<\/strong>\r\n<span id=\"eip-id1168469832615\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_047_img-07.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168469436943\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_047_img-08.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\r\n<td><span id=\"eip-id1168466286479\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_047_img-09.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168468558393\" style=\"width: 100%;\" summary=\"Step 1 says, \u201cRead the problem. Draw the figure and label it with the given information.\u201d The word \u201cread\u201d is in bold. Beside this is a picture of two angles that together form a right angle. One is labeled as c degrees, the other as 40 degrees. Step 2 says, \u201cIdentify what you are looking for.\u201d The word \u201cidentify\u201d is in bold. Beside this it says, \u201cthe complement of a 40 degree angle.\u201d Step 3 says, \u201cName. Choose a variable to represent it.\u201d The word \u201cname\u201d is in bold. Beside this it says, \u201clet c equal the measure of the complement.\u201d Step 4 says, \u201cTranslate. Write the appropriate formula for the situation and substitute in the given information.\u201d The word \u201ctranslate\u201d is in bold. Beside this is the equation c plus 40 equals 90. Step 5 says, \u201cSolve the equation.\u201d The word \u201csolve\u201d is in bold. Beside this is c equals 50. Step 6 says \u201cCheck\u201d in bold. Beside this is 50 plus 40 followed by an equal sign with a question mark, then 90. The next line says 90 equals 90 with a check mark. Step 7 says, \u201cAnswer the question.\u201d The word \u201canswer\u201d is in bold. Beside this is the answer, \u201cThe complement of the 40 degree angle is 50 degrees.\u201d\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>b)<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 1. <strong data-effect=\"bold\">Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n<td><span id=\"eip-id1168467117413\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_048_img-01.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong data-effect=\"bold\">Identify<\/strong> what you are looking for.<\/td>\r\n<td><span id=\"eip-id1168466083057\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_048_img-02.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong data-effect=\"bold\">Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td><span id=\"eip-id1168466051411\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_048_img-03.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong data-effect=\"bold\">Translate.<\/strong>\r\nWrite the appropriate formula for the situation and substitute in the given information.<\/td>\r\n<td><span id=\"eip-id1168469696180\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_048_img-04.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong data-effect=\"bold\">Solve<\/strong> the equation.<\/td>\r\n<td><span id=\"eip-id1168469428078\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_048_img-05.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168466186330\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_048_img-06.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong data-effect=\"bold\">Check:<\/strong>\r\n<span id=\"eip-id1168469478162\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_048_img-07.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168469726019\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_048_img-08.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\r\n<td><span id=\"eip-id1168469778644\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_048_img-09.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 1.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1280259\" data-type=\"problem\">\r\n<p id=\"fs-id1037597\">An angle measures \\(\\text{25}\\)\u00b0. Find its: a) supplement b) complement.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1296496\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<ol id=\"eip-id1168469681484\" class=\"circled\" type=\"a\">\r\n \t<li>155\u00b0<\/li>\r\n \t<li>65\u00b0<\/li>\r\n<\/ol>\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1322685\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1467711\" data-type=\"exercise\">\r\n<div id=\"fs-id1296496\" data-type=\"solution\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 1.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1318314\" data-type=\"problem\">\r\n<p id=\"fs-id1699286\">An angle measures \\(\\text{77}\\)\u00b0. Find its: a) supplement b) complement.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1315018\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<ol id=\"eip-id1168469790669\" class=\"circled\" type=\"a\">\r\n \t<li>103\u00b0<\/li>\r\n \t<li>13\u00b0<\/li>\r\n<\/ol>\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\nDid you notice that the words complementary and supplementary are in alphabetical order just like \\(90\\) and \\(180\\) are in numerical order?\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1280482\" data-type=\"problem\">\r\n<p id=\"fs-id1296703\">Two angles are supplementary. The larger angle is \\(\\text{30}\\)\u00b0 more than the smaller angle. Find the measure of both angles.<\/p>\r\n<strong>Solution<\/strong>\r\n\r\n<\/div>\r\n<div id=\"fs-id1435893\" data-type=\"solution\">\r\n<table id=\"eip-id1168468587362\" style=\"width: 100%;\" summary=\"Step 1 says, \u201cRead the problem. Draw the figure and label it with the given information.\u201d The word \u201cread\u201d is in bold. Beside this is a picture of two angles that together form a straight line. One is labeled as a plus 30 degrees, the other as a degrees. Step 2 says, \u201cIdentify what you are looking for.\u201d The word \u201cidentify\u201d is in bold. Beside this it says, \u201cthe measures of both angles.\u201d Step 3 says, \u201cName. Choose a variable to represent it. \u201c The word \u201cname\u201d is in bold. Beside this it says, \u201clet a equal the measure of the smaller angle.\u201d Below this, it says, \u201cThe larger angle is 30 degrees more than the smaller angle.\u201d Beside this is a plus 30 equals measure of larger angle. Step 4 says, \u201cTranslate. Write the appropriate formula and substitute.\u201d The word \u201ctranslate\u201d is in bold. Beside this is the equation measure of angle A plus measure of angle B equals 180. Step 5 says, \u201cSolve the equation.\u201d The word \u201csolve\u201d is in bold. Beside this is a plus 30 plus a equals 180, then 2a plus 30 equals 180, then 2a equals 150, then a equals 75. Step 6 says \u201cCheck\u201d in bold. Beside this is measure of angle A plus measure of angle B equals 180, then 75 plus 105 followed by an equal sign with a question mark, then 180. The next line says 180 equals 180 with a check mark. Step 7 says, \u201cAnswer the question.\u201d The word \u201canswer\u201d is in bold. Beside this is the answer, \u201cThe measures of the angles are 75 degrees and 105 degrees.\u201d\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong data-effect=\"bold\">Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n<td><span id=\"eip-id1168468587428\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-01.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong data-effect=\"bold\">Identify<\/strong> what you are looking for.<\/td>\r\n<td><span id=\"eip-id1168467187831\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-02.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong data-effect=\"bold\">Name.<\/strong> Choose a variable to represent it.\r\nThe larger angle is 30\u00b0 more than the smaller angle.<\/td>\r\n<td><span id=\"eip-id1168468428731\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-03.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168467451148\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-04.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong data-effect=\"bold\">Translate.<\/strong>\r\nWrite the appropriate formula and substitute.<\/td>\r\n<td><span id=\"eip-id1168467419147\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-05.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\">Step 5. <strong data-effect=\"bold\">Solve<\/strong> the equation.<\/td>\r\n<td><span id=\"eip-id1168467248476\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-06.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168467248485\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-07.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168467248494\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-08.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168468773960\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-09.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168468773969\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-10.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168468773978\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-11.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168468451141\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-12.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong data-effect=\"bold\">Check:<\/strong>\r\n<span id=\"eip-id1168467263637\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-13.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168467263646\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-14.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168467263655\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-15.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\r\n<td><span id=\"eip-id1168468458480\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-16.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 2.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1141164\" data-type=\"problem\">\r\n<p id=\"fs-id1546937\">Two angles are supplementary. The larger angle is \\(\\text{100}\\)\u00b0 more than the smaller angle. Find the measures of both angles.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1716497\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1676187\">40\u00b0, 140\u00b0<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1399790\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1431406\" data-type=\"exercise\">\r\n<div id=\"fs-id1716497\" data-type=\"solution\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 2.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1172591784744\" data-type=\"problem\">\r\n<p id=\"fs-id754602\">Two angles are complementary. The larger angle is \\(\\text{40}\\)\u00b0 more than the smaller angle. Find the measures of both angles.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1172599718234\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170856\">25\u00b0, 65\u00b0<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h1>Use the Properties of Triangles<\/h1>\r\n<p id=\"fs-id1415833\">What do you already know about triangles? Triangle have three sides and three angles. Triangles are named by their vertices. The triangle in <a class=\"autogenerated-content\" href=\"#CNX_BMath_Figure_09_03_011\">(Figure 5)<\/a> is called \\(\\Delta ABC\\), read \u2018triangle \\(\\text{ABC}\\)\u2019. We label each side with a lower case letter to match the upper case letter of the opposite vertex.<\/p>\r\n\r\n<div id=\"CNX_BMath_Figure_09_03_011\" class=\"bc-figure figure\">\r\n<div class=\"bc-figcaption figcaption\">\\(\\Delta ABC\\) has vertices \\(A,B,\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}C\\) and sides \\(a,b,\\text{and}\\phantom{\\rule{0.2em}{0ex}}c\\text{.}\\)<\/div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"215\"]<img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_011.jpg\" alt=\"The vertices of the triangle on the left are labeled A, B, and C. The sides are labeled a, b, and c.\" width=\"215\" height=\"148\" data-media-type=\"image\/jpeg\" \/> Figure 5[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1172599773381\">The three angles of a triangle are related in a special way. The sum of their measures is \\(\\text{180}\\)\u00b0.<\/p>\r\n\r\n<div id=\"fs-id1171104037911\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(m\\angle A+m\\angle B+m\\angle C=\\text{180}\\)\u00b0<\/div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Sum of the Measures of the Angles of a Triangle<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1413952\">For any \\(\\Delta ABC\\), the sum of the measures of the angles is \\(\\text{180}\\)\u00b0.<\/p>\r\n\r\n<div id=\"fs-id1171102911830\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(m\\angle A+m\\angle B+m\\angle C=\\text{180}\\)\u00b0<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1171102911830\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1334312\" data-type=\"problem\">\r\n<p id=\"fs-id1371881\">The measures of two angles of a triangle are \\(\\text{55}\\)\u00b0 and \\(\\text{82}\\)\u00b0. Find the measure of the third angle.<\/p>\r\n<strong>Solution<\/strong>\r\n\r\n<\/div>\r\n<div id=\"fs-id1172591771964\" data-type=\"solution\">\r\n<table id=\"eip-id1168467289823\" style=\"width: 100%;\" summary=\"Step 1 says, \u201cRead the problem. Draw the figure and label it with the given information.\u201d The word \u201cread\u201d is in bold. Beside this is a picture of a triangle with angles x, 82 degrees, and 55 degrees. Step 2 says, \u201cIdentify what you are looking for.\u201d The word \u201cidentify\u201d is in bold. Beside this it says, \u201cthe measure of the third angle in a triangle.\u201d Step 3 says, \u201cName. Choose a variable to represent it. \u201c The word \u201cname\u201d is in bold. Beside this it says, \u201clet x equal the measure of the angle.\u201d Step 4 says, \u201cTranslate. Write the appropriate formula and substitute.\u201d The word \u201ctranslate\u201d is in bold. Beside this is the equation measure of angle A plus measure of angle B plus measure of angle C equals 180. Step 5 says, \u201cSolve the equation.\u201d The word \u201csolve\u201d is in bold. Beside this is 55 plus 82 plus x equals 180, then 137 plus x equals 180, then x equals 43. Step 6 says \u201cCheck\u201d in bold. Beside this is 55 plus 82 plus 43 followed by an equal sign with a question mark, then 180. The next line says 180 equals 180 with a check mark. Step 7 says, \u201cAnswer the question.\u201d The word \u201canswer\u201d is in bold. Beside this is the answer, \u201cThe measure of the third angle is 43 degrees.\u201d\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong data-effect=\"bold\">Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n<td><span id=\"eip-id1168467362989\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_050_img-01.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong data-effect=\"bold\">Identify<\/strong> what you are looking for.<\/td>\r\n<td><span id=\"eip-id1168468505614\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_050_img-02.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong data-effect=\"bold\">Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td><span id=\"eip-id1168468707381\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_050_img-03.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong data-effect=\"bold\">Translate.<\/strong>\r\nWrite the appropriate formula and substitute.<\/td>\r\n<td><span id=\"eip-id1168468607629\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_050_img-04.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\">Step 5. <strong data-effect=\"bold\">Solve<\/strong> the equation.<\/td>\r\n<td><span id=\"eip-id1168468607650\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_050_img-05.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168467381406\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_050_img-06.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168469791076\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_050_img-07.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong data-effect=\"bold\">Check:<\/strong>\r\n<span id=\"eip-id1168468304554\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_050_img-08.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168468304564\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_050_img-09.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\r\n<td><span id=\"eip-id1168467164198\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_050_img-10.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 3.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1205986\" data-type=\"problem\">\r\n<p id=\"fs-id1172599718040\">The measures of two angles of a triangle are \\(\\text{31}\\)\u00b0 and \\(\\text{128}\\)\u00b0. Find the measure of the third angle.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1126761\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1335560\">21\u00b0<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1725798\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1326165\" data-type=\"exercise\">\r\n<div id=\"fs-id1126761\" data-type=\"solution\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 3.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id805484\" data-type=\"problem\">\r\n<p id=\"fs-id1276835\">A triangle has angles of \\(\\text{49}\\)\u00b0 and \\(\\text{75}\\)\u00b0. Find the measure of the third angle.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1446328\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1316328\">56\u00b0<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h1>Right Triangles<\/h1>\r\n<div id=\"fs-id1190071\" class=\"bc-section section\" data-depth=\"1\">\r\n<p data-type=\"title\">Some triangles have special names. We will look first at the right triangle. A right triangle has one \\(\\text{90}\\)\u00b0 angle, which is often marked with the symbol shown in <a class=\"autogenerated-content\" href=\"#CNX_BMath_Figure_09_03_015\">(Figure 6)<\/a>.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"109\"]<img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_015_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box and labeled 90 degrees.\" width=\"109\" height=\"86\" data-media-type=\"image\/jpeg\" \/> Figure 6[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1142543\">If we know that a triangle is a right triangle, we know that one angle measures \\(\\text{90}\\)\u00b0 so we only need the measure of one of the other angles in order to determine the measure of the third angle.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1304195\" data-type=\"problem\">\r\n<p id=\"fs-id1401181\">One angle of a right triangle measures \\(\\text{28}\\)\u00b0. What is the measure of the third angle?<\/p>\r\n<strong>Solution<\/strong>\r\n\r\n<\/div>\r\n<div id=\"fs-id1609548\" data-type=\"solution\">\r\n<table id=\"eip-id1168468725720\" style=\"width: 100%;\" summary=\"Step 1 says, \u201cRead the problem. Draw the figure and label it with the given information.\u201d The word \u201cread\u201d is in bold. Beside this is a picture of a triangle with angles x, 90 degrees, and 28 degrees. Step 2 says, \u201cIdentify what you are looking for.\u201d The word \u201cidentify\u201d is in bold. Beside this it says, \u201cthe measure of an angle.\u201d Step 3 says, \u201cName. Choose a variable to represent it. \u201c The word \u201cname\u201d is in bold. Beside this it says, \u201clet x equal the measure of the angle.\u201d Step 4 says, \u201cTranslate. Write the appropriate formula and substitute.\u201d The word \u201ctranslate\u201d is in bold. Beside this is the equation measure of angle A plus measure of angle B plus measure of angle C equals 180, then x plus 90 plus 28 equals 180. Step 5 says, \u201cSolve the equation.\u201d The word \u201csolve\u201d is in bold. Beside this is x plus 118 equals 180, then x equals 62. Step 6 says \u201cCheck\u201d in bold. Beside this is 180 followed by an equal sign with a question mark, then 90 plus 28 plus 62. The next line says 180 equals 180 with a check mark. Step 7 says, \u201cAnswer the question.\u201d The word \u201canswer\u201d is in bold. Beside this is the answer, \u201cThe measure of the third angle is 62 degrees.\u201d\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong data-effect=\"bold\">Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n<td><span id=\"eip-id1168468725760\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_051_img-01.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong data-effect=\"bold\">Identify<\/strong> what you are looking for.<\/td>\r\n<td><span id=\"eip-id1168468297279\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_051_img-02.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong data-effect=\"bold\">Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td><span id=\"eip-id1168468297301\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_051_img-03.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong data-effect=\"bold\">Translate.<\/strong>\r\nWrite the appropriate formula and substitute.<\/td>\r\n<td><span id=\"eip-id1168468254449\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_051_img-04.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong data-effect=\"bold\">Solve<\/strong> the equation.<\/td>\r\n<td><span id=\"eip-id1168468254472\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_051_img-05.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168468254481\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_051_img-06.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168468287884\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_051_img-07.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong data-effect=\"bold\">Check:<\/strong>\r\n<span id=\"eip-id1168468287906\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_051_img-08.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168468287915\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_051_img-09.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\r\n<td><span id=\"eip-id1168467267556\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_051_img-10.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 4.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1437460\" data-type=\"problem\">\r\n<p id=\"fs-id1546221\">One angle of a right triangle measures \\(\\text{56}\\)\u00b0. What is the measure of the other angle?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1596284\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1257352\">34\u00b0<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1172339\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id963740\" data-type=\"exercise\">\r\n<div id=\"fs-id1596284\" data-type=\"solution\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 4.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1665526\" data-type=\"problem\">\r\n<p id=\"fs-id1332188\">One angle of a right triangle measures \\(\\text{45}\\)\u00b0. What is the measure of the other angle?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1182217\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1231568\">45\u00b0<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\nIn the examples so far, we could draw a figure and label it directly after reading the problem. In the next example, we will have to define one angle in terms of another. So we will wait to draw the figure until we write expressions for all the angles we are looking for.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 5<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1172580513515\" data-type=\"problem\">\r\n<p id=\"fs-id1151760\">The measure of one angle of a right triangle is \\(\\text{20}\\)\u00b0 more than the measure of the smallest angle. Find the measures of all three angles.<\/p>\r\n<strong>Solution<\/strong>\r\n\r\n<\/div>\r\n<div id=\"fs-id1580719\" data-type=\"solution\">\r\n<table id=\"eip-id1168466077134\" style=\"width: 100%;\" summary=\"Step 1 says, \u201cRead the problem.\u201d The word \u201cread\u201d is in bold. Beside this is a picture of a triangle with angles a, 90 degrees, and a plus 20 degrees. Step 2 says, \u201cIdentify what you are looking for.\u201d The word \u201cidentify\u201d is in bold. Beside this it says, \u201cthe measures of all three angles.\u201d Step 3 says, \u201cName. Choose a variable to represent it. \u201c The word \u201cname\u201d is in bold. Beside this it says, \u201cLet d equal first angle. d plus 20 equals second angle. 90 equals third angle (the right angle).\u201d Step 4 says, \u201cTranslate. Write the appropriate formula and substitute into the formula.\u201d The word \u201ctranslate\u201d is in bold. Beside this is the equation measure of angle A plus measure of angle B plus measure of angle C equals 180, then d plus d plus 20 plus 90 equals 180. Step 5 says, \u201cSolve the equation.\u201d The word \u201csolve\u201d is in bold. Beside this is 2d plus 110 equals 180, then 2d equals 70, then d equals 35 first angle, then d plus 20 second angle, then 35 plus 20, then 55, then 90 third angle. Step 6 says \u201cCheck\u201d in bold. Beside this is 35 plus 55 plus 90 followed by an equal sign with a question mark, then 180. The next line says 180 equals 180 with a check mark. Step 7 says, \u201cAnswer the question.\u201d The word \u201canswer\u201d is in bold. Beside this is the answer, \u201cThe three angles measure 35 degrees, 55 degrees, and 90 degrees.\u201d\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong data-effect=\"bold\">Read<\/strong> the problem.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong data-effect=\"bold\">Identify<\/strong> what you are looking for.<\/td>\r\n<td>the measures of all three angles<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\">Step 3. <strong data-effect=\"bold\">Name.<\/strong> Choose a variable to represent it.\r\n\r\nNow draw the figure and label it with the given information.<\/td>\r\n<td><span id=\"eip-id1168469701816\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-01.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168466426323\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-02.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168469683819\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-03.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168466096904\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-04.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong data-effect=\"bold\">Translate.<\/strong>\r\nWrite the appropriate formula and substitute into the formula.<\/td>\r\n<td><span id=\"eip-id1168466113853\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-05.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168466113862\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-06.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong data-effect=\"bold\">Solve<\/strong> the equation.<\/td>\r\n<td><span id=\"eip-id1168469634538\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-07.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168469829064\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-08.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168466047198\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-09.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168466047207\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-10.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168469613957\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-11.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168469691852\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-12.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168469659677\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-13.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong data-effect=\"bold\">Check:<\/strong>\r\n<span id=\"eip-id1168469846595\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-14.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<span id=\"eip-id1168466076009\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-15.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\r\n<td><span id=\"eip-id1168466395444\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-16.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 5.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1736498\" data-type=\"problem\">\r\n<p id=\"fs-id1172580513385\">The measure of one angle of a right triangle is \\(\\text{50}\\)\u00b0 more than the measure of the smallest angle. Find the measures of all three angles.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1333588\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1572312\">20\u00b0, 70\u00b0, 90\u00b0<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1156706\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1414965\" data-type=\"exercise\">\r\n<div id=\"fs-id1333588\" data-type=\"solution\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 5.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1221217\" data-type=\"problem\">\r\n<p id=\"fs-id1172596666838\">The measure of one angle of a right triangle is \\(\\text{30}\\)\u00b0 more than the measure of the smallest angle. Find the measures of all three angles.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1425384\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1364756\">30\u00b0, 60\u00b0, 90\u00b0<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h1>Similar Triangles<\/h1>\r\n<p id=\"fs-id1247950\">When we use a map to plan a trip, a sketch to build a bookcase, or a pattern to sew a dress, we are working with similar figures. In geometry, if two figures have exactly the same shape but different sizes, we say they are similar figures. One is a scale model of the other. The corresponding sides of the two figures have the same ratio, and all their corresponding angles are have the same measures.<\/p>\r\n<p id=\"fs-id1159290\">The two triangles in <a class=\"autogenerated-content\" href=\"#CNX_BMath_Figure_09_03_020\">(Figure 7)<\/a> are similar. Each side of \\(\\Delta ABC\\) is four times the length of the corresponding side of \\(\\Delta XYZ\\) and their corresponding angles have equal measures.<\/p>\r\n\\(\\Delta ABC\\) and \\(\\Delta XYZ\\) are similar triangles. Their corresponding sides have the same ratio and the corresponding angles have the same measure.\r\n<div id=\"CNX_BMath_Figure_09_03_020\" class=\"bc-figure figure\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"522\"]<img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_020.jpg\" alt=\"Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled 16, the side across from B is labeled 20, and the side across from C is labeled 12. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 4, the side across from Y is labeled 5, and the side across from Z is labeled 3. Beside the triangles, it says that the measure of angle A equals the measure of angle X, the measure of angle B equals the measure of angle Y, and the measure of angle C equals the measure of angle Z. Below this is the proportion 16 over 4 equals 20 over 5 equals 12 over 3.\" width=\"522\" height=\"140\" data-media-type=\"image\/jpeg\" \/> Figure 7[\/caption]\r\n\r\n<\/div>\r\n<div id=\"eip-13\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Properties of Similar Triangles<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"eip-id1170323803303\">If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths are in the same ratio.<\/p>\r\n<span id=\"fs-id1182340538\" data-type=\"media\" data-alt=\"...\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_056_img-1.jpg\" alt=\"...\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\nThe length of a side of a triangle may be referred to by its endpoints, two vertices of the triangle. For example, in \\(\\Delta ABC\\text{:}\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1171104085009\">\\(\\phantom{\\rule{2em}{0ex}}\\begin{array}{c}\\text{the length}\\phantom{\\rule{0.2em}{0ex}}a\\phantom{\\rule{0.2em}{0ex}}\\text{can also be written}\\phantom{\\rule{0.2em}{0ex}}BC\\hfill \\\\ \\text{the length}\\phantom{\\rule{0.2em}{0ex}}b\\phantom{\\rule{0.2em}{0ex}}\\text{can also be written}\\phantom{\\rule{0.2em}{0ex}}AC\\hfill \\\\ \\text{the length}\\phantom{\\rule{0.2em}{0ex}}c\\phantom{\\rule{0.2em}{0ex}}\\text{can also be written}\\phantom{\\rule{0.2em}{0ex}}AB\\hfill \\end{array}\\)<\/p>\r\n<p id=\"fs-id1614427\">We will often use this notation when we solve similar triangles because it will help us match up the corresponding side lengths.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 6<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1725972\" data-type=\"problem\">\r\n<p id=\"fs-id1424229\">\\(\\Delta ABC\\) and \\(\\Delta XYZ\\) are similar triangles. The lengths of two sides of each triangle are shown. Find the lengths of the third side of each triangle.<\/p>\r\n<span id=\"fs-id1372781\" data-type=\"media\" data-alt=\"Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 3.2, and the side across from C is labeled 4. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 4.5, the side across from Y is labeled y, and the side across from Z is labeled 3.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_022.jpg\" alt=\"Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 3.2, and the side across from C is labeled 4. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 4.5, the side across from Y is labeled y, and the side across from Z is labeled 3.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n\r\n<strong>Solution<\/strong>\r\n\r\n<\/div>\r\n<div id=\"fs-id1188208\" data-type=\"solution\">\r\n<table id=\"eip-id1168466690156\" style=\"width: 100%; height: 129px;\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr style=\"height: 16px;\">\r\n<td style=\"height: 16px; width: 34.82%;\">Step 1. <strong data-effect=\"bold\">Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n<td style=\"height: 16px; width: 65.18%;\">The figure is provided.<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16px;\">\r\n<td style=\"height: 16px; width: 34.82%;\">Step 2. <strong data-effect=\"bold\">Identify<\/strong> what you are looking for.<\/td>\r\n<td style=\"height: 16px; width: 65.18%;\">The length of the sides of similar triangles<\/td>\r\n<\/tr>\r\n<tr style=\"height: 49px;\">\r\n<td style=\"height: 49px; width: 34.82%;\">Step 3. <strong data-effect=\"bold\">Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td style=\"height: 49px; width: 65.18%;\">Let\r\n<em data-effect=\"italics\">a<\/em> = length of the third side of \\(\\Delta ABC\\)\r\n<em data-effect=\"italics\">y<\/em> = length of the third side \\(\\Delta XYZ\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16px;\">\r\n<td style=\"height: 16px; width: 34.82%;\">Step 4.<strong> Translate.<\/strong><\/td>\r\n<td style=\"height: 16px; width: 65.18%;\">The triangles are similar, so the corresponding sides are in the same ratio. So \\(\\frac{AB}{XY}=\\frac{BC}{YZ}=\\frac{AC}{XZ}\\)\r\n\r\nSince the side \\(AB=4\\) corresponds to the side \\(XY=3\\), we will use the ratio \\(\\frac{\\mathrm{AB}}{\\mathrm{XY}}=\\frac{4}{3}\\) to find the other sides.\r\n\r\nBe careful to match up corresponding sides correctly.\r\n\r\n<img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_057_img-01.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 34.82%;\">Step 5. <strong data-effect=\"bold\">Solve<\/strong> the equation.<span id=\"eip-id1168466188711\" data-type=\"media\" data-alt=\".\"><\/span><\/td>\r\n<td style=\"width: 65.18%;\">&nbsp;\r\n\r\n<span id=\"eip-id1168466188711\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_057_img-02.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 34.82%;\"><span id=\"eip-id1168466188711\" data-type=\"media\" data-alt=\".\"><\/span>Step 6. <strong data-effect=\"bold\">Check.<\/strong>\r\n<span id=\"eip-id1168469628162\" data-type=\"media\" data-alt=\".\"><\/span><\/td>\r\n<td style=\"width: 65.18%;\"><span id=\"eip-id1168469628162\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_057_img-03.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 34.82%;\">Step 7. <strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\r\n<td style=\"width: 65.18%;\"><span id=\"eip-id1168469628162\" data-type=\"media\" data-alt=\".\">The third side of \\(\\Delta ABC\\) is 6 and the third side of \\(\\Delta XYZ\\) is 2.4.<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 6.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1732444\" data-type=\"problem\">\r\n<p id=\"fs-id1172584151393\">\\(\\Delta ABC\\) is similar to \\(\\Delta XYZ\\). Find \\(a\\).<\/p>\r\n<span id=\"fs-id1263455\" data-type=\"media\" data-alt=\"Two triangles are shown. They appear to be the same shape, but the triangle on the right is larger The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 15, and the side across from C is labeled 17. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 12, the side across from Y is labeled y, and the side across from Z is labeled 25.5.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_023.jpg\" alt=\"Two triangles are shown. They appear to be the same shape, but the triangle on the right is larger The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 15, and the side across from C is labeled 17. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 12, the side across from Y is labeled y, and the side across from Z is labeled 25.5.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1172591854682\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1172591854684\">8<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1303468\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1388517\" data-type=\"exercise\">\r\n<div id=\"fs-id1172591854682\" data-type=\"solution\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 6.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1172591772188\" data-type=\"problem\">\r\n<p id=\"fs-id742499\">\\(\\Delta ABC\\) is similar to \\(\\Delta XYZ\\). Find \\(y\\).<\/p>\r\n<span id=\"fs-id1172591786353\" data-type=\"media\" data-alt=\"Two triangles are shown. They appear to be the same shape, but the triangle on the right is larger The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 15, and the side across from C is labeled 17. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 12, the side across from Y is labeled y, and the side across from Z is labeled 25.5.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_023.jpg\" alt=\"Two triangles are shown. They appear to be the same shape, but the triangle on the right is larger The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 15, and the side across from C is labeled 17. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 12, the side across from Y is labeled y, and the side across from Z is labeled 25.5.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1841277\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1172591854687\">22.5<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h1 id=\"fs-id1172591772188\" data-type=\"problem\">Use the Pythagorean Theorem<\/h1>\r\n<p id=\"fs-id1410155\">The Pythagorean Theorem is a special property of right triangles that has been used since ancient times. It is named after the Greek philosopher and mathematician Pythagoras who lived around \\(500\\) BCE.<\/p>\r\n<p id=\"fs-id1410106\">Remember that a right triangle has a \\(\\text{90}\\)\u00b0 angle, which we usually mark with a small square in the corner. The side of the triangle opposite the \\(\\text{90}\\)\u00b0 angle is called the hypotenuse, and the other two sides are called the legs. See <a class=\"autogenerated-content\" href=\"#CNX_BMath_Figure_09_03_024\">(Figure 8)<\/a>.<\/p>\r\nIn a right triangle, the side opposite the \\(\\text{90}\\)\u00b0 angle is called the hypotenuse and each of the other sides is called a leg.\r\n<div id=\"CNX_BMath_Figure_09_03_024\" class=\"bc-figure figure\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"561\"]<img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_024.jpg\" alt=\"Three right triangles are shown. Each has a box representing the right angle. The first one has the right angle in the lower left corner, the next in the upper left corner, and the last one at the top. The two sides touching the right angle are labeled \u201cleg\u201d in each triangle. The sides across from the right angles are labeled \u201chypotenuse.\u201d\" width=\"561\" height=\"121\" data-media-type=\"image\/jpeg\" \/> Figure 8[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1550190\">The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse.<\/p>\r\n\r\n<div id=\"fs-id1701775\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">The Pythagorean Theorem<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1548386\">In any right triangle \\(\\Delta ABC\\),<\/p>\r\n\r\n<div id=\"fs-id1171104156908\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\({a}^{2}+{b}^{2}={c}^{2}\\)<\/div>\r\n<p id=\"fs-id1422647\">where \\(c\\) is the length of the hypotenuse \\(a\\) and \\(b\\) are the lengths of the legs.<\/p>\r\n<span id=\"fs-id1723708\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked a and b.\"><img class=\"alignnone\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_025.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked a and b.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\nTo solve problems that use the Pythagorean Theorem, we will need to find square roots. We defined the notation \\(\\sqrt{m}\\) in this way:\r\n\r\n\\(\\text{If}\\phantom{\\rule{0.2em}{0ex}}m={n}^{2},\\phantom{\\rule{0.2em}{0ex}}\\text{then}\\phantom{\\rule{0.2em}{0ex}}\\sqrt{m}=n\\phantom{\\rule{0.2em}{0ex}}\\text{for}\\phantom{\\rule{0.2em}{0ex}}n\\ge 0\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1432808\">For example, we found that \\(\\sqrt{25}\\) is \\(5\\) because \\({5}^{2}=25\\).<\/p>\r\n<p id=\"fs-id1538008\">We will use this definition of square roots to solve for the length of a side in a right triangle.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 7<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1419626\" data-type=\"problem\">\r\n<p id=\"fs-id1420675\">Use the Pythagorean Theorem to find the length of the hypotenuse.<\/p>\r\n<span id=\"fs-id1411841\" data-type=\"media\" data-alt=\"Right triangle with legs labeled as 3 and 4.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_026_img.jpg\" alt=\"Right triangle with legs labeled as 3 and 4.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n\r\n<strong>Solution<\/strong>\r\n\r\n<\/div>\r\n<div id=\"fs-id1172599766488\" data-type=\"solution\">\r\n<table id=\"eip-id1168469450887\" style=\"width: 100%;\" summary=\"Step 1 says, \u201cRead the problem.\u201d The word \u201cread\u201d is in bold. Beside this is a picture of a right triangle. The side across from the right angle is labeled c, the sides touching the right angle are labeled 3 and 4. Step 2 says, \u201cIdentify what you are looking for.\u201d The word \u201cidentify\u201d is in bold. Beside this it says, \u201cthe length of the hypotenuse of the triangle.\u201d Step 3 says, \u201cName. Choose a variable to represent it. \u201c The word \u201cname\u201d is in bold. Beside this it says, \u201cLet c equal the length of the hypotenuse.\u201d Step 4 says, \u201cTranslate. Write the appropriate formula. Substitute.\u201d The word \u201ctranslate\u201d is in bold. Beside this is the equation a squared plus b squared equals c squared, then 3 squared plus 4 squared equals c squared. Step 5 says, \u201cSolve the equation. Simplify. Use the definition of square root. Simplify.\u201d The word \u201csolve\u201d is in bold. Beside this is 9 plus 16 equals c squared, then 25 equals c squared, then square root of 25 equals c, then 5 equals c. Step 6 says \u201cCheck\u201d in bold. Beside this is 3 squared plus 4 squared followed by an equal sign with a question mark, then a red 5 squared. Below this is 9 plus 16 followed by an equal sign with a question mark, then 25. The next line says 25 equals 25 with a check mark. Step 7 says, \u201cAnswer the question.\u201d The word \u201canswer\u201d is in bold. Beside this is the answer, \u201cThe length of the hypotenuse is 5.\u201d\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong data-effect=\"bold\">Read<\/strong> the problem.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong data-effect=\"bold\">Identify<\/strong> what you are looking for.<\/td>\r\n<td>the length of the hypotenuse of the triangle<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong data-effect=\"bold\">Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td>Let \\(c=\\text{the length of the hypotenuse}\\)\r\n<span id=\"eip-id1168469449113\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_053_img-01.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong data-effect=\"bold\">Translate.<\/strong>\r\nWrite the appropriate formula.\r\nSubstitute.<\/td>\r\n<td><span id=\"eip-id1168469890812\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_053_img-02.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\">Step 5. <strong data-effect=\"bold\">Solve<\/strong> the equation.<\/td>\r\n<td><span id=\"eip-id1168469890834\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_053_img-03.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong data-effect=\"bold\">Check:<\/strong>\r\n<span id=\"eip-id1168469481160\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_053_img-04.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\r\n<td>The length of the hypotenuse is 5.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 7.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1542055\" data-type=\"problem\">\r\n<p id=\"fs-id1542057\">Use the Pythagorean Theorem to find the length of the hypotenuse.<\/p>\r\n<span id=\"fs-id1321685\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked 6 and 8.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_029_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked 6 and 8.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1727689\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1259642\">10<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1546550\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1172599768654\" data-type=\"exercise\">\r\n<div id=\"fs-id1727689\" data-type=\"solution\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 7.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1260564\" data-type=\"problem\">\r\n<p id=\"fs-id1260566\">Use the Pythagorean Theorem to find the length of the hypotenuse.<\/p>\r\n<span id=\"fs-id1573251\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as c. One of the sides touching the right angle is labeled as 15, the other is labeled \u201c8\u201d.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_030_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as c. One of the sides touching the right angle is labeled as 15, the other is labeled \u201c8\u201d.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1256380\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1172574418263\">17<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1403907\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1172591967221\" data-type=\"exercise\">\r\n<div id=\"fs-id1256380\" data-type=\"solution\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 8<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1172574467640\" data-type=\"problem\">\r\n<p id=\"fs-id1172591785037\">Use the Pythagorean Theorem to find the length of the longer leg.<\/p>\r\n<span id=\"fs-id1172591785041\" data-type=\"media\" data-alt=\"Right triangle is shown with one leg labeled as 5 and hypotenuse labeled as 13.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_031_img.jpg\" alt=\"Right triangle is shown with one leg labeled as 5 and hypotenuse labeled as 13.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n\r\n<strong>Solution<\/strong>\r\n\r\n<\/div>\r\n<div id=\"fs-id1172573419851\" data-type=\"solution\">\r\n<table id=\"eip-id1168467480162\" style=\"width: 100%;\" summary=\"Step 1 says, \u201cRead the problem.\u201d The word \u201cread\u201d is in bold. Step 2 says, \u201cIdentify what you are looking for.\u201d The word \u201cidentify\u201d is in bold. Beside this it says, \u201cthe length of the leg of the triangle.\u201d Step 3 says, \u201cName. Choose a variable to represent it. \u201c The word \u201cname\u201d is in bold. Beside this it says, \u201cLet b equal the leg of the triangle. Label side b.\u201d Beside this is a picture of a right triangle. The side across from the right angle is labeled 13, the sides touching the right angle are labeled 5 and b. Step 4 says, \u201cTranslate. Write the appropriate formula. Substitute.\u201d The word \u201ctranslate\u201d is in bold. Beside this is the equation a squared plus b squared equals c squared, then 5 squared plus b squared equals 13 squared. Step 5 says, \u201cSolve the equation. Isolate the variable term. Use the definition of square root. Simplify.\u201d The word \u201csolve\u201d is in bold. Beside this is 25 plus b squared equals 169, then b squared equals 144, then b equals the square root of 144, then b equals 12. Step 6 says \u201cCheck\u201d in bold. Beside this is 5 squared plus a red 12 squared followed by an equal sign with a question mark, then 13 squared. Below this is 25 plus 144 followed by an equal sign with a question mark, then 169. The next line says 169 equals 169 with a check mark. Step 7 says, \u201cAnswer the question.\u201d The word \u201canswer\u201d is in bold. Beside this is the answer, \u201cThe length of the leg is 12.\u201d\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong data-effect=\"bold\">Read<\/strong> the problem.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong data-effect=\"bold\">Identify<\/strong> what you are looking for.<\/td>\r\n<td>The length of the leg of the triangle<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\">Step 3. <strong data-effect=\"bold\">Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td>Let \\(b=\\text{the leg of the triangle}\\)\r\nLabel side <em data-effect=\"italics\">b<\/em>\r\n<span id=\"eip-id1168467480249\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_054_img-01.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong data-effect=\"bold\">Translate.<\/strong>\r\nWrite the appropriate formula. Substitute.<\/td>\r\n<td><span id=\"eip-id1168466021991\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_054_img-02.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\">Step 5. <strong data-effect=\"bold\">Solve<\/strong> the equation. Isolate the variable term. Use the definition of the square root.\r\nSimplify.<\/td>\r\n<td><span id=\"eip-id1168469557654\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_054_img-03.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong data-effect=\"bold\">Check:<\/strong><\/td>\r\n<td><span id=\"eip-id1168468687314\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_054_img-04.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\r\n<td>The length of the leg is 12.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1172597113282\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1172597113285\" data-type=\"exercise\">\r\n<div id=\"fs-id1402124\" data-type=\"problem\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 8.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1402124\" data-type=\"problem\">\r\n<p id=\"fs-id1402126\">Use the Pythagorean Theorem to find the length of the leg.<\/p>\r\n<span id=\"fs-id1421940\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 15, the other is labeled \u201cb\u201d.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_034_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 15, the other is labeled \u201cb\u201d.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1172599716643\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1172591773440\">8<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1172599716643\" data-type=\"solution\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 8.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1172599716594\" data-type=\"problem\">\r\n<p id=\"fs-id1172599716596\">Use the Pythagorean Theorem to find the length of the leg.<\/p>\r\n<span id=\"fs-id1340466\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 15. One of the sides touching the right angle is labeled as 9, the other is labeled \u201cb\u201d.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_035_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 15. One of the sides touching the right angle is labeled as 9, the other is labeled \u201cb\u201d.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1403242\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1715840\">12<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1435399\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1415446\" data-type=\"exercise\">\r\n<div id=\"fs-id1403242\" data-type=\"solution\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 9<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1230268\" data-type=\"problem\">\r\n<p id=\"fs-id1432662\">Kelvin is building a gazebo and wants to brace each corner by placing a \\(\\text{10-inch}\\) wooden bracket diagonally as shown. How far below the corner should he fasten the bracket if he wants the distances from the corner to each end of the bracket to be equal? Approximate to the nearest tenth of an inch.<\/p>\r\n<span id=\"fs-id1172591785993\" data-type=\"media\" data-alt=\"A picture of a gazebo is shown. Beneath the roof is a rectangular shape. There are two braces from the top to each side. The brace on the left is labeled as 10 inches. From where the brace hits the side to the roof is labeled as x.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_036.jpg\" alt=\"A picture of a gazebo is shown. Beneath the roof is a rectangular shape. There are two braces from the top to each side. The brace on the left is labeled as 10 inches. From where the brace hits the side to the roof is labeled as x.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n\r\n<strong>Solution<\/strong>\r\n\r\n<\/div>\r\n<div id=\"fs-id1727579\" data-type=\"solution\">\r\n<table id=\"eip-id1168468762748\" style=\"width: 100%;\" summary=\"Step 1 says, \u201cRead the problem.\u201d The word \u201cread\u201d is in bold. Step 2 says, \u201cIdentify what you are looking for.\u201d The word \u201cidentify\u201d is in bold. Beside this it says, \u201cthe distance from the corner that the bracket should be attached.\u201d Below this is a picture of a right triangle with the side across from the right angle labeled as 10 inches and the other 2 sides labeled as x. Step 3 says, \u201cName. Choose a variable to represent it. \u201c The word \u201cname\u201d is in bold. Beside this it says, \u201cLet x equal the distance from the corner.\u201d Step 4 says, \u201cTranslate. Write the appropriate formula. Substitute.\u201d The word \u201ctranslate\u201d is in bold. Beside this is the equation a squared plus b squared equals c squared, then x squared plus x squared equals 10 squared. Step 5 says, \u201cSolve the equation. Isolate the variable term. Use the definition of square root. Simplify. Approximate to the nearest tenth.\u201d The word \u201csolve\u201d is in bold. Beside this is 2 times x squared equals 100, then x squared equals 50, then x equals the square root of 50, then x is approximately 7.1. Step 6 says \u201cCheck\u201d in bold. Beside this is a squared plus b squared equals c squared, then a red 7.1 squared plus a red 7.1 squared followed by an approximate sign with a question mark, then 10 squared. Below this, it says \u201cYes.\u201d Step 7 says, \u201cAnswer the question.\u201d The word \u201canswer\u201d is in bold. Beside this is the answer, \u201cKelvin should fasten each piece of wood approximately 7.1 inches from the corner.\u201d\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong data-effect=\"bold\">Read<\/strong> the problem.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\">Step 2. <strong data-effect=\"bold\">Identify<\/strong> what you are looking for.<\/td>\r\n<td>the distance from the corner that the bracket should be attached<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong data-effect=\"bold\">Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td>Let <em data-effect=\"italics\">x<\/em> = the distance from the corner\r\n<span id=\"eip-id1168467381315\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_055_img-01.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong data-effect=\"bold\">Translate.<\/strong>\r\nWrite the appropriate formula.\r\nSubstitute.<\/td>\r\n<td><span id=\"eip-id1168469480877\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_055_img-02.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong data-effect=\"bold\">Solve<\/strong> the equation.\r\nIsolate the variable.\r\nUse the definition of the square root.\r\nSimplify. Approximate to the nearest tenth.<\/td>\r\n<td><span id=\"eip-id1168466339030\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_055_img-03.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong data-effect=\"bold\">Check:<\/strong>\r\n<span id=\"eip-id1168465147070\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_055_img-04.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span>\r\nYes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\r\n<td>Kelvin should fasten each piece of wood approximately 7.1\" from the corner.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 9.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1172591855349\" data-type=\"problem\">\r\n<p id=\"fs-id1172591855351\">John puts the base of a \\(\\text{13-ft}\\) ladder \\(5\\) feet from the wall of his house. How far up the wall does the ladder reach?<\/p>\r\n<span id=\"fs-id1543199\" data-type=\"media\" data-alt=\"A picture of a house is shown. There is a ladder leaning against the side of the house. The ladder is labeled 13 feet. The horizontal distance from the ladder's base to the house is labeled 5 feet.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_039_img.jpg\" alt=\"A picture of a house is shown. There is a ladder leaning against the side of the house. The ladder is labeled 13 feet. The horizontal distance from the ladder's base to the house is labeled 5 feet.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1660058\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1431748\">12 feet<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1405087\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1207430\" data-type=\"exercise\">\r\n<div id=\"fs-id1660058\" data-type=\"solution\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 9.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1536639\" data-type=\"problem\">\r\n<p id=\"fs-id1536641\">Randy wants to attach a \\(\\text{17-ft}\\) string of lights to the top of the \\(\\text{15-ft}\\) mast of his sailboat. How far from the base of the mast should he attach the end of the light string?<\/p>\r\n<span id=\"fs-id1172591861163\" data-type=\"media\" data-alt=\"A picture of a boat is shown. The height of the centre pole is labeled 15 feet. The string of lights is at a diagonal from the top of the pole and is labeled 17 feet.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_040_img.jpg\" alt=\"A picture of a boat is shown. The height of the centre pole is labeled 15 feet. The string of lights is at a diagonal from the top of the pole and is labeled 17 feet.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1172596641002\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1172596641004\">8 feet<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h1>Key Concepts<\/h1>\r\n<ul id=\"eip-655\">\r\n \t<li><strong>Supplementary and Complementary Angles<\/strong>\r\n<ul id=\"eip-id1170324960756\">\r\n \t<li>If the sum of the measures of two angles is 180\u00b0, then the angles are supplementary.<\/li>\r\n \t<li>If \\(\\angle A\\) and \\(\\angle B\\) are supplementary, then \\(m\\angle A+m\\angle B=180\\).<\/li>\r\n \t<li>If the sum of the measures of two angles is 90\u00b0, then the angles are complementary.<\/li>\r\n \t<li>If \\(\\angle A\\) and \\(\\angle B\\) are complementary, then \\(m\\angle A+m\\angle B=90\\).<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Solve Geometry Applications<\/strong>\r\n<ol id=\"eip-id1170326094648\" class=\"stepwise\" type=\"1\">\r\n \t<li>Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.<\/li>\r\n \t<li>Identify what you are looking for.<\/li>\r\n \t<li>Name what you are looking for and choose a variable to represent it.<\/li>\r\n \t<li>Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.<\/li>\r\n \t<li>Solve the equation using good algebra techniques.<\/li>\r\n \t<li>Check the answer in the problem and make sure it makes sense.<\/li>\r\n \t<li>Answer the question with a complete sentence.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li><span id=\"fs-id11722323361163\" data-type=\"media\" data-alt=\"..\"><img class=\"alignright\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_012.jpg\" alt=\"..\" data-media-type=\"image\/jpeg\" \/><\/span><strong>Sum of the Measures of the Angles of a Triangle<\/strong>\r\n<ul id=\"eip-id1170322666796\">\r\n \t<li>For any \\(\\Delta ABC\\), the sum of the measures is 180\u00b0<\/li>\r\n \t<li>\\(m\\angle A+m\\angle B=180\\)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><span id=\"eip-id5408054\" data-type=\"media\" data-alt=\".\"><img class=\"alignright\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_015_img.jpg\" alt=\".\" data-media-type=\"image\/jpg\" \/><\/span><strong>Right Triangle<\/strong>\r\n<ul id=\"eip-id1170323029556\">\r\n \t<li>A right triangle is a triangle that has one 90\u00b0 angle, which is often marked with a \u299c symbol.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Properties of Similar Triangles<\/strong>\r\n<ul id=\"eip-id1170324076236\">\r\n \t<li>If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths have the same ratio.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<h1>Glossary<\/h1>\r\n<div class=\"textbox shaded\">\r\n<dl id=\"fs-id1648881\">\r\n \t<dt>angle<\/dt>\r\n \t<dd id=\"fs-id1327370\">An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1327374\">\r\n \t<dt>complementary angles<\/dt>\r\n \t<dd id=\"fs-id1544988\">If the sum of the measures of two angles is \\(\\text{90}\\)\u00b0, then they are called complementary angles.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170322667662\">\r\n \t<dt>hypotenuse<\/dt>\r\n \t<dd id=\"fs-id1170325427844\">The side of the triangle opposite the 90\u00b0 angle is called the hypotenuse.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1444791\">\r\n \t<dt>legs of a right triangle<\/dt>\r\n \t<dd id=\"fs-id1540654\">The sides of a right triangle adjacent to the right angle are called the legs.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1172591862696\">\r\n \t<dt>right triangle<\/dt>\r\n \t<dd id=\"fs-id1172596661577\">A right triangle is a triangle that has one \\(\\text{90}\\)\u00b0 angle.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1172591572065\">\r\n \t<dt>similar figures<\/dt>\r\n \t<dd id=\"fs-id1236955\">In geometry, if two figures have exactly the same shape but different sizes, we say they are similar figures.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1300144\">\r\n \t<dt>supplementary angles<\/dt>\r\n \t<dd id=\"fs-id1434532\">If the sum of the measures of two angles is \\(\\text{180}\\)\u00b0, then they are called supplementary angles.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1334464\">\r\n \t<dt>triangle<\/dt>\r\n \t<dd id=\"fs-id1172599767842\">A triangle is a geometric figure with three sides and three angles.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1172599767847\">\r\n \t<dt>vertex of an angle<\/dt>\r\n \t<dd id=\"fs-id1402834\">When two rays meet to form an angle, the common endpoint is called the vertex of the angle.<\/dd>\r\n<\/dl>\r\n<\/div>\r\n<h1>Practice Makes Perfect<\/h1>\r\n<div id=\"fs-id1309715\" class=\"practice-perfect\" data-depth=\"2\">\r\n<h2 id=\"fs-id1172591855673\">Use the Properties of Angles<\/h2>\r\n<p id=\"eip-294\">In the following exercises, find a) the supplement and b) the complement of the given angle.<\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\">1. \\(\\text{53}\\)\u00b0<\/td>\r\n<td style=\"width: 50%;\">2. \\(\\text{16}\\)\u00b0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">3. \\(\\text{29}\\)\u00b0<\/td>\r\n<td style=\"width: 50%;\">4. \\(\\text{72}\\)\u00b0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1537917\">In the following exercises, use the properties of angles to solve.<\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 100%; height: 56px;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 50%; height: 14px;\">5. Find the supplement of a \\(\\text{135}\\)\u00b0 angle.<\/td>\r\n<td style=\"width: 50%; height: 14px;\">6. Find the complement of a \\(\\text{38}\\)\u00b0 angle.<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 50%; height: 14px;\">7. Find the complement of a \\(27.5\\)\u00b0 angle.<\/td>\r\n<td style=\"width: 50%; height: 14px;\">8. Find the supplement of a \\(109.5\\)\u00b0 angle.<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 50%; height: 14px;\">9. Two angles are supplementary. The larger angle is \\(\\text{56}\\)\u00b0 more than the smaller angle. Find the measures of both angles.<\/td>\r\n<td style=\"width: 50%; height: 14px;\">10. Two angles are supplementary. The smaller angle is \\(\\text{36}\\)\u00b0 less than the larger angle. Find the measures of both angles.<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 50%; height: 14px;\">11. Two angles are complementary. The smaller angle is \\(\\text{34}\\)\u00b0 less than the larger angle. Find the measures of both angles.<\/td>\r\n<td style=\"width: 50%; height: 14px;\">12. Two angles are complementary. The larger angle is \\(\\text{52}\\)\u00b0 more than the smaller angle. Find the measures of both angles.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2 id=\"fs-id1296727\">Use the Properties of Triangles<\/h2>\r\n<p id=\"eip-759\">In the following exercises, solve using properties of triangles.<\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 100%; height: 144px;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"width: 50%; height: 30px;\">13. The measures of two angles of a triangle are \\(\\text{26}\\)\u00b0 and \\(\\text{98}\\)\u00b0. Find the measure of the third angle.<\/td>\r\n<td style=\"width: 50%; height: 30px;\">14. The measures of two angles of a triangle are \\(\\text{61}\\)\u00b0 and \\(\\text{84}\\)\u00b0. Find the measure of the third angle.<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"width: 50%; height: 30px;\">15. The measures of two angles of a triangle are \\(\\text{105}\\)\u00b0 and \\(\\text{31}\\)\u00b0. Find the measure of the third angle.<\/td>\r\n<td style=\"width: 50%; height: 30px;\">16. The measures of two angles of a triangle are \\(\\text{47}\\)\u00b0 and \\(\\text{72}\\)\u00b0. Find the measure of the third angle.<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 50%; height: 14px;\">17. One angle of a right triangle measures \\(\\text{33}\\)\u00b0. What is the measure of the other angle?<\/td>\r\n<td style=\"width: 50%; height: 14px;\">18. One angle of a right triangle measures \\(\\text{51}\\)\u00b0. What is the measure of the other angle?<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 50%; height: 14px;\">19. One angle of a right triangle measures \\(22.5\\)\u00b0. What is the measure of the other angle?<\/td>\r\n<td style=\"width: 50%; height: 14px;\">20. One angle of a right triangle measures \\(36.5\\)\u00b0. What is the measure of the other angle?<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 50%; height: 14px;\">21. The two smaller angles of a right triangle have equal measures. Find the measures of all three angles.<\/td>\r\n<td style=\"width: 50%; height: 14px;\">22. The measure of the smallest angle of a right triangle is \\(\\text{20}\\)\u00b0 less than the measure of the other small angle. Find the measures of all three angles.<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 50%; height: 14px;\">23. The angles in a triangle are such that the measure of one angle is twice the measure of the smallest angle, while the measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles.<\/td>\r\n<td style=\"width: 50%; height: 14px;\">24. The angles in a triangle are such that the measure of one angle is \\(\\text{20}\\)\u00b0 more than the measure of the smallest angle, while the measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h3 id=\"fs-id1548145\">Find the Length of the Missing Side<\/h3>\r\n<p id=\"eip-480\">In the following exercises, \\(\\Delta ABC\\) is similar to \\(\\Delta XYZ\\). Find the length of the indicated side.<\/p>\r\n<span id=\"fs-id1547556\" data-type=\"media\" data-alt=\"Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled 9, the side across from B is labeled b, and the side across from C is labeled 15. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled x, the side across from Y is labeled 8, and the side across from Z is labeled 10.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_201_img.jpg\" alt=\"Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled 9, the side across from B is labeled b, and the side across from C is labeled 15. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled x, the side across from Y is labeled 8, and the side across from Z is labeled 10.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<div id=\"fs-id1547228\" data-type=\"exercise\">\r\n<div id=\"fs-id1547230\" data-type=\"problem\">\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\">25. side \\(b\\)<\/td>\r\n<td style=\"width: 50%;\">26. side \\(x\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1297695\"><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"eip-941\">On a map, San Francisco, Las Vegas, and Los Angeles form a triangle whose sides are shown in the figure below. The actual distance from Los Angeles to Las Vegas is \\(270\\) miles.<\/p>\r\n<span id=\"fs-id1260455\" data-type=\"media\" data-alt=\"A triangle is shown. The vertices are labeled San Francisco, Las Vegas, and Los Angeles. The side across from San Francisco is labeled 1 inch, the side across from Las Vegas is labeled 1.3 inches, and the side across from Los Angeles is labeled 2.1 inches.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_202_img.jpg\" alt=\"A triangle is shown. The vertices are labeled San Francisco, Las Vegas, and Los Angeles. The side across from San Francisco is labeled 1 inch, the side across from Las Vegas is labeled 1.3 inches, and the side across from Los Angeles is labeled 2.1 inches.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\">27. Find the distance from Los Angeles to San Francisco.<\/td>\r\n<td style=\"width: 50%;\">28. Find the distance from San Francisco to Las Vegas.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2 id=\"fs-id1413042\">Use the Pythagorean Theorem<\/h2>\r\nIn the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.\r\n<table style=\"border-collapse: collapse; width: 100%; height: 189px;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 175px;\">\r\n<td style=\"width: 50%; height: 175px;\"><span id=\"fs-id1545681\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 9, the other as 12.\">29. <img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_203_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 9, the other as 12.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<td style=\"width: 50%; height: 175px;\"><span id=\"fs-id1238083\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 16, the other as 12.\">30. <img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_204_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 16, the other as 12.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 50%; height: 14px;\"><span id=\"fs-id1543755\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 15, the other as 20.\">31. <img class=\"\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_205_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 15, the other as 20.\" width=\"217\" height=\"163\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<td style=\"width: 50%; height: 14px;\"><span id=\"fs-id1428325\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 5, the other as 12.\">32. <img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_206_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 5, the other as 12.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h3 id=\"fs-id1263576\">Find the Length of the Missing Side<\/h3>\r\n<p id=\"eip-851\">In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.<\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 100%; height: 152px;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 110px;\">\r\n<td style=\"width: 50%; height: 110px;\"><span id=\"fs-id1576553\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 10. One of the sides touching the right angle is labeled as 6.\">33. <img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_207_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 10. One of the sides touching the right angle is labeled as 6.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<td style=\"width: 50%; height: 110px;\"><span id=\"fs-id1547852\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 8.\">34. <img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_208_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 8.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 50%; height: 14px;\"><span id=\"fs-id1701813\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 13. One of the sides touching the right angle is labeled as 5.\">35. <img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_209_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 13. One of the sides touching the right angle is labeled as 5.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<td style=\"width: 50%; height: 14px;\"><span id=\"fs-id1439355\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 20. One of the sides touching the right angle is labeled as 16.\">36. <img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_210_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 20. One of the sides touching the right angle is labeled as 16.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 50%; height: 14px;\"><span id=\"fs-id1172599767679\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 13. One of the sides touching the right angle is labeled as 8.\">37. <img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_211_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 13. One of the sides touching the right angle is labeled as 8.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<td style=\"width: 50%; height: 14px;\"><span id=\"fs-id1704712\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. Both of the sides touching the right angle are labeled as 6.\">38. <img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_212_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. Both of the sides touching the right angle are labeled as 6.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 50%; height: 14px;\"><span id=\"fs-id1614367\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 15.\">39. <img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_213_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 15.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<td style=\"width: 50%; height: 14px;\"><span id=\"fs-id1537416\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 7. One of the sides touching the right angle is labeled as 5.\">40. <img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_214_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 7. One of the sides touching the right angle is labeled as 5.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1428288\">In the following exercises, solve. Approximate to the nearest tenth, if necessary.<\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\">\r\n<p id=\"fs-id1538067\">41. A \\(\\text{13-foot}\\) string of lights will be attached to the top of a \\(\\text{12-foot}\\) pole for a holiday display. How far from the base of the pole should the end of the string of lights be anchored?<\/p>\r\n<span id=\"fs-id1279182\" data-type=\"media\" data-alt=\"A vertical pole is shown with a string of lights going from the top of the pole to the ground. The pole is labeled 12 feet. The string of lights is labeled 13 feet.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_215_img.jpg\" alt=\"A vertical pole is shown with a string of lights going from the top of the pole to the ground. The pole is labeled 12 feet. The string of lights is labeled 13 feet.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<td style=\"width: 50%;\">\r\n<p id=\"fs-id1259700\">42. Pam wants to put a banner across her garage door to congratulate her son on his college graduation. The garage door is \\(12\\) feet high and \\(16\\) feet wide. How long should the banner be to fit the garage door?<\/p>\r\n<span id=\"fs-id1172597125751\" data-type=\"media\" data-alt=\"A picture of a house is shown. The rectangular garage is 12 feet high and 16 feet wide. A blue banner goes diagonally across the garage.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_03_04_214.jpg\" alt=\"A picture of a house is shown. The rectangular garage is 12 feet high and 16 feet wide. A blue banner goes diagonally across the garage.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">\r\n<p id=\"fs-id1543256\">43. Chi is planning to put a path of paving stones through her flower garden. The flower garden is a square with sides of \\(10\\) feet. What will the length of the path be?<\/p>\r\n<span id=\"fs-id1542270\" data-type=\"media\" data-alt=\"A square garden is shown. One side is labeled as 10 feet. There is a diagonal path of blue circular stones going from the lower left corner to the upper right corner.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_03_04_215.jpg\" alt=\"A square garden is shown. One side is labeled as 10 feet. There is a diagonal path of blue circular stones going from the lower left corner to the upper right corner.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<td style=\"width: 50%;\">\r\n<p id=\"fs-id1332146\">44. Brian borrowed a \\(\\text{20-foot}\\) extension ladder to paint his house. If he sets the base of the ladder \\(6\\) feet from the house, how far up will the top of the ladder reach?<\/p>\r\n<span id=\"fs-id1459140\" data-type=\"media\" data-alt=\"A picture of a house is shown with a ladder leaning against it. The ladder is labeled 20 feet tall. The horizontal distance from the house to the base of the ladder is 6 feet.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_03_04_216.jpg\" alt=\"A picture of a house is shown with a ladder leaning against it. The ladder is labeled 20 feet tall. The horizontal distance from the house to the base of the ladder is 6 feet.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1420696\" class=\"everyday\" data-depth=\"2\">\r\n<h2 data-type=\"title\">Everyday Math<\/h2>\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\">45. <strong data-effect=\"bold\">Building a scale model<\/strong> Joe wants to build a doll house for his daughter. He wants the doll house to look just like his house. His house is \\(30\\) feet wide and \\(35\\) feet tall at the highest point of the roof. If the dollhouse will be \\(2.5\\) feet wide, how tall will its highest point be?<\/td>\r\n<td style=\"width: 50%;\">\r\n<p id=\"fs-id1172597157720\">46. <strong data-effect=\"bold\">Measurement<\/strong> A city engineer plans to build a footbridge across a lake from point \\(\\text{X}\\) to point \\(\\text{Y}\\), as shown in the picture below. To find the length of the footbridge, she draws a right triangle \\(\\text{XYZ}\\), with right angle at \\(\\text{X}\\). She measures the distance from \\(\\text{X}\\) to \\(\\text{Z},800\\) feet, and from \\(\\text{Y}\\) to \\(\\text{Z},1,000\\) feet. How long will the bridge be?<\/p>\r\n<span id=\"fs-id1545751\" data-type=\"media\" data-alt=\"A lake is shown. Point Y is on one side of the lake, directly across from point X. Point Z is on the same side of the lake as point X.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_219_img.jpg\" alt=\"A lake is shown. Point Y is on one side of the lake, directly across from point X. Point Z is on the same side of the lake as point X.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1580956\" class=\"writing\" data-depth=\"2\">\r\n<h2 data-type=\"title\">Writing Exercises<\/h2>\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\">47. Write three of the properties of triangles from this section and then explain each in your own words.<\/td>\r\n<td style=\"width: 50%;\">\r\n<p id=\"fs-id1458581\">48. Explain how the figure below illustrates the Pythagorean Theorem for a triangle with legs of length \\(3\\) and \\(4\\).<\/p>\r\n<span id=\"fs-id1399992\" data-type=\"media\" data-alt=\"Three squares are shown, forming a right triangle in the center. Each square is divided into smaller squares. The smallest square is divided into 9 small squares. The medium square is divided into 16 small squares. The large square is divided into 25 small squares.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_220_img.jpg\" alt=\"Three squares are shown, forming a right triangle in the center. Each square is divided into smaller squares. The smallest square is divided into 9 small squares. The medium square is divided into 16 small squares. The large square is divided into 25 small squares.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<h1>Answers<\/h1>\r\n<table style=\"border-collapse: collapse; width: 100%; height: 257px;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 104px;\">\r\n<td style=\"width: 33.3333%; height: 104px;\">1.\r\n\r\na) 127\u00b0\r\n\r\nb) 37\u00b0<\/td>\r\n<td style=\"width: 33.3333%; height: 104px;\">3.\r\n<div id=\"fs-id1393950\" data-type=\"exercise\">\r\n<div id=\"fs-id1415688\" data-type=\"solution\">\r\n\r\na) 151\u00b0\r\n\r\nb) 61\u00b0\r\n\r\n<\/div>\r\n<\/div><\/td>\r\n<td style=\"width: 33.3333%; height: 104px;\">5. 45\u00b0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"width: 33.3333%; height: 17px;\">7. 62.5\u00b0<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">9. 62\u00b0, 118\u00b0<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">11. 62\u00b0, 28\u00b0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"width: 33.3333%; height: 17px;\">13. 56\u00b0<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">15. 44\u00b0<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">17. 57\u00b0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"width: 33.3333%; height: 17px;\">19. 67.5\u00b0<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">21. 45\u00b0, 45\u00b0, 90\u00b0<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">23. 30\u00b0, 60\u00b0, 90\u00b0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"width: 33.3333%; height: 17px;\">25. 12<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">27. 351 miles<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">29. 15<\/td>\r\n<\/tr>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"width: 33.3333%; height: 17px;\">31. 25<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">33. 8<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">35. 12<\/td>\r\n<\/tr>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"width: 33.3333%; height: 17px;\">37. 10.2<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">39. 8<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">41. 5 feet<\/td>\r\n<\/tr>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"width: 33.3333%; height: 17px;\">43. 14.1 feet<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">45. 2.9 feet<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">47.\u00a0Answers will vary.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h1>Attributions<\/h1>\r\nThis chapter has been adapted from \u201cUse Properties of Angles, Triangles, and the Pythagorean Theorem\u201d in <a href=\"https:\/\/openstax.org\/details\/books\/prealgebra-2e\"><em>Prealgebra<\/em><\/a> (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a <a href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY 4.0 Licence<\/a>. Adapted by Izabela Mazur. See the Copyright page for more information.","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Use the properties of angles<\/li>\n<li>Use the properties of triangles<\/li>\n<li>Use the Pythagorean Theorem<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h1 data-type=\"title\">Use the Properties of Angles<\/h1>\n<p id=\"fs-id1172574714298\">Are you familiar with the phrase \u2018do a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b45f2f7293eeae5e345f6bbda29cc20a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#39;&#63;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: -1px;\" \/> It means to make a full turn so that you face the opposite direction. It comes from the fact that the measure of an angle that makes a straight line is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6f62ea9815d82070371ef56ddaabdc0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/> degrees. See <a class=\"autogenerated-content\" href=\"#CNX_BMath_Figure_09_03_046\">(Figure 1)<\/a>.<\/p>\n<div id=\"CNX_BMath_Figure_09_03_046\" class=\"bc-figure figure\">\n<figure style=\"width: 152px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2020\/06\/CNX_BMath_Figure_09_03_046_img.jpg\" alt=\"The image is a straight line with an arrow on each end. There is a dot in the center. There is an arrow pointing from one side of the dot to the other, and the angle is marked as 180 degrees.\" width=\"152\" height=\"47\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 1<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1172591978152\">An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle and the common endpoint is called the vertex. An angle is named by its vertex. In <a class=\"autogenerated-content\" href=\"#CNX_BMath_Figure_09_03_001\">(Figure 2)<\/a>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c9644d971857e5b6fb1370bb8de85b92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: 0px;\" \/> is the angle with vertex at point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>. The measure of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c9644d971857e5b6fb1370bb8de85b92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: 0px;\" \/> is written <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9ded6157fc6f46979909bf47611442bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<div id=\"CNX_BMath_Figure_09_03_001\" class=\"bc-figure figure\">\n<div class=\"bc-figcaption figcaption\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c9644d971857e5b6fb1370bb8de85b92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: 0px;\" \/> is the angle with vertex at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-bfb092dce41be7ae8b4334208525852d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#111;&#105;&#110;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"57\" style=\"vertical-align: -4px;\" \/>.<\/div>\n<figure style=\"width: 201px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_001.jpg\" alt=\"The image is an angle made up of two rays. The angle is labeled with letter A.\" width=\"201\" height=\"73\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 2<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1172584183371\">We measure angles in degrees, and use the symbol \u00b0\u00a0 to represent degrees. We use the abbreviation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\" \/> to for the <em data-effect=\"italics\">measure<\/em> of an angle. So if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c9644d971857e5b6fb1370bb8de85b92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: 0px;\" \/> is 27\u00b0, we would write <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ee0bd93ede48f39112a87a6342226465_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#65;&#61;&#50;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"83\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<p id=\"fs-id1411514\">If the sum of the measures of two angles is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1371cfb17a414d739b40e633931613a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/>\u00b0, then they are called supplementary angles. In <a class=\"autogenerated-content\" href=\"#CNX_BMath_Figure_09_03_003\">(Figure 3)<\/a>, each pair of angles is supplementary because their measures add to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1371cfb17a414d739b40e633931613a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/>\u00b0. Each angle is the <em data-effect=\"italics\">supplement<\/em> of the other.<\/p>\n<div id=\"CNX_BMath_Figure_09_03_003\" class=\"bc-figure figure\">\n<div class=\"bc-figcaption figcaption\">The sum of the measures of supplementary angles is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1371cfb17a414d739b40e633931613a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/>\u00b0.<\/div>\n<figure style=\"width: 574px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_003.jpg\" alt=\"Part a shows a 120 degree angle next to a 60 degree angle. Together, the angles form a straight line. Below the image, it reads 120 degrees plus 60 degrees equals 180 degrees. Part b shows a 45 degree angle attached to a 135 degree angle. Together, the angles form a straight line. Below the image, it reads 45 degrees plus 135 degrees equals 180 degrees.\" width=\"574\" height=\"177\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 3<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1277381\">If the sum of the measures of two angles is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-dc16a99348931a3823eff3764a961940_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0, then the angles are complementary angles. In <a class=\"autogenerated-content\" href=\"#CNX_BMath_Figure_09_03_004\">(Figure 4)<\/a>, each pair of angles is complementary, because their measures add to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-dc16a99348931a3823eff3764a961940_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0. Each angle is the <em data-effect=\"italics\">complement<\/em> of the other.<\/p>\n<div id=\"CNX_BMath_Figure_09_03_004\" class=\"bc-figure figure\">\n<div class=\"bc-figcaption figcaption\">The sum of the measures of complementary angles is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-dc16a99348931a3823eff3764a961940_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0.<\/div>\n<figure style=\"width: 286px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_004.jpg\" alt=\"Part a shows a 50 degree angle next to a 40 degree angle. Together, the angles form a right angle. Below the image, it reads 50 degrees plus 40 degrees equals 90 degrees. Part b shows a 60 degree angle attached to a 30 degree angle. Together, the angles form a right angle. Below the image, it reads 60 degrees plus 30 degrees equals 90 degrees.\" width=\"286\" height=\"185\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 4<\/figcaption><\/figure>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Supplementary and Complementary Angles<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1136622\">If the sum of the measures of two angles is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1371cfb17a414d739b40e633931613a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/>\u00b0, then the angles are supplementary.<\/p>\n<p id=\"eip-173\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c9644d971857e5b6fb1370bb8de85b92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6150531321f68bd917536a4352de756f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> are supplementary, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-00dc3804bf0a4cde6d2046aa6968882e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#65;&#43;&#109;&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#66;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"157\" style=\"vertical-align: -2px;\" \/>\u00b0.<\/p>\n<p id=\"eip-id1165148725040\">If the sum of the measures of two angles is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-dc16a99348931a3823eff3764a961940_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0, then the angles are complementary.<\/p>\n<p id=\"eip-id1165144076222\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c9644d971857e5b6fb1370bb8de85b92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6150531321f68bd917536a4352de756f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> are complementary, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2e0daee2fbc03259949f4eb72835a997_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#65;&#43;&#109;&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#66;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"148\" style=\"vertical-align: -2px;\" \/>\u00b0.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1610226\">In this section and the next, you will be introduced to some common geometry formulas. We will adapt our Problem Solving Strategy for Geometry Applications. The geometry formula will name the variables and give us the equation to solve.<\/p>\n<p id=\"fs-id1172596651337\">In addition, since these applications will all involve geometric shapes, it will be helpful to draw a figure and then label it with the information from the problem. We will include this step in the Problem Solving Strategy for Geometry Applications.<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">HOW TO: Use a Problem Solving Strategy for Geometry Applications<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol id=\"eip-id1170325410595\" class=\"stepwise\" type=\"1\">\n<li><strong data-effect=\"bold\">Read<\/strong> the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.<\/li>\n<li><strong>Identify<\/strong> what you are looking for.<\/li>\n<li><strong>Name<\/strong> what you are looking for and choose a variable to represent it.<\/li>\n<li><strong data-effect=\"bold\">Translate<\/strong> into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.<\/li>\n<li><strong data-effect=\"bold\">Solve<\/strong> the equation using good algebra techniques.<\/li>\n<li><strong data-effect=\"bold\">Check<\/strong> the answer in the problem and make sure it makes sense.<\/li>\n<li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p>The next example will show how you can use the Problem Solving Strategy for Geometry Applications to answer questions about supplementary and complementary angles.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1172599768030\" data-type=\"problem\">\n<p id=\"fs-id1468569\">An angle measures <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b5452daee4be3e74f7f618e26ea9f8fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0. Find a) its supplement, and b) its complement.<\/p>\n<\/div>\n<div id=\"fs-id1222879\" data-type=\"solution\">\n<div data-type=\"title\">Solution<\/div>\n<table id=\"eip-id1168469720851\" style=\"width: 100%;\" summary=\"Step 1 says, \u201cRead the problem. Draw the figure and label it with the given information.\u201d The word \u201cread\u201d is in bold. Beside this is a picture of two angles that together form a right angle. One is labeled as c degrees, the other as 40 degrees. Step 2 says, \u201cIdentify what you are looking for.\u201d The word \u201cidentify\u201d is in bold. Beside this it says, \u201cthe complement of a 40 degree angle.\u201d Step 3 says, \u201cName. Choose a variable to represent it.\u201d The word \u201cname\u201d is in bold. Beside this it says, \u201clet c equal the measure of the complement.\u201d Step 4 says, \u201cTranslate. Write the appropriate formula for the situation and substitute in the given information.\u201d The word \u201ctranslate\u201d is in bold. Beside this is the equation c plus 40 equals 90. Step 5 says, \u201cSolve the equation.\u201d The word \u201csolve\u201d is in bold. Beside this is c equals 50. Step 6 says \u201cCheck\u201d in bold. Beside this is 50 plus 40 followed by an equal sign with a question mark, then 90. The next line says 90 equals 90 with a check mark. Step 7 says, \u201cAnswer the question.\u201d The word \u201canswer\u201d is in bold. Beside this is the answer, \u201cThe complement of the 40 degree angle is 50 degrees.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td>a)<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 1. <strong data-effect=\"bold\">Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td><span id=\"eip-id1168466323010\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_047_img-01.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong data-effect=\"bold\">Identify<\/strong> what you are looking for.<\/td>\n<td><span id=\"eip-id1168469768902\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_047_img-02.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong data-effect=\"bold\">Name.<\/strong> Choose a variable to represent it.<\/td>\n<td><span id=\"eip-id1168469696175\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_047_img-03.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong data-effect=\"bold\">Translate.<\/strong><br \/>\nWrite the appropriate formula for the situation and substitute in the given information.<\/td>\n<td><span id=\"eip-id1168466314136\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_047_img-04.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168469631749\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_047_img-05.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong data-effect=\"bold\">Solve<\/strong> the equation.<\/td>\n<td><span id=\"eip-id1168469420724\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_047_img-06.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong data-effect=\"bold\">Check:<\/strong><br \/>\n<span id=\"eip-id1168469832615\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_047_img-07.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168469436943\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_047_img-08.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\n<td><span id=\"eip-id1168466286479\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_047_img-09.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168468558393\" style=\"width: 100%;\" summary=\"Step 1 says, \u201cRead the problem. Draw the figure and label it with the given information.\u201d The word \u201cread\u201d is in bold. Beside this is a picture of two angles that together form a right angle. One is labeled as c degrees, the other as 40 degrees. Step 2 says, \u201cIdentify what you are looking for.\u201d The word \u201cidentify\u201d is in bold. Beside this it says, \u201cthe complement of a 40 degree angle.\u201d Step 3 says, \u201cName. Choose a variable to represent it.\u201d The word \u201cname\u201d is in bold. Beside this it says, \u201clet c equal the measure of the complement.\u201d Step 4 says, \u201cTranslate. Write the appropriate formula for the situation and substitute in the given information.\u201d The word \u201ctranslate\u201d is in bold. Beside this is the equation c plus 40 equals 90. Step 5 says, \u201cSolve the equation.\u201d The word \u201csolve\u201d is in bold. Beside this is c equals 50. Step 6 says \u201cCheck\u201d in bold. Beside this is 50 plus 40 followed by an equal sign with a question mark, then 90. The next line says 90 equals 90 with a check mark. Step 7 says, \u201cAnswer the question.\u201d The word \u201canswer\u201d is in bold. Beside this is the answer, \u201cThe complement of the 40 degree angle is 50 degrees.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td>b)<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 1. <strong data-effect=\"bold\">Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td><span id=\"eip-id1168467117413\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_048_img-01.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong data-effect=\"bold\">Identify<\/strong> what you are looking for.<\/td>\n<td><span id=\"eip-id1168466083057\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_048_img-02.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong data-effect=\"bold\">Name.<\/strong> Choose a variable to represent it.<\/td>\n<td><span id=\"eip-id1168466051411\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_048_img-03.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong data-effect=\"bold\">Translate.<\/strong><br \/>\nWrite the appropriate formula for the situation and substitute in the given information.<\/td>\n<td><span id=\"eip-id1168469696180\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_048_img-04.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong data-effect=\"bold\">Solve<\/strong> the equation.<\/td>\n<td><span id=\"eip-id1168469428078\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_048_img-05.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168466186330\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_048_img-06.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong data-effect=\"bold\">Check:<\/strong><br \/>\n<span id=\"eip-id1168469478162\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_048_img-07.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168469726019\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_048_img-08.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\n<td><span id=\"eip-id1168469778644\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_048_img-09.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 1.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1280259\" data-type=\"problem\">\n<p id=\"fs-id1037597\">An angle measures <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-fccc29223e5ed0f314c0ff5b2d4395aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: 0px;\" \/>\u00b0. Find its: a) supplement b) complement.<\/p>\n<\/div>\n<div id=\"fs-id1296496\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<ol id=\"eip-id1168469681484\" class=\"circled\" type=\"a\">\n<li>155\u00b0<\/li>\n<li>65\u00b0<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1322685\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1467711\" data-type=\"exercise\">\n<div id=\"fs-id1296496\" data-type=\"solution\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 1.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1318314\" data-type=\"problem\">\n<p id=\"fs-id1699286\">An angle measures <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4b602f9e49dcb217ffd62f0186bb3dfb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0. Find its: a) supplement b) complement.<\/p>\n<\/div>\n<div id=\"fs-id1315018\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<ol id=\"eip-id1168469790669\" class=\"circled\" type=\"a\">\n<li>103\u00b0<\/li>\n<li>13\u00b0<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p>Did you notice that the words complementary and supplementary are in alphabetical order just like <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5b5f1c3a0e1ca514e4e63519e4d0c016_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6f62ea9815d82070371ef56ddaabdc0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/> are in numerical order?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1280482\" data-type=\"problem\">\n<p id=\"fs-id1296703\">Two angles are supplementary. The larger angle is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-92eb5a081f1b277fd071110bb208255d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0 more than the smaller angle. Find the measure of both angles.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<\/div>\n<div id=\"fs-id1435893\" data-type=\"solution\">\n<table id=\"eip-id1168468587362\" style=\"width: 100%;\" summary=\"Step 1 says, \u201cRead the problem. Draw the figure and label it with the given information.\u201d The word \u201cread\u201d is in bold. Beside this is a picture of two angles that together form a straight line. One is labeled as a plus 30 degrees, the other as a degrees. Step 2 says, \u201cIdentify what you are looking for.\u201d The word \u201cidentify\u201d is in bold. Beside this it says, \u201cthe measures of both angles.\u201d Step 3 says, \u201cName. Choose a variable to represent it. \u201c The word \u201cname\u201d is in bold. Beside this it says, \u201clet a equal the measure of the smaller angle.\u201d Below this, it says, \u201cThe larger angle is 30 degrees more than the smaller angle.\u201d Beside this is a plus 30 equals measure of larger angle. Step 4 says, \u201cTranslate. Write the appropriate formula and substitute.\u201d The word \u201ctranslate\u201d is in bold. Beside this is the equation measure of angle A plus measure of angle B equals 180. Step 5 says, \u201cSolve the equation.\u201d The word \u201csolve\u201d is in bold. Beside this is a plus 30 plus a equals 180, then 2a plus 30 equals 180, then 2a equals 150, then a equals 75. Step 6 says \u201cCheck\u201d in bold. Beside this is measure of angle A plus measure of angle B equals 180, then 75 plus 105 followed by an equal sign with a question mark, then 180. The next line says 180 equals 180 with a check mark. Step 7 says, \u201cAnswer the question.\u201d The word \u201canswer\u201d is in bold. Beside this is the answer, \u201cThe measures of the angles are 75 degrees and 105 degrees.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td>Step 1. <strong data-effect=\"bold\">Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td><span id=\"eip-id1168468587428\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-01.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong data-effect=\"bold\">Identify<\/strong> what you are looking for.<\/td>\n<td><span id=\"eip-id1168467187831\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-02.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong data-effect=\"bold\">Name.<\/strong> Choose a variable to represent it.<br \/>\nThe larger angle is 30\u00b0 more than the smaller angle.<\/td>\n<td><span id=\"eip-id1168468428731\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-03.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168467451148\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-04.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong data-effect=\"bold\">Translate.<\/strong><br \/>\nWrite the appropriate formula and substitute.<\/td>\n<td><span id=\"eip-id1168467419147\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-05.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Step 5. <strong data-effect=\"bold\">Solve<\/strong> the equation.<\/td>\n<td><span id=\"eip-id1168467248476\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-06.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168467248485\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-07.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168467248494\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-08.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168468773960\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-09.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168468773969\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-10.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168468773978\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-11.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168468451141\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-12.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong data-effect=\"bold\">Check:<\/strong><br \/>\n<span id=\"eip-id1168467263637\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-13.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168467263646\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-14.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168467263655\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-15.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\n<td><span id=\"eip-id1168468458480\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_049_img-16.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 2.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1141164\" data-type=\"problem\">\n<p id=\"fs-id1546937\">Two angles are supplementary. The larger angle is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-27f8d995528d175111be7511ae4bee05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/>\u00b0 more than the smaller angle. Find the measures of both angles.<\/p>\n<\/div>\n<div id=\"fs-id1716497\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1676187\">40\u00b0, 140\u00b0<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1399790\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1431406\" data-type=\"exercise\">\n<div id=\"fs-id1716497\" data-type=\"solution\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 2.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1172591784744\" data-type=\"problem\">\n<p id=\"fs-id754602\">Two angles are complementary. The larger angle is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b5452daee4be3e74f7f618e26ea9f8fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0 more than the smaller angle. Find the measures of both angles.<\/p>\n<\/div>\n<div id=\"fs-id1172599718234\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170856\">25\u00b0, 65\u00b0<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Use the Properties of Triangles<\/h1>\n<p id=\"fs-id1415833\">What do you already know about triangles? Triangle have three sides and three angles. Triangles are named by their vertices. The triangle in <a class=\"autogenerated-content\" href=\"#CNX_BMath_Figure_09_03_011\">(Figure 5)<\/a> is called <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-23f00feef91cb9ba3057f9e15d82ff39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#65;&#66;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/>, read \u2018triangle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c569357d9fb5d1ac87acab8cf2632f8b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#66;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"38\" style=\"vertical-align: -1px;\" \/>\u2019. We label each side with a lower case letter to match the upper case letter of the opposite vertex.<\/p>\n<div id=\"CNX_BMath_Figure_09_03_011\" class=\"bc-figure figure\">\n<div class=\"bc-figcaption figcaption\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-23f00feef91cb9ba3057f9e15d82ff39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#65;&#66;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/> has vertices <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a71d89b38876413f37b166d62d45daf6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#44;&#66;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"93\" style=\"vertical-align: -4px;\" \/> and sides <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b09b4a14842bd5986a79473d90b35756_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#98;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#99;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"77\" style=\"vertical-align: -4px;\" \/><\/div>\n<figure style=\"width: 215px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_011.jpg\" alt=\"The vertices of the triangle on the left are labeled A, B, and C. The sides are labeled a, b, and c.\" width=\"215\" height=\"148\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 5<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1172599773381\">The three angles of a triangle are related in a special way. The sum of their measures is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1371cfb17a414d739b40e633931613a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/>\u00b0.<\/p>\n<div id=\"fs-id1171104037911\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-88e2dad49ec6bc160c882e0577bbcf78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#65;&#43;&#109;&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#66;&#43;&#109;&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#67;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"221\" style=\"vertical-align: -2px;\" \/>\u00b0<\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Sum of the Measures of the Angles of a Triangle<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1413952\">For any <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-23f00feef91cb9ba3057f9e15d82ff39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#65;&#66;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/>, the sum of the measures of the angles is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1371cfb17a414d739b40e633931613a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/>\u00b0.<\/p>\n<div id=\"fs-id1171102911830\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-88e2dad49ec6bc160c882e0577bbcf78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#65;&#43;&#109;&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#66;&#43;&#109;&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#67;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"221\" style=\"vertical-align: -2px;\" \/>\u00b0<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1171102911830\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1334312\" data-type=\"problem\">\n<p id=\"fs-id1371881\">The measures of two angles of a triangle are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-358172ccd658c98b2ed5f4b3233b7fb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: 0px;\" \/>\u00b0 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6d47645cb621ef5598cc3ab3716391f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"17\" style=\"vertical-align: 0px;\" \/>\u00b0. Find the measure of the third angle.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<\/div>\n<div id=\"fs-id1172591771964\" data-type=\"solution\">\n<table id=\"eip-id1168467289823\" style=\"width: 100%;\" summary=\"Step 1 says, \u201cRead the problem. Draw the figure and label it with the given information.\u201d The word \u201cread\u201d is in bold. Beside this is a picture of a triangle with angles x, 82 degrees, and 55 degrees. Step 2 says, \u201cIdentify what you are looking for.\u201d The word \u201cidentify\u201d is in bold. Beside this it says, \u201cthe measure of the third angle in a triangle.\u201d Step 3 says, \u201cName. Choose a variable to represent it. \u201c The word \u201cname\u201d is in bold. Beside this it says, \u201clet x equal the measure of the angle.\u201d Step 4 says, \u201cTranslate. Write the appropriate formula and substitute.\u201d The word \u201ctranslate\u201d is in bold. Beside this is the equation measure of angle A plus measure of angle B plus measure of angle C equals 180. Step 5 says, \u201cSolve the equation.\u201d The word \u201csolve\u201d is in bold. Beside this is 55 plus 82 plus x equals 180, then 137 plus x equals 180, then x equals 43. Step 6 says \u201cCheck\u201d in bold. Beside this is 55 plus 82 plus 43 followed by an equal sign with a question mark, then 180. The next line says 180 equals 180 with a check mark. Step 7 says, \u201cAnswer the question.\u201d The word \u201canswer\u201d is in bold. Beside this is the answer, \u201cThe measure of the third angle is 43 degrees.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td>Step 1. <strong data-effect=\"bold\">Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td><span id=\"eip-id1168467362989\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_050_img-01.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong data-effect=\"bold\">Identify<\/strong> what you are looking for.<\/td>\n<td><span id=\"eip-id1168468505614\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_050_img-02.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong data-effect=\"bold\">Name.<\/strong> Choose a variable to represent it.<\/td>\n<td><span id=\"eip-id1168468707381\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_050_img-03.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong data-effect=\"bold\">Translate.<\/strong><br \/>\nWrite the appropriate formula and substitute.<\/td>\n<td><span id=\"eip-id1168468607629\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_050_img-04.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Step 5. <strong data-effect=\"bold\">Solve<\/strong> the equation.<\/td>\n<td><span id=\"eip-id1168468607650\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_050_img-05.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168467381406\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_050_img-06.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168469791076\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_050_img-07.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong data-effect=\"bold\">Check:<\/strong><br \/>\n<span id=\"eip-id1168468304554\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_050_img-08.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168468304564\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_050_img-09.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\n<td><span id=\"eip-id1168467164198\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_050_img-10.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 3.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1205986\" data-type=\"problem\">\n<p id=\"fs-id1172599718040\">The measures of two angles of a triangle are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d43d80b38a17ad1f60ee096179357a72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"17\" style=\"vertical-align: 0px;\" \/>\u00b0 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3f318a811b662dd82cb75e9bbf837411_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/>\u00b0. Find the measure of the third angle.<\/p>\n<\/div>\n<div id=\"fs-id1126761\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1335560\">21\u00b0<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1725798\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1326165\" data-type=\"exercise\">\n<div id=\"fs-id1126761\" data-type=\"solution\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 3.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id805484\" data-type=\"problem\">\n<p id=\"fs-id1276835\">A triangle has angles of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8aa381f435aa743d15144a973f1c3d43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-58fd615195b8c31477ef6feb82d62466_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: 0px;\" \/>\u00b0. Find the measure of the third angle.<\/p>\n<\/div>\n<div id=\"fs-id1446328\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1316328\">56\u00b0<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Right Triangles<\/h1>\n<div id=\"fs-id1190071\" class=\"bc-section section\" data-depth=\"1\">\n<p data-type=\"title\">Some triangles have special names. We will look first at the right triangle. A right triangle has one <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-dc16a99348931a3823eff3764a961940_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0 angle, which is often marked with the symbol shown in <a class=\"autogenerated-content\" href=\"#CNX_BMath_Figure_09_03_015\">(Figure 6)<\/a>.<\/p>\n<figure style=\"width: 109px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_015_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box and labeled 90 degrees.\" width=\"109\" height=\"86\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 6<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1142543\">If we know that a triangle is a right triangle, we know that one angle measures <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-dc16a99348931a3823eff3764a961940_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0 so we only need the measure of one of the other angles in order to determine the measure of the third angle.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1304195\" data-type=\"problem\">\n<p id=\"fs-id1401181\">One angle of a right triangle measures <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-54850e9dcf92ce1f38a0b614ade1b93f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0. What is the measure of the third angle?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<\/div>\n<div id=\"fs-id1609548\" data-type=\"solution\">\n<table id=\"eip-id1168468725720\" style=\"width: 100%;\" summary=\"Step 1 says, \u201cRead the problem. Draw the figure and label it with the given information.\u201d The word \u201cread\u201d is in bold. Beside this is a picture of a triangle with angles x, 90 degrees, and 28 degrees. Step 2 says, \u201cIdentify what you are looking for.\u201d The word \u201cidentify\u201d is in bold. Beside this it says, \u201cthe measure of an angle.\u201d Step 3 says, \u201cName. Choose a variable to represent it. \u201c The word \u201cname\u201d is in bold. Beside this it says, \u201clet x equal the measure of the angle.\u201d Step 4 says, \u201cTranslate. Write the appropriate formula and substitute.\u201d The word \u201ctranslate\u201d is in bold. Beside this is the equation measure of angle A plus measure of angle B plus measure of angle C equals 180, then x plus 90 plus 28 equals 180. Step 5 says, \u201cSolve the equation.\u201d The word \u201csolve\u201d is in bold. Beside this is x plus 118 equals 180, then x equals 62. Step 6 says \u201cCheck\u201d in bold. Beside this is 180 followed by an equal sign with a question mark, then 90 plus 28 plus 62. The next line says 180 equals 180 with a check mark. Step 7 says, \u201cAnswer the question.\u201d The word \u201canswer\u201d is in bold. Beside this is the answer, \u201cThe measure of the third angle is 62 degrees.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td>Step 1. <strong data-effect=\"bold\">Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td><span id=\"eip-id1168468725760\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_051_img-01.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong data-effect=\"bold\">Identify<\/strong> what you are looking for.<\/td>\n<td><span id=\"eip-id1168468297279\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_051_img-02.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong data-effect=\"bold\">Name.<\/strong> Choose a variable to represent it.<\/td>\n<td><span id=\"eip-id1168468297301\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_051_img-03.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong data-effect=\"bold\">Translate.<\/strong><br \/>\nWrite the appropriate formula and substitute.<\/td>\n<td><span id=\"eip-id1168468254449\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_051_img-04.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong data-effect=\"bold\">Solve<\/strong> the equation.<\/td>\n<td><span id=\"eip-id1168468254472\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_051_img-05.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168468254481\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_051_img-06.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168468287884\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_051_img-07.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong data-effect=\"bold\">Check:<\/strong><br \/>\n<span id=\"eip-id1168468287906\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_051_img-08.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168468287915\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_051_img-09.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\n<td><span id=\"eip-id1168467267556\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_051_img-10.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 4.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1437460\" data-type=\"problem\">\n<p id=\"fs-id1546221\">One angle of a right triangle measures <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a5e88c9e5e627cee775eddae75df9393_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0. What is the measure of the other angle?<\/p>\n<\/div>\n<div id=\"fs-id1596284\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1257352\">34\u00b0<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1172339\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id963740\" data-type=\"exercise\">\n<div id=\"fs-id1596284\" data-type=\"solution\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 4.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1665526\" data-type=\"problem\">\n<p id=\"fs-id1332188\">One angle of a right triangle measures <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-be68bbb911da04517fa3266e6db046fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: 0px;\" \/>\u00b0. What is the measure of the other angle?<\/p>\n<\/div>\n<div id=\"fs-id1182217\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1231568\">45\u00b0<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p>In the examples so far, we could draw a figure and label it directly after reading the problem. In the next example, we will have to define one angle in terms of another. So we will wait to draw the figure until we write expressions for all the angles we are looking for.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1172580513515\" data-type=\"problem\">\n<p id=\"fs-id1151760\">The measure of one angle of a right triangle is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d647a5ecf47eb476cd54ba72ffd63b5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0 more than the measure of the smallest angle. Find the measures of all three angles.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<\/div>\n<div id=\"fs-id1580719\" data-type=\"solution\">\n<table id=\"eip-id1168466077134\" style=\"width: 100%;\" summary=\"Step 1 says, \u201cRead the problem.\u201d The word \u201cread\u201d is in bold. Beside this is a picture of a triangle with angles a, 90 degrees, and a plus 20 degrees. Step 2 says, \u201cIdentify what you are looking for.\u201d The word \u201cidentify\u201d is in bold. Beside this it says, \u201cthe measures of all three angles.\u201d Step 3 says, \u201cName. Choose a variable to represent it. \u201c The word \u201cname\u201d is in bold. Beside this it says, \u201cLet d equal first angle. d plus 20 equals second angle. 90 equals third angle (the right angle).\u201d Step 4 says, \u201cTranslate. Write the appropriate formula and substitute into the formula.\u201d The word \u201ctranslate\u201d is in bold. Beside this is the equation measure of angle A plus measure of angle B plus measure of angle C equals 180, then d plus d plus 20 plus 90 equals 180. Step 5 says, \u201cSolve the equation.\u201d The word \u201csolve\u201d is in bold. Beside this is 2d plus 110 equals 180, then 2d equals 70, then d equals 35 first angle, then d plus 20 second angle, then 35 plus 20, then 55, then 90 third angle. Step 6 says \u201cCheck\u201d in bold. Beside this is 35 plus 55 plus 90 followed by an equal sign with a question mark, then 180. The next line says 180 equals 180 with a check mark. Step 7 says, \u201cAnswer the question.\u201d The word \u201canswer\u201d is in bold. Beside this is the answer, \u201cThe three angles measure 35 degrees, 55 degrees, and 90 degrees.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td>Step 1. <strong data-effect=\"bold\">Read<\/strong> the problem.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong data-effect=\"bold\">Identify<\/strong> what you are looking for.<\/td>\n<td>the measures of all three angles<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Step 3. <strong data-effect=\"bold\">Name.<\/strong> Choose a variable to represent it.<\/p>\n<p>Now draw the figure and label it with the given information.<\/td>\n<td><span id=\"eip-id1168469701816\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-01.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168466426323\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-02.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168469683819\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-03.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168466096904\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-04.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong data-effect=\"bold\">Translate.<\/strong><br \/>\nWrite the appropriate formula and substitute into the formula.<\/td>\n<td><span id=\"eip-id1168466113853\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-05.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168466113862\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-06.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong data-effect=\"bold\">Solve<\/strong> the equation.<\/td>\n<td><span id=\"eip-id1168469634538\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-07.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168469829064\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-08.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168466047198\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-09.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168466047207\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-10.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168469613957\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-11.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168469691852\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-12.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168469659677\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-13.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong data-effect=\"bold\">Check:<\/strong><br \/>\n<span id=\"eip-id1168469846595\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-14.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\n<span id=\"eip-id1168466076009\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-15.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\n<td><span id=\"eip-id1168466395444\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_052_img-16.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 5.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1736498\" data-type=\"problem\">\n<p id=\"fs-id1172580513385\">The measure of one angle of a right triangle is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-411cb15b691d010abfe998424e6a80a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0 more than the measure of the smallest angle. Find the measures of all three angles.<\/p>\n<\/div>\n<div id=\"fs-id1333588\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1572312\">20\u00b0, 70\u00b0, 90\u00b0<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1156706\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1414965\" data-type=\"exercise\">\n<div id=\"fs-id1333588\" data-type=\"solution\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 5.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1221217\" data-type=\"problem\">\n<p id=\"fs-id1172596666838\">The measure of one angle of a right triangle is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-92eb5a081f1b277fd071110bb208255d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0 more than the measure of the smallest angle. Find the measures of all three angles.<\/p>\n<\/div>\n<div id=\"fs-id1425384\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1364756\">30\u00b0, 60\u00b0, 90\u00b0<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Similar Triangles<\/h1>\n<p id=\"fs-id1247950\">When we use a map to plan a trip, a sketch to build a bookcase, or a pattern to sew a dress, we are working with similar figures. In geometry, if two figures have exactly the same shape but different sizes, we say they are similar figures. One is a scale model of the other. The corresponding sides of the two figures have the same ratio, and all their corresponding angles are have the same measures.<\/p>\n<p id=\"fs-id1159290\">The two triangles in <a class=\"autogenerated-content\" href=\"#CNX_BMath_Figure_09_03_020\">(Figure 7)<\/a> are similar. Each side of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-23f00feef91cb9ba3057f9e15d82ff39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#65;&#66;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/> is four times the length of the corresponding side of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-bfb544ee50aa52cf4503ca937feda154_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#88;&#89;&#90;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"58\" style=\"vertical-align: 0px;\" \/> and their corresponding angles have equal measures.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-23f00feef91cb9ba3057f9e15d82ff39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#65;&#66;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-bfb544ee50aa52cf4503ca937feda154_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#88;&#89;&#90;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"58\" style=\"vertical-align: 0px;\" \/> are similar triangles. Their corresponding sides have the same ratio and the corresponding angles have the same measure.<\/p>\n<div id=\"CNX_BMath_Figure_09_03_020\" class=\"bc-figure figure\">\n<figure style=\"width: 522px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_020.jpg\" alt=\"Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled 16, the side across from B is labeled 20, and the side across from C is labeled 12. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 4, the side across from Y is labeled 5, and the side across from Z is labeled 3. Beside the triangles, it says that the measure of angle A equals the measure of angle X, the measure of angle B equals the measure of angle Y, and the measure of angle C equals the measure of angle Z. Below this is the proportion 16 over 4 equals 20 over 5 equals 12 over 3.\" width=\"522\" height=\"140\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 7<\/figcaption><\/figure>\n<\/div>\n<div id=\"eip-13\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Properties of Similar Triangles<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"eip-id1170323803303\">If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths are in the same ratio.<\/p>\n<p><span id=\"fs-id1182340538\" data-type=\"media\" data-alt=\"...\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_056_img-1.jpg\" alt=\"...\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<p>The length of a side of a triangle may be referred to by its endpoints, two vertices of the triangle. For example, in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3d25051ee43d411c0c92aa7b938bf013_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#65;&#66;&#67;&#92;&#116;&#101;&#120;&#116;&#123;&#58;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"60\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1171104085009\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8f8a87e0a6b0a8e3d3757be65d388b5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#108;&#101;&#110;&#103;&#116;&#104;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#97;&#110;&#32;&#97;&#108;&#115;&#111;&#32;&#98;&#101;&#32;&#119;&#114;&#105;&#116;&#116;&#101;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#66;&#67;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#108;&#101;&#110;&#103;&#116;&#104;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#97;&#110;&#32;&#97;&#108;&#115;&#111;&#32;&#98;&#101;&#32;&#119;&#114;&#105;&#116;&#116;&#101;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#65;&#67;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#108;&#101;&#110;&#103;&#116;&#104;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#99;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#97;&#110;&#32;&#97;&#108;&#115;&#111;&#32;&#98;&#101;&#32;&#119;&#114;&#105;&#116;&#116;&#101;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#65;&#66;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"60\" width=\"275\" style=\"vertical-align: -25px;\" \/><\/p>\n<p id=\"fs-id1614427\">We will often use this notation when we solve similar triangles because it will help us match up the corresponding side lengths.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1725972\" data-type=\"problem\">\n<p id=\"fs-id1424229\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-23f00feef91cb9ba3057f9e15d82ff39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#65;&#66;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-bfb544ee50aa52cf4503ca937feda154_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#88;&#89;&#90;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"58\" style=\"vertical-align: 0px;\" \/> are similar triangles. The lengths of two sides of each triangle are shown. Find the lengths of the third side of each triangle.<\/p>\n<p><span id=\"fs-id1372781\" data-type=\"media\" data-alt=\"Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 3.2, and the side across from C is labeled 4. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 4.5, the side across from Y is labeled y, and the side across from Z is labeled 3.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_022.jpg\" alt=\"Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 3.2, and the side across from C is labeled 4. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 4.5, the side across from Y is labeled y, and the side across from Z is labeled 3.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p><strong>Solution<\/strong><\/p>\n<\/div>\n<div id=\"fs-id1188208\" data-type=\"solution\">\n<table id=\"eip-id1168466690156\" style=\"width: 100%; height: 129px;\" summary=\".\" data-label=\"\">\n<tbody>\n<tr style=\"height: 16px;\">\n<td style=\"height: 16px; width: 34.82%;\">Step 1. <strong data-effect=\"bold\">Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td style=\"height: 16px; width: 65.18%;\">The figure is provided.<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"height: 16px; width: 34.82%;\">Step 2. <strong data-effect=\"bold\">Identify<\/strong> what you are looking for.<\/td>\n<td style=\"height: 16px; width: 65.18%;\">The length of the sides of similar triangles<\/td>\n<\/tr>\n<tr style=\"height: 49px;\">\n<td style=\"height: 49px; width: 34.82%;\">Step 3. <strong data-effect=\"bold\">Name.<\/strong> Choose a variable to represent it.<\/td>\n<td style=\"height: 49px; width: 65.18%;\">Let<br \/>\n<em data-effect=\"italics\">a<\/em> = length of the third side of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-23f00feef91cb9ba3057f9e15d82ff39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#65;&#66;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/><br \/>\n<em data-effect=\"italics\">y<\/em> = length of the third side <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-bfb544ee50aa52cf4503ca937feda154_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#88;&#89;&#90;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"58\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"height: 16px; width: 34.82%;\">Step 4.<strong> Translate.<\/strong><\/td>\n<td style=\"height: 16px; width: 65.18%;\">The triangles are similar, so the corresponding sides are in the same ratio. So <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-0de3d58b31c5a7b2eae0a8750f2ef0b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#65;&#66;&#125;&#123;&#88;&#89;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#66;&#67;&#125;&#123;&#89;&#90;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#65;&#67;&#125;&#123;&#88;&#90;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"123\" style=\"vertical-align: -6px;\" \/><\/p>\n<p>Since the side <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-499c06d1fd6bc714961ad70029f60415_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#66;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"60\" style=\"vertical-align: -1px;\" \/> corresponds to the side <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c31a08367ad037e788bfc95e1f4528b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#88;&#89;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"63\" style=\"vertical-align: 0px;\" \/>, we will use the ratio <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-bd16575cc557552c52b00aadf57e24a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#65;&#66;&#125;&#125;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#88;&#89;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"55\" style=\"vertical-align: -7px;\" \/> to find the other sides.<\/p>\n<p>Be careful to match up corresponding sides correctly.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_057_img-01.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 34.82%;\">Step 5. <strong data-effect=\"bold\">Solve<\/strong> the equation.<span id=\"eip-id1168466188711\" data-type=\"media\" data-alt=\".\"><\/span><\/td>\n<td style=\"width: 65.18%;\">&nbsp;<\/p>\n<p><span id=\"eip-id1168466188711\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_057_img-02.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 34.82%;\"><span id=\"eip-id1168466188711\" data-type=\"media\" data-alt=\".\"><\/span>Step 6. <strong data-effect=\"bold\">Check.<\/strong><br \/>\n<span id=\"eip-id1168469628162\" data-type=\"media\" data-alt=\".\"><\/span><\/td>\n<td style=\"width: 65.18%;\"><span id=\"eip-id1168469628162\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_057_img-03.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 34.82%;\">Step 7. <strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\n<td style=\"width: 65.18%;\"><span id=\"eip-id1168469628162\" data-type=\"media\" data-alt=\".\">The third side of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-23f00feef91cb9ba3057f9e15d82ff39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#65;&#66;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/> is 6 and the third side of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-bfb544ee50aa52cf4503ca937feda154_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#88;&#89;&#90;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"58\" style=\"vertical-align: 0px;\" \/> is 2.4.<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 6.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1732444\" data-type=\"problem\">\n<p id=\"fs-id1172584151393\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-23f00feef91cb9ba3057f9e15d82ff39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#65;&#66;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/> is similar to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-bfb544ee50aa52cf4503ca937feda154_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#88;&#89;&#90;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"58\" style=\"vertical-align: 0px;\" \/>. Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<p><span id=\"fs-id1263455\" data-type=\"media\" data-alt=\"Two triangles are shown. They appear to be the same shape, but the triangle on the right is larger The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 15, and the side across from C is labeled 17. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 12, the side across from Y is labeled y, and the side across from Z is labeled 25.5.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_023.jpg\" alt=\"Two triangles are shown. They appear to be the same shape, but the triangle on the right is larger The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 15, and the side across from C is labeled 17. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 12, the side across from Y is labeled y, and the side across from Z is labeled 25.5.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1172591854682\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1172591854684\">8<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1303468\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1388517\" data-type=\"exercise\">\n<div id=\"fs-id1172591854682\" data-type=\"solution\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 6.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1172591772188\" data-type=\"problem\">\n<p id=\"fs-id742499\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-23f00feef91cb9ba3057f9e15d82ff39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#65;&#66;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/> is similar to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-bfb544ee50aa52cf4503ca937feda154_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#88;&#89;&#90;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"58\" style=\"vertical-align: 0px;\" \/>. Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<p><span id=\"fs-id1172591786353\" data-type=\"media\" data-alt=\"Two triangles are shown. They appear to be the same shape, but the triangle on the right is larger The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 15, and the side across from C is labeled 17. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 12, the side across from Y is labeled y, and the side across from Z is labeled 25.5.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_023.jpg\" alt=\"Two triangles are shown. They appear to be the same shape, but the triangle on the right is larger The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 15, and the side across from C is labeled 17. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 12, the side across from Y is labeled y, and the side across from Z is labeled 25.5.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1841277\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1172591854687\">22.5<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1 id=\"fs-id1172591772188\" data-type=\"problem\">Use the Pythagorean Theorem<\/h1>\n<p id=\"fs-id1410155\">The Pythagorean Theorem is a special property of right triangles that has been used since ancient times. It is named after the Greek philosopher and mathematician Pythagoras who lived around <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8ce4f218c691da59dfd93c07b15d0dcf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"27\" style=\"vertical-align: 0px;\" \/> BCE.<\/p>\n<p id=\"fs-id1410106\">Remember that a right triangle has a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-dc16a99348931a3823eff3764a961940_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0 angle, which we usually mark with a small square in the corner. The side of the triangle opposite the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-dc16a99348931a3823eff3764a961940_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0 angle is called the hypotenuse, and the other two sides are called the legs. See <a class=\"autogenerated-content\" href=\"#CNX_BMath_Figure_09_03_024\">(Figure 8)<\/a>.<\/p>\n<p>In a right triangle, the side opposite the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-dc16a99348931a3823eff3764a961940_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0 angle is called the hypotenuse and each of the other sides is called a leg.<\/p>\n<div id=\"CNX_BMath_Figure_09_03_024\" class=\"bc-figure figure\">\n<figure style=\"width: 561px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_024.jpg\" alt=\"Three right triangles are shown. Each has a box representing the right angle. The first one has the right angle in the lower left corner, the next in the upper left corner, and the last one at the top. The two sides touching the right angle are labeled \u201cleg\u201d in each triangle. The sides across from the right angles are labeled \u201chypotenuse.\u201d\" width=\"561\" height=\"121\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 8<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1550190\">The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse.<\/p>\n<div id=\"fs-id1701775\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">The Pythagorean Theorem<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1548386\">In any right triangle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-23f00feef91cb9ba3057f9e15d82ff39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#65;&#66;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/>,<\/p>\n<div id=\"fs-id1171104156908\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7ea02600b9972a334e42686c1b0b4cf4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#99;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"93\" style=\"vertical-align: -2px;\" \/><\/div>\n<p id=\"fs-id1422647\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> is the length of the hypotenuse <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/> are the lengths of the legs.<\/p>\n<p><span id=\"fs-id1723708\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked a and b.\"><img decoding=\"async\" class=\"alignnone\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_025.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked a and b.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<p>To solve problems that use the Pythagorean Theorem, we will need to find square roots. We defined the notation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ff6e04829c3778dcab4453317cfa77ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"30\" style=\"vertical-align: -4px;\" \/> in this way:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ac70d9c243da0f81e62eed022d7197cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#109;&#61;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#109;&#125;&#61;&#110;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#110;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"259\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1432808\">For example, we found that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-0c6a36f543576cfd4fa34a645e3eff65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8995e2084120c1c1f4b53f490d281bc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/> because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-fdc53830c9614b4d54be741e3490a7a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#53;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"57\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<p id=\"fs-id1538008\">We will use this definition of square roots to solve for the length of a side in a right triangle.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1419626\" data-type=\"problem\">\n<p id=\"fs-id1420675\">Use the Pythagorean Theorem to find the length of the hypotenuse.<\/p>\n<p><span id=\"fs-id1411841\" data-type=\"media\" data-alt=\"Right triangle with legs labeled as 3 and 4.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_026_img.jpg\" alt=\"Right triangle with legs labeled as 3 and 4.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p><strong>Solution<\/strong><\/p>\n<\/div>\n<div id=\"fs-id1172599766488\" data-type=\"solution\">\n<table id=\"eip-id1168469450887\" style=\"width: 100%;\" summary=\"Step 1 says, \u201cRead the problem.\u201d The word \u201cread\u201d is in bold. Beside this is a picture of a right triangle. The side across from the right angle is labeled c, the sides touching the right angle are labeled 3 and 4. Step 2 says, \u201cIdentify what you are looking for.\u201d The word \u201cidentify\u201d is in bold. Beside this it says, \u201cthe length of the hypotenuse of the triangle.\u201d Step 3 says, \u201cName. Choose a variable to represent it. \u201c The word \u201cname\u201d is in bold. Beside this it says, \u201cLet c equal the length of the hypotenuse.\u201d Step 4 says, \u201cTranslate. Write the appropriate formula. Substitute.\u201d The word \u201ctranslate\u201d is in bold. Beside this is the equation a squared plus b squared equals c squared, then 3 squared plus 4 squared equals c squared. Step 5 says, \u201cSolve the equation. Simplify. Use the definition of square root. Simplify.\u201d The word \u201csolve\u201d is in bold. Beside this is 9 plus 16 equals c squared, then 25 equals c squared, then square root of 25 equals c, then 5 equals c. Step 6 says \u201cCheck\u201d in bold. Beside this is 3 squared plus 4 squared followed by an equal sign with a question mark, then a red 5 squared. Below this is 9 plus 16 followed by an equal sign with a question mark, then 25. The next line says 25 equals 25 with a check mark. Step 7 says, \u201cAnswer the question.\u201d The word \u201canswer\u201d is in bold. Beside this is the answer, \u201cThe length of the hypotenuse is 5.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td>Step 1. <strong data-effect=\"bold\">Read<\/strong> the problem.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong data-effect=\"bold\">Identify<\/strong> what you are looking for.<\/td>\n<td>the length of the hypotenuse of the triangle<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong data-effect=\"bold\">Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-710f083184f6961892c6e045dda64ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#108;&#101;&#110;&#103;&#116;&#104;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#104;&#121;&#112;&#111;&#116;&#101;&#110;&#117;&#115;&#101;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"254\" style=\"vertical-align: -3px;\" \/><br \/>\n<span id=\"eip-id1168469449113\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_053_img-01.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong data-effect=\"bold\">Translate.<\/strong><br \/>\nWrite the appropriate formula.<br \/>\nSubstitute.<\/td>\n<td><span id=\"eip-id1168469890812\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_053_img-02.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Step 5. <strong data-effect=\"bold\">Solve<\/strong> the equation.<\/td>\n<td><span id=\"eip-id1168469890834\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_053_img-03.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong data-effect=\"bold\">Check:<\/strong><br \/>\n<span id=\"eip-id1168469481160\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_053_img-04.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\n<td>The length of the hypotenuse is 5.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 7.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1542055\" data-type=\"problem\">\n<p id=\"fs-id1542057\">Use the Pythagorean Theorem to find the length of the hypotenuse.<\/p>\n<p><span id=\"fs-id1321685\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked 6 and 8.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_029_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked 6 and 8.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1727689\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1259642\">10<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1546550\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1172599768654\" data-type=\"exercise\">\n<div id=\"fs-id1727689\" data-type=\"solution\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 7.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1260564\" data-type=\"problem\">\n<p id=\"fs-id1260566\">Use the Pythagorean Theorem to find the length of the hypotenuse.<\/p>\n<p><span id=\"fs-id1573251\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as c. One of the sides touching the right angle is labeled as 15, the other is labeled \u201c8\u201d.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_030_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as c. One of the sides touching the right angle is labeled as 15, the other is labeled \u201c8\u201d.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1256380\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1172574418263\">17<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1403907\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1172591967221\" data-type=\"exercise\">\n<div id=\"fs-id1256380\" data-type=\"solution\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1172574467640\" data-type=\"problem\">\n<p id=\"fs-id1172591785037\">Use the Pythagorean Theorem to find the length of the longer leg.<\/p>\n<p><span id=\"fs-id1172591785041\" data-type=\"media\" data-alt=\"Right triangle is shown with one leg labeled as 5 and hypotenuse labeled as 13.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_031_img.jpg\" alt=\"Right triangle is shown with one leg labeled as 5 and hypotenuse labeled as 13.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p><strong>Solution<\/strong><\/p>\n<\/div>\n<div id=\"fs-id1172573419851\" data-type=\"solution\">\n<table id=\"eip-id1168467480162\" style=\"width: 100%;\" summary=\"Step 1 says, \u201cRead the problem.\u201d The word \u201cread\u201d is in bold. Step 2 says, \u201cIdentify what you are looking for.\u201d The word \u201cidentify\u201d is in bold. Beside this it says, \u201cthe length of the leg of the triangle.\u201d Step 3 says, \u201cName. Choose a variable to represent it. \u201c The word \u201cname\u201d is in bold. Beside this it says, \u201cLet b equal the leg of the triangle. Label side b.\u201d Beside this is a picture of a right triangle. The side across from the right angle is labeled 13, the sides touching the right angle are labeled 5 and b. Step 4 says, \u201cTranslate. Write the appropriate formula. Substitute.\u201d The word \u201ctranslate\u201d is in bold. Beside this is the equation a squared plus b squared equals c squared, then 5 squared plus b squared equals 13 squared. Step 5 says, \u201cSolve the equation. Isolate the variable term. Use the definition of square root. Simplify.\u201d The word \u201csolve\u201d is in bold. Beside this is 25 plus b squared equals 169, then b squared equals 144, then b equals the square root of 144, then b equals 12. Step 6 says \u201cCheck\u201d in bold. Beside this is 5 squared plus a red 12 squared followed by an equal sign with a question mark, then 13 squared. Below this is 25 plus 144 followed by an equal sign with a question mark, then 169. The next line says 169 equals 169 with a check mark. Step 7 says, \u201cAnswer the question.\u201d The word \u201canswer\u201d is in bold. Beside this is the answer, \u201cThe length of the leg is 12.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td>Step 1. <strong data-effect=\"bold\">Read<\/strong> the problem.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong data-effect=\"bold\">Identify<\/strong> what you are looking for.<\/td>\n<td>The length of the leg of the triangle<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Step 3. <strong data-effect=\"bold\">Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f94142334e8897c30594de5a65d69016_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#108;&#101;&#103;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#116;&#114;&#105;&#97;&#110;&#103;&#108;&#101;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"200\" style=\"vertical-align: -3px;\" \/><br \/>\nLabel side <em data-effect=\"italics\">b<\/em><br \/>\n<span id=\"eip-id1168467480249\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_054_img-01.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong data-effect=\"bold\">Translate.<\/strong><br \/>\nWrite the appropriate formula. Substitute.<\/td>\n<td><span id=\"eip-id1168466021991\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_054_img-02.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Step 5. <strong data-effect=\"bold\">Solve<\/strong> the equation. Isolate the variable term. Use the definition of the square root.<br \/>\nSimplify.<\/td>\n<td><span id=\"eip-id1168469557654\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_054_img-03.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong data-effect=\"bold\">Check:<\/strong><\/td>\n<td><span id=\"eip-id1168468687314\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_054_img-04.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\n<td>The length of the leg is 12.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1172597113282\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1172597113285\" data-type=\"exercise\">\n<div id=\"fs-id1402124\" data-type=\"problem\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 8.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1402124\" data-type=\"problem\">\n<p id=\"fs-id1402126\">Use the Pythagorean Theorem to find the length of the leg.<\/p>\n<p><span id=\"fs-id1421940\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 15, the other is labeled \u201cb\u201d.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_034_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 15, the other is labeled \u201cb\u201d.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1172599716643\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1172591773440\">8<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1172599716643\" data-type=\"solution\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 8.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1172599716594\" data-type=\"problem\">\n<p id=\"fs-id1172599716596\">Use the Pythagorean Theorem to find the length of the leg.<\/p>\n<p><span id=\"fs-id1340466\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 15. One of the sides touching the right angle is labeled as 9, the other is labeled \u201cb\u201d.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_035_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 15. One of the sides touching the right angle is labeled as 9, the other is labeled \u201cb\u201d.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1403242\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1715840\">12<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1435399\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1415446\" data-type=\"exercise\">\n<div id=\"fs-id1403242\" data-type=\"solution\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 9<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1230268\" data-type=\"problem\">\n<p id=\"fs-id1432662\">Kelvin is building a gazebo and wants to brace each corner by placing a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e4bd492f6d145ce706a985d4ea19add8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#45;&#105;&#110;&#99;&#104;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"55\" style=\"vertical-align: -1px;\" \/> wooden bracket diagonally as shown. How far below the corner should he fasten the bracket if he wants the distances from the corner to each end of the bracket to be equal? Approximate to the nearest tenth of an inch.<\/p>\n<p><span id=\"fs-id1172591785993\" data-type=\"media\" data-alt=\"A picture of a gazebo is shown. Beneath the roof is a rectangular shape. There are two braces from the top to each side. The brace on the left is labeled as 10 inches. From where the brace hits the side to the roof is labeled as x.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_036.jpg\" alt=\"A picture of a gazebo is shown. Beneath the roof is a rectangular shape. There are two braces from the top to each side. The brace on the left is labeled as 10 inches. From where the brace hits the side to the roof is labeled as x.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p><strong>Solution<\/strong><\/p>\n<\/div>\n<div id=\"fs-id1727579\" data-type=\"solution\">\n<table id=\"eip-id1168468762748\" style=\"width: 100%;\" summary=\"Step 1 says, \u201cRead the problem.\u201d The word \u201cread\u201d is in bold. Step 2 says, \u201cIdentify what you are looking for.\u201d The word \u201cidentify\u201d is in bold. Beside this it says, \u201cthe distance from the corner that the bracket should be attached.\u201d Below this is a picture of a right triangle with the side across from the right angle labeled as 10 inches and the other 2 sides labeled as x. Step 3 says, \u201cName. Choose a variable to represent it. \u201c The word \u201cname\u201d is in bold. Beside this it says, \u201cLet x equal the distance from the corner.\u201d Step 4 says, \u201cTranslate. Write the appropriate formula. Substitute.\u201d The word \u201ctranslate\u201d is in bold. Beside this is the equation a squared plus b squared equals c squared, then x squared plus x squared equals 10 squared. Step 5 says, \u201cSolve the equation. Isolate the variable term. Use the definition of square root. Simplify. Approximate to the nearest tenth.\u201d The word \u201csolve\u201d is in bold. Beside this is 2 times x squared equals 100, then x squared equals 50, then x equals the square root of 50, then x is approximately 7.1. Step 6 says \u201cCheck\u201d in bold. Beside this is a squared plus b squared equals c squared, then a red 7.1 squared plus a red 7.1 squared followed by an approximate sign with a question mark, then 10 squared. Below this, it says \u201cYes.\u201d Step 7 says, \u201cAnswer the question.\u201d The word \u201canswer\u201d is in bold. Beside this is the answer, \u201cKelvin should fasten each piece of wood approximately 7.1 inches from the corner.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td>Step 1. <strong data-effect=\"bold\">Read<\/strong> the problem.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Step 2. <strong data-effect=\"bold\">Identify<\/strong> what you are looking for.<\/td>\n<td>the distance from the corner that the bracket should be attached<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong data-effect=\"bold\">Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let <em data-effect=\"italics\">x<\/em> = the distance from the corner<br \/>\n<span id=\"eip-id1168467381315\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_055_img-01.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong data-effect=\"bold\">Translate.<\/strong><br \/>\nWrite the appropriate formula.<br \/>\nSubstitute.<\/td>\n<td><span id=\"eip-id1168469480877\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_055_img-02.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong data-effect=\"bold\">Solve<\/strong> the equation.<br \/>\nIsolate the variable.<br \/>\nUse the definition of the square root.<br \/>\nSimplify. Approximate to the nearest tenth.<\/td>\n<td><span id=\"eip-id1168466339030\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_055_img-03.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong data-effect=\"bold\">Check:<\/strong><br \/>\n<span id=\"eip-id1168465147070\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_055_img-04.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><br \/>\nYes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\n<td>Kelvin should fasten each piece of wood approximately 7.1&#8243; from the corner.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 9.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1172591855349\" data-type=\"problem\">\n<p id=\"fs-id1172591855351\">John puts the base of a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d50466835e4643efbfb1306b3f9607e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#45;&#102;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"34\" style=\"vertical-align: -1px;\" \/> ladder <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8995e2084120c1c1f4b53f490d281bc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/> feet from the wall of his house. How far up the wall does the ladder reach?<\/p>\n<p><span id=\"fs-id1543199\" data-type=\"media\" data-alt=\"A picture of a house is shown. There is a ladder leaning against the side of the house. The ladder is labeled 13 feet. The horizontal distance from the ladder's base to the house is labeled 5 feet.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_039_img.jpg\" alt=\"A picture of a house is shown. There is a ladder leaning against the side of the house. The ladder is labeled 13 feet. The horizontal distance from the ladder's base to the house is labeled 5 feet.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1660058\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1431748\">12 feet<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1405087\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1207430\" data-type=\"exercise\">\n<div id=\"fs-id1660058\" data-type=\"solution\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 9.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1536639\" data-type=\"problem\">\n<p id=\"fs-id1536641\">Randy wants to attach a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c3d9e0deebf7245fc90ab48628612c18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#55;&#45;&#102;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"34\" style=\"vertical-align: -1px;\" \/> string of lights to the top of the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2c968ffc4e3b36843e92449c2e0206e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&#45;&#102;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"34\" style=\"vertical-align: -1px;\" \/> mast of his sailboat. How far from the base of the mast should he attach the end of the light string?<\/p>\n<p><span id=\"fs-id1172591861163\" data-type=\"media\" data-alt=\"A picture of a boat is shown. The height of the centre pole is labeled 15 feet. The string of lights is at a diagonal from the top of the pole and is labeled 17 feet.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_040_img.jpg\" alt=\"A picture of a boat is shown. The height of the centre pole is labeled 15 feet. The string of lights is at a diagonal from the top of the pole and is labeled 17 feet.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1172596641002\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1172596641004\">8 feet<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Key Concepts<\/h1>\n<ul id=\"eip-655\">\n<li><strong>Supplementary and Complementary Angles<\/strong>\n<ul id=\"eip-id1170324960756\">\n<li>If the sum of the measures of two angles is 180\u00b0, then the angles are supplementary.<\/li>\n<li>If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c9644d971857e5b6fb1370bb8de85b92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6150531321f68bd917536a4352de756f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> are supplementary, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c2568b474c5db6ab6cc65b067483c09f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#65;&#43;&#109;&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#66;&#61;&#49;&#56;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"157\" style=\"vertical-align: -2px;\" \/>.<\/li>\n<li>If the sum of the measures of two angles is 90\u00b0, then the angles are complementary.<\/li>\n<li>If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c9644d971857e5b6fb1370bb8de85b92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6150531321f68bd917536a4352de756f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> are complementary, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7f0bb809a83776578a07793d80c05ed5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#65;&#43;&#109;&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#66;&#61;&#57;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"148\" style=\"vertical-align: -2px;\" \/>.<\/li>\n<\/ul>\n<\/li>\n<li><strong>Solve Geometry Applications<\/strong>\n<ol id=\"eip-id1170326094648\" class=\"stepwise\" type=\"1\">\n<li>Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.<\/li>\n<li>Identify what you are looking for.<\/li>\n<li>Name what you are looking for and choose a variable to represent it.<\/li>\n<li>Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.<\/li>\n<li>Solve the equation using good algebra techniques.<\/li>\n<li>Check the answer in the problem and make sure it makes sense.<\/li>\n<li>Answer the question with a complete sentence.<\/li>\n<\/ol>\n<\/li>\n<li><span id=\"fs-id11722323361163\" data-type=\"media\" data-alt=\"..\"><img decoding=\"async\" class=\"alignright\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_012.jpg\" alt=\"..\" data-media-type=\"image\/jpeg\" \/><\/span><strong>Sum of the Measures of the Angles of a Triangle<\/strong>\n<ul id=\"eip-id1170322666796\">\n<li>For any <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-23f00feef91cb9ba3057f9e15d82ff39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#65;&#66;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/>, the sum of the measures is 180\u00b0<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c2568b474c5db6ab6cc65b067483c09f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#65;&#43;&#109;&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#66;&#61;&#49;&#56;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"157\" style=\"vertical-align: -2px;\" \/><\/li>\n<\/ul>\n<\/li>\n<li><span id=\"eip-id5408054\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" class=\"alignright\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_015_img.jpg\" alt=\".\" data-media-type=\"image\/jpg\" \/><\/span><strong>Right Triangle<\/strong>\n<ul id=\"eip-id1170323029556\">\n<li>A right triangle is a triangle that has one 90\u00b0 angle, which is often marked with a \u299c symbol.<\/li>\n<\/ul>\n<\/li>\n<li><strong>Properties of Similar Triangles<\/strong>\n<ul id=\"eip-id1170324076236\">\n<li>If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths have the same ratio.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h1>Glossary<\/h1>\n<div class=\"textbox shaded\">\n<dl id=\"fs-id1648881\">\n<dt>angle<\/dt>\n<dd id=\"fs-id1327370\">An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle.<\/dd>\n<\/dl>\n<dl id=\"fs-id1327374\">\n<dt>complementary angles<\/dt>\n<dd id=\"fs-id1544988\">If the sum of the measures of two angles is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-dc16a99348931a3823eff3764a961940_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0, then they are called complementary angles.<\/dd>\n<\/dl>\n<dl id=\"fs-id1170322667662\">\n<dt>hypotenuse<\/dt>\n<dd id=\"fs-id1170325427844\">The side of the triangle opposite the 90\u00b0 angle is called the hypotenuse.<\/dd>\n<\/dl>\n<dl id=\"fs-id1444791\">\n<dt>legs of a right triangle<\/dt>\n<dd id=\"fs-id1540654\">The sides of a right triangle adjacent to the right angle are called the legs.<\/dd>\n<\/dl>\n<dl id=\"fs-id1172591862696\">\n<dt>right triangle<\/dt>\n<dd id=\"fs-id1172596661577\">A right triangle is a triangle that has one <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-dc16a99348931a3823eff3764a961940_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0 angle.<\/dd>\n<\/dl>\n<dl id=\"fs-id1172591572065\">\n<dt>similar figures<\/dt>\n<dd id=\"fs-id1236955\">In geometry, if two figures have exactly the same shape but different sizes, we say they are similar figures.<\/dd>\n<\/dl>\n<dl id=\"fs-id1300144\">\n<dt>supplementary angles<\/dt>\n<dd id=\"fs-id1434532\">If the sum of the measures of two angles is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1371cfb17a414d739b40e633931613a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/>\u00b0, then they are called supplementary angles.<\/dd>\n<\/dl>\n<dl id=\"fs-id1334464\">\n<dt>triangle<\/dt>\n<dd id=\"fs-id1172599767842\">A triangle is a geometric figure with three sides and three angles.<\/dd>\n<\/dl>\n<dl id=\"fs-id1172599767847\">\n<dt>vertex of an angle<\/dt>\n<dd id=\"fs-id1402834\">When two rays meet to form an angle, the common endpoint is called the vertex of the angle.<\/dd>\n<\/dl>\n<\/div>\n<h1>Practice Makes Perfect<\/h1>\n<div id=\"fs-id1309715\" class=\"practice-perfect\" data-depth=\"2\">\n<h2 id=\"fs-id1172591855673\">Use the Properties of Angles<\/h2>\n<p id=\"eip-294\">In the following exercises, find a) the supplement and b) the complement of the given angle.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">1. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8718229001cf617c5587bb321641e740_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0<\/td>\n<td style=\"width: 50%;\">2. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d7212ca7081719f501da545167884c20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/>\u00b0<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">3. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4df3cd194a3c2b88749ea327908532a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0<\/td>\n<td style=\"width: 50%;\">4. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ea045a81487493e20568c63169e2b1d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: 0px;\" \/>\u00b0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1537917\">In the following exercises, use the properties of angles to solve.<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 56px;\">\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">5. Find the supplement of a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c8b0e53e684caf6385d1b040cbe159e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"25\" style=\"vertical-align: 0px;\" \/>\u00b0 angle.<\/td>\n<td style=\"width: 50%; height: 14px;\">6. Find the complement of a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2e5ceda806b8c209062fe8ce3df4528c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0 angle.<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">7. Find the complement of a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-22e997ba8761d3fdb658e707c17ee5be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#55;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\" \/>\u00b0 angle.<\/td>\n<td style=\"width: 50%; height: 14px;\">8. Find the supplement of a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8e344019d5a870692c07034c3703747e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#57;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"39\" style=\"vertical-align: 0px;\" \/>\u00b0 angle.<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">9. Two angles are supplementary. The larger angle is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a5e88c9e5e627cee775eddae75df9393_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0 more than the smaller angle. Find the measures of both angles.<\/td>\n<td style=\"width: 50%; height: 14px;\">10. Two angles are supplementary. The smaller angle is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-24928a945f1bd891ae4b1b2d2171c6fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0 less than the larger angle. Find the measures of both angles.<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">11. Two angles are complementary. The smaller angle is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4184edf594112c9a40fa10c2813a9bba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0 less than the larger angle. Find the measures of both angles.<\/td>\n<td style=\"width: 50%; height: 14px;\">12. Two angles are complementary. The larger angle is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d635f27535c5c26235d8673124b06d38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: 0px;\" \/>\u00b0 more than the smaller angle. Find the measures of both angles.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2 id=\"fs-id1296727\">Use the Properties of Triangles<\/h2>\n<p id=\"eip-759\">In the following exercises, solve using properties of triangles.<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 144px;\">\n<tbody>\n<tr style=\"height: 30px;\">\n<td style=\"width: 50%; height: 30px;\">13. The measures of two angles of a triangle are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a842d61007d6e20e8f05812630d602cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b9e0ff263997e56a21812b9a43879ae0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0. Find the measure of the third angle.<\/td>\n<td style=\"width: 50%; height: 30px;\">14. The measures of two angles of a triangle are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5acbaa54ca3014b500f7eefd5f7436da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"17\" style=\"vertical-align: 0px;\" \/>\u00b0 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a438d235819429cb643229cb961a7c5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0. Find the measure of the third angle.<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"width: 50%; height: 30px;\">15. The measures of two angles of a triangle are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1ea80194d8a873f318b0c1a6101c5274_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"25\" style=\"vertical-align: 0px;\" \/>\u00b0 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d43d80b38a17ad1f60ee096179357a72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"17\" style=\"vertical-align: 0px;\" \/>\u00b0. Find the measure of the third angle.<\/td>\n<td style=\"width: 50%; height: 30px;\">16. The measures of two angles of a triangle are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b4f92b8c2ec844259613bcf7b3e7d53d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ea045a81487493e20568c63169e2b1d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: 0px;\" \/>\u00b0. Find the measure of the third angle.<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">17. One angle of a right triangle measures <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3ecb939b09e29249cdd612a4c72c3791_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0. What is the measure of the other angle?<\/td>\n<td style=\"width: 50%; height: 14px;\">18. One angle of a right triangle measures <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c9d5abb00a4b243b75dfa716195c91f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: 0px;\" \/>\u00b0. What is the measure of the other angle?<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">19. One angle of a right triangle measures <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-15b2991d89c11accab930b55f9850fac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#50;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\" \/>\u00b0. What is the measure of the other angle?<\/td>\n<td style=\"width: 50%; height: 14px;\">20. One angle of a right triangle measures <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9d4b3af9422fbd469cebb4e9bf1299df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#54;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\" \/>\u00b0. What is the measure of the other angle?<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">21. The two smaller angles of a right triangle have equal measures. Find the measures of all three angles.<\/td>\n<td style=\"width: 50%; height: 14px;\">22. The measure of the smallest angle of a right triangle is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d647a5ecf47eb476cd54ba72ffd63b5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0 less than the measure of the other small angle. Find the measures of all three angles.<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">23. The angles in a triangle are such that the measure of one angle is twice the measure of the smallest angle, while the measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles.<\/td>\n<td style=\"width: 50%; height: 14px;\">24. The angles in a triangle are such that the measure of one angle is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d647a5ecf47eb476cd54ba72ffd63b5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0 more than the measure of the smallest angle, while the measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3 id=\"fs-id1548145\">Find the Length of the Missing Side<\/h3>\n<p id=\"eip-480\">In the following exercises, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-23f00feef91cb9ba3057f9e15d82ff39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#65;&#66;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/> is similar to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-bfb544ee50aa52cf4503ca937feda154_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#88;&#89;&#90;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"58\" style=\"vertical-align: 0px;\" \/>. Find the length of the indicated side.<\/p>\n<p><span id=\"fs-id1547556\" data-type=\"media\" data-alt=\"Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled 9, the side across from B is labeled b, and the side across from C is labeled 15. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled x, the side across from Y is labeled 8, and the side across from Z is labeled 10.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_201_img.jpg\" alt=\"Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled 9, the side across from B is labeled b, and the side across from C is labeled 15. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled x, the side across from Y is labeled 8, and the side across from Z is labeled 10.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<div id=\"fs-id1547228\" data-type=\"exercise\">\n<div id=\"fs-id1547230\" data-type=\"problem\">\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">25. side <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/td>\n<td style=\"width: 50%;\">26. side <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1297695\">\n<\/div>\n<\/div>\n<p id=\"eip-941\">On a map, San Francisco, Las Vegas, and Los Angeles form a triangle whose sides are shown in the figure below. The actual distance from Los Angeles to Las Vegas is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9b27ac687cb8b4a0f1c4fabda31aabf6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#55;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"27\" style=\"vertical-align: 0px;\" \/> miles.<\/p>\n<p><span id=\"fs-id1260455\" data-type=\"media\" data-alt=\"A triangle is shown. The vertices are labeled San Francisco, Las Vegas, and Los Angeles. The side across from San Francisco is labeled 1 inch, the side across from Las Vegas is labeled 1.3 inches, and the side across from Los Angeles is labeled 2.1 inches.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_202_img.jpg\" alt=\"A triangle is shown. The vertices are labeled San Francisco, Las Vegas, and Los Angeles. The side across from San Francisco is labeled 1 inch, the side across from Las Vegas is labeled 1.3 inches, and the side across from Los Angeles is labeled 2.1 inches.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">27. Find the distance from Los Angeles to San Francisco.<\/td>\n<td style=\"width: 50%;\">28. Find the distance from San Francisco to Las Vegas.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2 id=\"fs-id1413042\">Use the Pythagorean Theorem<\/h2>\n<p>In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 189px;\">\n<tbody>\n<tr style=\"height: 175px;\">\n<td style=\"width: 50%; height: 175px;\"><span id=\"fs-id1545681\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 9, the other as 12.\">29. <img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_203_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 9, the other as 12.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td style=\"width: 50%; height: 175px;\"><span id=\"fs-id1238083\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 16, the other as 12.\">30. <img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_204_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 16, the other as 12.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\"><span id=\"fs-id1543755\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 15, the other as 20.\">31. <img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_205_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 15, the other as 20.\" width=\"217\" height=\"163\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td style=\"width: 50%; height: 14px;\"><span id=\"fs-id1428325\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 5, the other as 12.\">32. <img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_206_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 5, the other as 12.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3 id=\"fs-id1263576\">Find the Length of the Missing Side<\/h3>\n<p id=\"eip-851\">In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 152px;\">\n<tbody>\n<tr style=\"height: 110px;\">\n<td style=\"width: 50%; height: 110px;\"><span id=\"fs-id1576553\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 10. One of the sides touching the right angle is labeled as 6.\">33. <img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_207_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 10. One of the sides touching the right angle is labeled as 6.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td style=\"width: 50%; height: 110px;\"><span id=\"fs-id1547852\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 8.\">34. <img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_208_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 8.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\"><span id=\"fs-id1701813\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 13. One of the sides touching the right angle is labeled as 5.\">35. <img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_209_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 13. One of the sides touching the right angle is labeled as 5.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td style=\"width: 50%; height: 14px;\"><span id=\"fs-id1439355\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 20. One of the sides touching the right angle is labeled as 16.\">36. <img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_210_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 20. One of the sides touching the right angle is labeled as 16.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\"><span id=\"fs-id1172599767679\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 13. One of the sides touching the right angle is labeled as 8.\">37. <img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_211_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 13. One of the sides touching the right angle is labeled as 8.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td style=\"width: 50%; height: 14px;\"><span id=\"fs-id1704712\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. Both of the sides touching the right angle are labeled as 6.\">38. <img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_212_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. Both of the sides touching the right angle are labeled as 6.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\"><span id=\"fs-id1614367\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 15.\">39. <img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_213_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 15.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td style=\"width: 50%; height: 14px;\"><span id=\"fs-id1537416\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 7. One of the sides touching the right angle is labeled as 5.\">40. <img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_214_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 7. One of the sides touching the right angle is labeled as 5.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1428288\">In the following exercises, solve. Approximate to the nearest tenth, if necessary.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">\n<p id=\"fs-id1538067\">41. A <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6677ef5cc558d4ef98a8afc5ea189831_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#45;&#102;&#111;&#111;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: -1px;\" \/> string of lights will be attached to the top of a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8b0c1a4a0b23172d8778a2cc6faa495d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#45;&#102;&#111;&#111;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: -1px;\" \/> pole for a holiday display. How far from the base of the pole should the end of the string of lights be anchored?<\/p>\n<p><span id=\"fs-id1279182\" data-type=\"media\" data-alt=\"A vertical pole is shown with a string of lights going from the top of the pole to the ground. The pole is labeled 12 feet. The string of lights is labeled 13 feet.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_215_img.jpg\" alt=\"A vertical pole is shown with a string of lights going from the top of the pole to the ground. The pole is labeled 12 feet. The string of lights is labeled 13 feet.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td style=\"width: 50%;\">\n<p id=\"fs-id1259700\">42. Pam wants to put a banner across her garage door to congratulate her son on his college graduation. The garage door is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-40f7428479801bf4ef27a5149e9df526_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"16\" style=\"vertical-align: -1px;\" \/> feet high and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8dde8d4cade60a4f3ca0e779512b974c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/> feet wide. How long should the banner be to fit the garage door?<\/p>\n<p><span id=\"fs-id1172597125751\" data-type=\"media\" data-alt=\"A picture of a house is shown. The rectangular garage is 12 feet high and 16 feet wide. A blue banner goes diagonally across the garage.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_03_04_214.jpg\" alt=\"A picture of a house is shown. The rectangular garage is 12 feet high and 16 feet wide. A blue banner goes diagonally across the garage.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">\n<p id=\"fs-id1543256\">43. Chi is planning to put a path of paving stones through her flower garden. The flower garden is a square with sides of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5450913cc453faf132dc56e5965ca797_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/> feet. What will the length of the path be?<\/p>\n<p><span id=\"fs-id1542270\" data-type=\"media\" data-alt=\"A square garden is shown. One side is labeled as 10 feet. There is a diagonal path of blue circular stones going from the lower left corner to the upper right corner.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_03_04_215.jpg\" alt=\"A square garden is shown. One side is labeled as 10 feet. There is a diagonal path of blue circular stones going from the lower left corner to the upper right corner.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td style=\"width: 50%;\">\n<p id=\"fs-id1332146\">44. Brian borrowed a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f72408cf9aa1bf775155d708df85eeb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#45;&#102;&#111;&#111;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"53\" style=\"vertical-align: -1px;\" \/> extension ladder to paint his house. If he sets the base of the ladder <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b5c3e12330dabaeec7413281aba0f134_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> feet from the house, how far up will the top of the ladder reach?<\/p>\n<p><span id=\"fs-id1459140\" data-type=\"media\" data-alt=\"A picture of a house is shown with a ladder leaning against it. The ladder is labeled 20 feet tall. The horizontal distance from the house to the base of the ladder is 6 feet.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_03_04_216.jpg\" alt=\"A picture of a house is shown with a ladder leaning against it. The ladder is labeled 20 feet tall. The horizontal distance from the house to the base of the ladder is 6 feet.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1420696\" class=\"everyday\" data-depth=\"2\">\n<h2 data-type=\"title\">Everyday Math<\/h2>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">45. <strong data-effect=\"bold\">Building a scale model<\/strong> Joe wants to build a doll house for his daughter. He wants the doll house to look just like his house. His house is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-874b63beceda768488ea76f12b9cf9e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/> feet wide and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6bff01d8c87d130da6950c561718ce13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: 0px;\" \/> feet tall at the highest point of the roof. If the dollhouse will be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-01948ea56b434f27f412043e785fe880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: 0px;\" \/> feet wide, how tall will its highest point be?<\/td>\n<td style=\"width: 50%;\">\n<p id=\"fs-id1172597157720\">46. <strong data-effect=\"bold\">Measurement<\/strong> A city engineer plans to build a footbridge across a lake from point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c23bf1defc91f554ffdf7fe3f4f5c835_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#88;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"13\" style=\"vertical-align: -1px;\" \/> to point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1b47852a682b1a6ec3a1901e958faadb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#89;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"14\" style=\"vertical-align: -1px;\" \/>, as shown in the picture below. To find the length of the footbridge, she draws a right triangle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d2afa579648fdd9becb1393ccf2182c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#88;&#89;&#90;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"36\" style=\"vertical-align: 0px;\" \/>, with right angle at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c23bf1defc91f554ffdf7fe3f4f5c835_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#88;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"13\" style=\"vertical-align: -1px;\" \/>. She measures the distance from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c23bf1defc91f554ffdf7fe3f4f5c835_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#88;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"13\" style=\"vertical-align: -1px;\" \/> to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9e244f3b78ae87434c1a15c17c9c2bf8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#90;&#125;&#44;&#56;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/> feet, and from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1b47852a682b1a6ec3a1901e958faadb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#89;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"14\" style=\"vertical-align: -1px;\" \/> to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-406077c69eda671e1406554019ae8e64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#90;&#125;&#44;&#49;&#44;&#48;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"62\" style=\"vertical-align: -4px;\" \/> feet. How long will the bridge be?<\/p>\n<p><span id=\"fs-id1545751\" data-type=\"media\" data-alt=\"A lake is shown. Point Y is on one side of the lake, directly across from point X. Point Z is on the same side of the lake as point X.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_219_img.jpg\" alt=\"A lake is shown. Point Y is on one side of the lake, directly across from point X. Point Z is on the same side of the lake as point X.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1580956\" class=\"writing\" data-depth=\"2\">\n<h2 data-type=\"title\">Writing Exercises<\/h2>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">47. Write three of the properties of triangles from this section and then explain each in your own words.<\/td>\n<td style=\"width: 50%;\">\n<p id=\"fs-id1458581\">48. Explain how the figure below illustrates the Pythagorean Theorem for a triangle with legs of length <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4a1d3ea4963f568cabd97329456036b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6111a899fd636b7a5238708f8679f6ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<p><span id=\"fs-id1399992\" data-type=\"media\" data-alt=\"Three squares are shown, forming a right triangle in the center. Each square is divided into smaller squares. The smallest square is divided into 9 small squares. The medium square is divided into 16 small squares. The large square is divided into 25 small squares.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_09_03_220_img.jpg\" alt=\"Three squares are shown, forming a right triangle in the center. Each square is divided into smaller squares. The smallest square is divided into 9 small squares. The medium square is divided into 16 small squares. The large square is divided into 25 small squares.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<h1>Answers<\/h1>\n<table style=\"border-collapse: collapse; width: 100%; height: 257px;\">\n<tbody>\n<tr style=\"height: 104px;\">\n<td style=\"width: 33.3333%; height: 104px;\">1.<\/p>\n<p>a) 127\u00b0<\/p>\n<p>b) 37\u00b0<\/td>\n<td style=\"width: 33.3333%; height: 104px;\">3.<\/p>\n<div id=\"fs-id1393950\" data-type=\"exercise\">\n<div id=\"fs-id1415688\" data-type=\"solution\">\n<p>a) 151\u00b0<\/p>\n<p>b) 61\u00b0<\/p>\n<\/div>\n<\/div>\n<\/td>\n<td style=\"width: 33.3333%; height: 104px;\">5. 45\u00b0<\/td>\n<\/tr>\n<tr style=\"height: 17px;\">\n<td style=\"width: 33.3333%; height: 17px;\">7. 62.5\u00b0<\/td>\n<td style=\"width: 33.3333%; height: 17px;\">9. 62\u00b0, 118\u00b0<\/td>\n<td style=\"width: 33.3333%; height: 17px;\">11. 62\u00b0, 28\u00b0<\/td>\n<\/tr>\n<tr style=\"height: 17px;\">\n<td style=\"width: 33.3333%; height: 17px;\">13. 56\u00b0<\/td>\n<td style=\"width: 33.3333%; height: 17px;\">15. 44\u00b0<\/td>\n<td style=\"width: 33.3333%; height: 17px;\">17. 57\u00b0<\/td>\n<\/tr>\n<tr style=\"height: 17px;\">\n<td style=\"width: 33.3333%; height: 17px;\">19. 67.5\u00b0<\/td>\n<td style=\"width: 33.3333%; height: 17px;\">21. 45\u00b0, 45\u00b0, 90\u00b0<\/td>\n<td style=\"width: 33.3333%; height: 17px;\">23. 30\u00b0, 60\u00b0, 90\u00b0<\/td>\n<\/tr>\n<tr style=\"height: 17px;\">\n<td style=\"width: 33.3333%; height: 17px;\">25. 12<\/td>\n<td style=\"width: 33.3333%; height: 17px;\">27. 351 miles<\/td>\n<td style=\"width: 33.3333%; height: 17px;\">29. 15<\/td>\n<\/tr>\n<tr style=\"height: 17px;\">\n<td style=\"width: 33.3333%; height: 17px;\">31. 25<\/td>\n<td style=\"width: 33.3333%; height: 17px;\">33. 8<\/td>\n<td style=\"width: 33.3333%; height: 17px;\">35. 12<\/td>\n<\/tr>\n<tr style=\"height: 17px;\">\n<td style=\"width: 33.3333%; height: 17px;\">37. 10.2<\/td>\n<td style=\"width: 33.3333%; height: 17px;\">39. 8<\/td>\n<td style=\"width: 33.3333%; height: 17px;\">41. 5 feet<\/td>\n<\/tr>\n<tr style=\"height: 17px;\">\n<td style=\"width: 33.3333%; height: 17px;\">43. 14.1 feet<\/td>\n<td style=\"width: 33.3333%; height: 17px;\">45. 2.9 feet<\/td>\n<td style=\"width: 33.3333%; height: 17px;\">47.\u00a0Answers will vary.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h1>Attributions<\/h1>\n<p>This chapter has been adapted from \u201cUse Properties of Angles, Triangles, and the Pythagorean Theorem\u201d in <a href=\"https:\/\/openstax.org\/details\/books\/prealgebra-2e\"><em>Prealgebra<\/em><\/a> (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a <a href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY 4.0 Licence<\/a>. Adapted by Izabela Mazur. See the Copyright page for more information.<\/p>\n","protected":false},"author":90,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2400","chapter","type-chapter","status-publish","hentry"],"part":2272,"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/pressbooks\/v2\/chapters\/2400","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/wp\/v2\/users\/90"}],"version-history":[{"count":8,"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/pressbooks\/v2\/chapters\/2400\/revisions"}],"predecessor-version":[{"id":2529,"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/pressbooks\/v2\/chapters\/2400\/revisions\/2529"}],"part":[{"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/pressbooks\/v2\/parts\/2272"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/pressbooks\/v2\/chapters\/2400\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/wp\/v2\/media?parent=2400"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2400"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/wp\/v2\/contributor?post=2400"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/wp\/v2\/license?post=2400"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}