{"id":10751,"date":"2017-06-05T15:46:15","date_gmt":"2017-06-05T15:46:15","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=10751"},"modified":"2024-04-30T16:46:21","modified_gmt":"2024-04-30T16:46:21","slug":"using-the-properties-of-triangles-to-solve-problems","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/using-the-properties-of-triangles-to-solve-problems\/","title":{"raw":"Using the Properties of Triangles to Solve Problems","rendered":"Using the Properties of Triangles to Solve Problems"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Given the measures of two angles of a triangle, find the third<\/li>\r\n \t<li>Use properties of similar triangles to find unknown side lengths<\/li>\r\n<\/ul>\r\n<\/div>\r\nWhat do you already know about triangles? Triangle have three sides and three angles. Triangles are named by their vertices. The triangle below\u00a0is called [latex]\\Delta ABC[\/latex], read \u2018triangle [latex]\\text{ABC}[\/latex] \u2019. We label each side with a lower case letter to match the upper case letter of the opposite vertex.\r\n\r\n[latex]\\Delta ABC[\/latex] has vertices [latex]A,B,\\text{ and }C[\/latex] and sides [latex]a,b,\\text{ and }c\\text{.}[\/latex]\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223629\/CNX_BMath_Figure_09_03_011.png\" alt=\"The vertices of the triangle on the left are labeled A, B, and C. The sides are labeled a, b, and c.\" \/>\r\nThe three angles of a triangle are related in a special way. The sum of their measures is [latex]\\text{180}^ \\circ[\/latex].\r\n\r\n[latex]m\\angle A+m\\angle B+m\\angle C=\\text{180}^ \\circ[\/latex]\r\n<div class=\"textbox shaded\">\r\n<h3>Sum of the Measures of the Angles of a Triangle<\/h3>\r\nFor any [latex]\\Delta ABC[\/latex], the sum of the measures of the angles is [latex]\\text{180}^ \\circ[\/latex].\r\n<p style=\"text-align: center;\">[latex]m\\angle A+m\\angle B+m\\angle C=\\text{180}^ \\circ[\/latex]<\/p>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nThe measures of two angles of a triangle are [latex]55^\\circ [\/latex] and [latex]82^\\circ [\/latex]. Find the measure of the third angle.\r\n\r\nSolution\r\n<table id=\"eip-id1168467289823\" class=\"unnumbered unstyled\" summary=\"Step 1 says, \">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223631\/CNX_BMath_Figure_09_03_050_img-01.png\" alt=\"A triangle of the following vertex, angle pairs: A is 82 degrees, B is 55 degrees, and C is x degrees.\" width=\"365\" height=\"192\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>The measure of the third angle in the triangle.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td>Let [latex]x=[\/latex]the measure of the angle.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong>Translate.<\/strong>\r\n\r\nWrite the appropriate formula and substitute.<\/td>\r\n<td>[latex]m\\angle A+m\\angle B+m\\angle C=180[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]55\u00b0+82\u00b0+x=180\u00b0[\/latex]\r\n\r\n[latex]137\u00b0+x=180\u00b0[\/latex]\r\n\r\n[latex]x=43\u00b0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong>\r\n\r\n[latex]55\u00b0+82\u00b0+43\u00b0\\stackrel{?}{=}180\u00b0[\/latex]\r\n\r\n[latex]180\u00b0=180\u00b0\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>The measure of the third angle is [latex]43\u00b0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146498[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we show an example of how to find the measure of an unknown angle in a triangle. In this example, we have two triangles who share a common side, and find two unknown interior angles.\r\n\r\nhttps:\/\/youtu.be\/3kRLkbU6-cI\r\n<h2>Right Triangles<\/h2>\r\nSome triangles have special names. We will look first at the right triangle. A right triangle has one [latex]90^\\circ[\/latex] angle, which is often marked with the symbol shown in the triangle below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223646\/CNX_BMath_Figure_09_03_015_img.png\" alt=\"A right triangle is shown. The right angle is marked with a box and labeled 90 degrees.\" \/>\r\nIf we know that a triangle is a right triangle, we know that one angle measures [latex]90^\\circ[\/latex] so we only need the measure of one of the other angles in order to determine the measure of the third angle.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nOne angle of a right triangle measures [latex]28^\\circ [\/latex]. What is the measure of the third angle?\r\n[reveal-answer q=\"738782\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"738782\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468725720\" class=\"unnumbered unstyled\" summary=\"Step 1 says, \">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223647\/CNX_BMath_Figure_09_03_051_img-01.png\" alt=\"A triangle of the following vertex, angle pairs: C is x degrees. A is 90 degrees. B is 28 degrees.\" width=\"272\" height=\"167\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>The measure of an angle.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td>Let [latex]x=[\/latex]the measure of the angle.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong>Translate.<\/strong>\r\n\r\nWrite the appropriate formula and substitute.<\/td>\r\n<td>[latex]m\\angle A+m\\angle B+m\\angle C=180\u00b0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]x+90\u00b0+28\u00b0=180\u00b0[\/latex]\r\n\r\n[latex]x+118\u00b0=180\u00b0[\/latex]\r\n\r\n[latex]x=62\u00b0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong>\r\n\r\n[latex]180\u00b0\\stackrel{?}{=}90\u00b0+28\u00b0+62\u00b0[\/latex]\r\n\r\n[latex]180\u00b0=180\u00b0\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>The measure of the third angle is [latex]62\u00b0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146499[\/ohm_question]\r\n\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<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nThe measure of one angle of a right triangle is [latex]20^\\circ [\/latex] more than the measure of the smallest angle. Find the measures of all three angles.\r\n[reveal-answer q=\"255227\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"255227\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466077134\" class=\"unnumbered unstyled\" summary=\"Step 1 says, \">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong>Read<\/strong> the problem.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong>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>Step 3. <strong>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>Let [latex]a={1}^{st}[\/latex] angle.\r\n\r\n[latex]a+20={2}^{nd}[\/latex] angle.\r\n\r\n[latex]90={3}^{rd}[\/latex] angle (the right angle).\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223706\/CNX_BMath_Figure_09_03_052_img-04.png\" alt=\"A triangle of the following vertex, angle pairs: C is 90 degrees. A is a degrees. B is a plus 20 degrees.\" width=\"212\" height=\"197\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong>Translate.<\/strong>\r\n\r\nWrite the appropriate formula and substitute into the formula.<\/td>\r\n<td>[latex]m\\angle A+m\\angle B+m\\angle C=180[\/latex]\r\n\r\n[latex]a+(a+20)+90=180[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]2a+110=180[\/latex]\r\n\r\n[latex]2a=70[\/latex]\r\n\r\n[latex]a=35[\/latex] first angle\r\n\r\n[latex]a+20[\/latex] second angle\r\n\r\n[latex]\\color{red}{35}+20[\/latex]\r\n\r\n[latex]55[\/latex]\r\n\r\n[latex]90[\/latex] third angle.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong>\r\n\r\n[latex]35\u00b0+55\u00b0+90\u00b0\\stackrel{?}{=}180\u00b0[\/latex]\r\n\r\n[latex]180\u00b0=180\u00b0\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>The three angles measure [latex]35\u00b0, 55\u00b0, 90\u00b0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146500[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Similar Triangles<\/h2>\r\nWhen 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.\r\n\r\nThe two triangles below\u00a0are similar. Each side of [latex]\\Delta ABC[\/latex] is four times the length of the corresponding side of [latex]\\Delta XYZ[\/latex] and their corresponding angles have equal measures.\r\n\r\n[latex]\\Delta ABC[\/latex] and [latex]\\Delta XYZ[\/latex] are similar triangles. Their corresponding sides have the same ratio and the corresponding angles have the same measure.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223723\/CNX_BMath_Figure_09_03_020.png\" 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.\" \/>\r\n<div class=\"textbox shaded\">\r\n<h3>Properties of Similar Triangles<\/h3>\r\nIf two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths are in the same ratio.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223725\/CNX_BMath_Figure_09_03_056_img-1.png\" alt=\"Two similar triangles adjacent to eachother. The first triangle has vertices A, B, and C with side lengths a, b, and c. Side lengths correspond to the side opposite the vertex of the same letter. Similarly, a smaller triangle to the right has vertices X, Y, and Z, and sides x, y, and z. The same can be said for the relationship between side and vertex names. Statements on the right express the congruency of angles A and X, B and Y, and C and Z. The ratio of side length a to side length x is equal to the ratio of side length b to side length y which is equal to the ratio of side length c to side length z.\" width=\"628\" height=\"176\" \/>\r\n\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 [latex]\\Delta ABC\\text{:}[\/latex]\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\text{the length }a\\text{ can also be written }BC\\hfill \\\\ \\text{the length }b\\text{ can also be written }AC\\hfill \\\\ \\text{ the length }c\\text{ can also be written }AB\\hfill \\end{array}[\/latex]<\/p>\r\nWe will often use this notation when we solve similar triangles because it will help us match up the corresponding side lengths.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\n[latex]\\Delta ABC[\/latex] and [latex]\\Delta XYZ[\/latex] are similar triangles. The lengths of two sides of each triangle are shown. Find the lengths of the third side of each triangle.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223727\/CNX_BMath_Figure_09_03_022.png\" 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.\" \/>\r\n[reveal-answer q=\"584050\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"584050\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466690156\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n<td>The figure is provided.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>The length of the sides of similar triangles<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td>Let\r\n\r\n[latex]a[\/latex] = length of the third side of [latex]\\Delta ABC[\/latex]\r\n\r\n[latex]y[\/latex] = length of the third side of [latex]\\Delta XYZ[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Step 4. <strong>Translate.<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">The triangles are similar, so the corresponding sides are in the same ratio. So\r\n\r\n[latex]{\\Large\\frac{AB}{XY}}={\\Large\\frac{BC}{YZ}}={\\Large\\frac{AC}{XZ}}[\/latex]\r\n\r\nSince the side [latex]AB=4[\/latex] corresponds to the side [latex]XY=3[\/latex] , we will use the ratio [latex]{\\Large\\frac{\\mathrm{AB}}{\\mathrm{XY}}}={\\Large\\frac{4}{3}}[\/latex] to find the other sides.\r\n\r\nBe careful to match up corresponding sides correctly.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223728\/CNX_BMath_Figure_09_03_057_img-01.png\" alt=\"Two adjacent solutions for finding side length a and side length y. The sides of the large triangle will be in the numerator and the sides of the smaller triangle will be in the denominator. To find side length a: angles A B divided by angles X Y equals angles B C divided by angles Y Z. Plugging in values, 4 thirds equals side length a divided by 4.5. To find side length y: angles A B divided by angles X Y equals angles A C divided by angles X Z. Plugging in values, 4 thirds equals 3.2 divided by side length y.\" width=\"565\" height=\"122\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223730\/CNX_BMath_Figure_09_03_057_img-02.png\" alt=\"The solution for side lengths a and y. 3 a equals 4 times 4.5, which simplifies to 3 a equals 18. a is 6. 4 y equals 3 times 3.2, which simplifies to 4 y equals 9.6. y equals 2.4\" width=\"269\" height=\"78\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong>\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223731\/CNX_BMath_Figure_09_03_057_img-03.png\" alt=\"Two checking sequences for the solutions for side length a and y. Does 4 thirds equal 6 divided by 4.5? Simplified, does 4 times 4.5 equal 6 times 3? Both sides equal 18. The check passes. Does 4 thirds equal 3.2 divided by 2.4? Simplified, does 4 times 2.4 equal 3.2 times 3? Both sides equal 9.6. The check passes.\" width=\"258\" height=\"111\" \/><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>The third side of [latex]\\Delta ABC[\/latex] is [latex]6[\/latex] and the third side of [latex]\\Delta XYZ[\/latex] is [latex]2.4[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146912[\/ohm_question]\r\n\r\n<\/div>\r\n<span style=\"color: #000000;\">In the video below we show an example of how to find the missing sides of two triangles that are similar. \u00a0Note that the measures of the sides in this example are whole numbers, and we use a cross product to solve the resulting proportions.<\/span>\r\n\r\nhttps:\/\/youtu.be\/FbtCUXgVA3A","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Given the measures of two angles of a triangle, find the third<\/li>\n<li>Use properties of similar triangles to find unknown side lengths<\/li>\n<\/ul>\n<\/div>\n<p>What do you already know about triangles? Triangle have three sides and three angles. Triangles are named by their vertices. The triangle below\u00a0is called [latex]\\Delta ABC[\/latex], read \u2018triangle [latex]\\text{ABC}[\/latex] \u2019. We label each side with a lower case letter to match the upper case letter of the opposite vertex.<\/p>\n<p>[latex]\\Delta ABC[\/latex] has vertices [latex]A,B,\\text{ and }C[\/latex] and sides [latex]a,b,\\text{ and }c\\text{.}[\/latex]<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223629\/CNX_BMath_Figure_09_03_011.png\" alt=\"The vertices of the triangle on the left are labeled A, B, and C. The sides are labeled a, b, and c.\" \/><br \/>\nThe three angles of a triangle are related in a special way. The sum of their measures is [latex]\\text{180}^ \\circ[\/latex].<\/p>\n<p>[latex]m\\angle A+m\\angle B+m\\angle C=\\text{180}^ \\circ[\/latex]<\/p>\n<div class=\"textbox shaded\">\n<h3>Sum of the Measures of the Angles of a Triangle<\/h3>\n<p>For any [latex]\\Delta ABC[\/latex], the sum of the measures of the angles is [latex]\\text{180}^ \\circ[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]m\\angle A+m\\angle B+m\\angle C=\\text{180}^ \\circ[\/latex]<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>The measures of two angles of a triangle are [latex]55^\\circ[\/latex] and [latex]82^\\circ[\/latex]. Find the measure of the third angle.<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168467289823\" class=\"unnumbered unstyled\" summary=\"Step 1 says,\">\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223631\/CNX_BMath_Figure_09_03_050_img-01.png\" alt=\"A triangle of the following vertex, angle pairs: A is 82 degrees, B is 55 degrees, and C is x degrees.\" width=\"365\" height=\"192\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>The measure of the third angle in the triangle.<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let [latex]x=[\/latex]the measure of the angle.<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula and substitute.<\/td>\n<td>[latex]m\\angle A+m\\angle B+m\\angle C=180[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]55\u00b0+82\u00b0+x=180\u00b0[\/latex]<\/p>\n<p>[latex]137\u00b0+x=180\u00b0[\/latex]<\/p>\n<p>[latex]x=43\u00b0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong><\/p>\n<p>[latex]55\u00b0+82\u00b0+43\u00b0\\stackrel{?}{=}180\u00b0[\/latex]<\/p>\n<p>[latex]180\u00b0=180\u00b0\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The measure of the third angle is [latex]43\u00b0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146498\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146498&theme=oea&iframe_resize_id=ohm146498&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show an example of how to find the measure of an unknown angle in a triangle. In this example, we have two triangles who share a common side, and find two unknown interior angles.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 2B: Find the Measure of an Interior Angle of a Triangle\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/3kRLkbU6-cI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Right Triangles<\/h2>\n<p>Some triangles have special names. We will look first at the right triangle. A right triangle has one [latex]90^\\circ[\/latex] angle, which is often marked with the symbol shown in the triangle below.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223646\/CNX_BMath_Figure_09_03_015_img.png\" alt=\"A right triangle is shown. The right angle is marked with a box and labeled 90 degrees.\" \/><br \/>\nIf we know that a triangle is a right triangle, we know that one angle measures [latex]90^\\circ[\/latex] 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 exercises\">\n<h3>example<\/h3>\n<p>One angle of a right triangle measures [latex]28^\\circ[\/latex]. What is the measure of the third angle?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q738782\">Show Solution<\/span><\/p>\n<div id=\"q738782\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468725720\" class=\"unnumbered unstyled\" summary=\"Step 1 says,\">\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223647\/CNX_BMath_Figure_09_03_051_img-01.png\" alt=\"A triangle of the following vertex, angle pairs: C is x degrees. A is 90 degrees. B is 28 degrees.\" width=\"272\" height=\"167\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>The measure of an angle.<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let [latex]x=[\/latex]the measure of the angle.<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula and substitute.<\/td>\n<td>[latex]m\\angle A+m\\angle B+m\\angle C=180\u00b0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]x+90\u00b0+28\u00b0=180\u00b0[\/latex]<\/p>\n<p>[latex]x+118\u00b0=180\u00b0[\/latex]<\/p>\n<p>[latex]x=62\u00b0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong><\/p>\n<p>[latex]180\u00b0\\stackrel{?}{=}90\u00b0+28\u00b0+62\u00b0[\/latex]<\/p>\n<p>[latex]180\u00b0=180\u00b0\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The measure of the third angle is [latex]62\u00b0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146499\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146499&theme=oea&iframe_resize_id=ohm146499&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\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 class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>The measure of one angle of a right triangle is [latex]20^\\circ[\/latex] more than the measure of the smallest angle. Find the measures of all three angles.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q255227\">Show Solution<\/span><\/p>\n<div id=\"q255227\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466077134\" class=\"unnumbered unstyled\" summary=\"Step 1 says,\">\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the measures of all three angles<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>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>Let [latex]a={1}^{st}[\/latex] angle.<\/p>\n<p>[latex]a+20={2}^{nd}[\/latex] angle.<\/p>\n<p>[latex]90={3}^{rd}[\/latex] angle (the right angle).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223706\/CNX_BMath_Figure_09_03_052_img-04.png\" alt=\"A triangle of the following vertex, angle pairs: C is 90 degrees. A is a degrees. B is a plus 20 degrees.\" width=\"212\" height=\"197\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula and substitute into the formula.<\/td>\n<td>[latex]m\\angle A+m\\angle B+m\\angle C=180[\/latex]<\/p>\n<p>[latex]a+(a+20)+90=180[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]2a+110=180[\/latex]<\/p>\n<p>[latex]2a=70[\/latex]<\/p>\n<p>[latex]a=35[\/latex] first angle<\/p>\n<p>[latex]a+20[\/latex] second angle<\/p>\n<p>[latex]\\color{red}{35}+20[\/latex]<\/p>\n<p>[latex]55[\/latex]<\/p>\n<p>[latex]90[\/latex] third angle.<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong><\/p>\n<p>[latex]35\u00b0+55\u00b0+90\u00b0\\stackrel{?}{=}180\u00b0[\/latex]<\/p>\n<p>[latex]180\u00b0=180\u00b0\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The three angles measure [latex]35\u00b0, 55\u00b0, 90\u00b0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146500\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146500&theme=oea&iframe_resize_id=ohm146500&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Similar Triangles<\/h2>\n<p>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>The two triangles below\u00a0are similar. Each side of [latex]\\Delta ABC[\/latex] is four times the length of the corresponding side of [latex]\\Delta XYZ[\/latex] and their corresponding angles have equal measures.<\/p>\n<p>[latex]\\Delta ABC[\/latex] and [latex]\\Delta XYZ[\/latex] are similar triangles. Their corresponding sides have the same ratio and the corresponding angles have the same measure.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223723\/CNX_BMath_Figure_09_03_020.png\" 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.\" \/><\/p>\n<div class=\"textbox shaded\">\n<h3>Properties of Similar Triangles<\/h3>\n<p>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><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223725\/CNX_BMath_Figure_09_03_056_img-1.png\" alt=\"Two similar triangles adjacent to eachother. The first triangle has vertices A, B, and C with side lengths a, b, and c. Side lengths correspond to the side opposite the vertex of the same letter. Similarly, a smaller triangle to the right has vertices X, Y, and Z, and sides x, y, and z. The same can be said for the relationship between side and vertex names. Statements on the right express the congruency of angles A and X, B and Y, and C and Z. The ratio of side length a to side length x is equal to the ratio of side length b to side length y which is equal to the ratio of side length c to side length z.\" width=\"628\" height=\"176\" \/><\/p>\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 [latex]\\Delta ABC\\text{:}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\text{the length }a\\text{ can also be written }BC\\hfill \\\\ \\text{the length }b\\text{ can also be written }AC\\hfill \\\\ \\text{ the length }c\\text{ can also be written }AB\\hfill \\end{array}[\/latex]<\/p>\n<p>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 exercises\">\n<h3>example<\/h3>\n<p>[latex]\\Delta ABC[\/latex] and [latex]\\Delta XYZ[\/latex] 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><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223727\/CNX_BMath_Figure_09_03_022.png\" 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.\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q584050\">Show Solution<\/span><\/p>\n<div id=\"q584050\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466690156\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td>The figure is provided.<\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>The length of the sides of similar triangles<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let<\/p>\n<p>[latex]a[\/latex] = length of the third side of [latex]\\Delta ABC[\/latex]<\/p>\n<p>[latex]y[\/latex] = length of the third side of [latex]\\Delta XYZ[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Step 4. <strong>Translate.<\/strong><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">The triangles are similar, so the corresponding sides are in the same ratio. So<\/p>\n<p>[latex]{\\Large\\frac{AB}{XY}}={\\Large\\frac{BC}{YZ}}={\\Large\\frac{AC}{XZ}}[\/latex]<\/p>\n<p>Since the side [latex]AB=4[\/latex] corresponds to the side [latex]XY=3[\/latex] , we will use the ratio [latex]{\\Large\\frac{\\mathrm{AB}}{\\mathrm{XY}}}={\\Large\\frac{4}{3}}[\/latex] to find the other sides.<\/p>\n<p>Be careful to match up corresponding sides correctly.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223728\/CNX_BMath_Figure_09_03_057_img-01.png\" alt=\"Two adjacent solutions for finding side length a and side length y. The sides of the large triangle will be in the numerator and the sides of the smaller triangle will be in the denominator. To find side length a: angles A B divided by angles X Y equals angles B C divided by angles Y Z. Plugging in values, 4 thirds equals side length a divided by 4.5. To find side length y: angles A B divided by angles X Y equals angles A C divided by angles X Z. Plugging in values, 4 thirds equals 3.2 divided by side length y.\" width=\"565\" height=\"122\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223730\/CNX_BMath_Figure_09_03_057_img-02.png\" alt=\"The solution for side lengths a and y. 3 a equals 4 times 4.5, which simplifies to 3 a equals 18. a is 6. 4 y equals 3 times 3.2, which simplifies to 4 y equals 9.6. y equals 2.4\" width=\"269\" height=\"78\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223731\/CNX_BMath_Figure_09_03_057_img-03.png\" alt=\"Two checking sequences for the solutions for side length a and y. Does 4 thirds equal 6 divided by 4.5? Simplified, does 4 times 4.5 equal 6 times 3? Both sides equal 18. The check passes. Does 4 thirds equal 3.2 divided by 2.4? Simplified, does 4 times 2.4 equal 3.2 times 3? Both sides equal 9.6. The check passes.\" width=\"258\" height=\"111\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The third side of [latex]\\Delta ABC[\/latex] is [latex]6[\/latex] and the third side of [latex]\\Delta XYZ[\/latex] is [latex]2.4[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146912\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146912&theme=oea&iframe_resize_id=ohm146912&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><span style=\"color: #000000;\">In the video below we show an example of how to find the missing sides of two triangles that are similar. \u00a0Note that the measures of the sides in this example are whole numbers, and we use a cross product to solve the resulting proportions.<\/span><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex 1: Find the Length of a Side of a Triangle Using Similar Triangles\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/FbtCUXgVA3A?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-10751\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Question ID 146912, 146498, 146499, 146500. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex 2B: Find the Measure of an Interior Angle of a Triangle. <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/3kRLkbU6-cI\">https:\/\/youtu.be\/3kRLkbU6-cI<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"original\",\"description\":\"Question ID 146912, 146498, 146499, 146500\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 2B: Find the Measure of an Interior Angle of a Triangle\",\"author\":\"James Sousa (mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/3kRLkbU6-cI\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"0abe40ae7c7b4d9bb490a70e0ab41de2, 70cd612a307249cca2e3fdb5e8679b75, 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