{"id":2981,"date":"2023-05-17T18:18:07","date_gmt":"2023-05-17T18:18:07","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=2981"},"modified":"2025-08-26T03:24:33","modified_gmt":"2025-08-26T03:24:33","slug":"triangles-learn-it-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/triangles-learn-it-2\/","title":{"raw":"Triangles: Learn It 2","rendered":"Triangles: Learn It 2"},"content":{"raw":"<h2>Finding the Measure of the Third Angle in a Triangle<\/h2>\r\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>\r\n<center>[latex]\\Delta ABC[\/latex] has vertices [latex]A,B,\\text{ and }C[\/latex] and sides [latex]a,b,\\text{ and }c\\text{.}[\/latex]<\/center>\r\n<p>&nbsp;<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"215\"]<img 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 capital A, capital B, and capital C. The sides are labeled lowercase a, lowercase b, and lowercase c.\" width=\"215\" height=\"148\" \/> Figure 1. This triangle is labeled with its vertices and sides[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>The three angles of a triangle are related in a special way. The sum of their measures is [latex]\\text{180}^ \\circ[\/latex].<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>sum of the measures of the angles of a triangle<\/h3>\r\n\r\nFor any [latex]\\Delta ABC[\/latex], the sum of the measures of the angles is [latex]\\text{180}^ \\circ[\/latex].\r\n\r\n<p>&nbsp;<\/p>\r\n<p style=\"text-align: center;\">[latex]m\\angle A+m\\angle B+m\\angle C=\\text{180}^ \\circ[\/latex]<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox example\">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. [reveal-answer q=\"57883\"]Show Solution[\/reveal-answer] [hidden-answer a=\"57883\"]\r\n\r\n<table id=\"eip-id1168467289823\" class=\"unnumbered unstyled alignleft\" 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><center><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 with vertex A labeled 82 degrees, vertex B labeled 55 degrees, and vertex C labeled x\" width=\"365\" height=\"192\" \/><\/center>\r\n<p>&nbsp;<\/p>\r\n<\/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> Write 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+82+x=180[\/latex] [latex]137+x=180[\/latex] [latex]x=43[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check.<\/strong><\/td>\r\n<td>[latex]55+82+43\\stackrel{?}{=}180[\/latex] [latex]180=180\\checkmark[\/latex]<\/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\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]7023[\/ohm2_question]<\/section>\r\n<h3>Right Triangles<\/h3>\r\n<p>Sometimes you will be asked to find the unknown angle in a right triangle. As we just learned, a right triangle has one [latex]90^\\circ[\/latex] angle, which is often marked with the symbol shown in the triangle below.<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"109\"]<img 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.\" width=\"109\" height=\"86\" \/> Figure 2. The angle marks a right triangle[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>If 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>\r\n<section class=\"textbox example\">One angle of a right triangle measures [latex]28^\\circ [\/latex]. What is the measure of the third angle? [reveal-answer q=\"738782\"]Show Solution[\/reveal-answer] [hidden-answer a=\"738782\"]\r\n\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 right triangle with vertex A labeled 90 degrees, vertex B labeled 28 degrees, and vertex C labeled x.\" 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> Write 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]x+90+28=180[\/latex] [latex]x+118=180[\/latex] [latex]x=62[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check.<\/strong><\/td>\r\n<td>[latex]180\\stackrel{?}{=}90+28+62[\/latex] [latex]180=180\\checkmark[\/latex]<\/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\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]7024[\/ohm2_question]<\/section>\r\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>\r\n<section class=\"textbox example\">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. [reveal-answer q=\"255227\"]Show Solution[\/reveal-answer] [hidden-answer a=\"255227\"]\r\n\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>&nbsp;<\/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. Now draw the figure and label it with the given information.<\/td>\r\n<td>Let [latex]a={1}^{st}[\/latex] angle. [latex]a+20={2}^{nd}[\/latex] angle. [latex]90={3}^{rd}[\/latex] angle (the right angle).<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 right triangle with vertex A labeled a, vertex B labeled a + 20, and vertex C labeled 90 degrees.\" width=\"212\" height=\"197\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong>Translate.<\/strong> Write the appropriate formula and substitute into the formula.<\/td>\r\n<td>[latex]m\\angle A+m\\angle B+m\\angle C=180[\/latex] [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] [latex]2a=70[\/latex] [latex]a=35[\/latex] first angle [latex]a+20[\/latex] second angle [latex]\\color{red}{35}+20[\/latex] [latex]55[\/latex] second angle [latex]90[\/latex] third angle.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check.<\/strong><\/td>\r\n<td>[latex]35+55+90\\stackrel{?}{=}180[\/latex] [latex]180=180\\checkmark[\/latex]<\/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\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]7025[\/ohm2_question]<\/section>","rendered":"<h2>Finding the Measure of the Third Angle in a Triangle<\/h2>\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<div style=\"text-align: center;\">[latex]\\Delta ABC[\/latex] has vertices [latex]A,B,\\text{ and }C[\/latex] and sides [latex]a,b,\\text{ and }c\\text{.}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 215px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" 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 capital A, capital B, and capital C. The sides are labeled lowercase a, lowercase b, and lowercase c.\" width=\"215\" height=\"148\" \/><figcaption class=\"wp-caption-text\">Figure 1. This triangle is labeled with its vertices and sides<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The three angles of a triangle are related in a special way. The sum of their measures is [latex]\\text{180}^ \\circ[\/latex].<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\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>&nbsp;<\/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<\/section>\n<section class=\"textbox example\">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<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q57883\">Show Solution<\/button> <\/p>\n<div id=\"q57883\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168467289823\" class=\"unnumbered unstyled alignleft\" 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>\n<div style=\"text-align: center;\"><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 with vertex A labeled 82 degrees, vertex B labeled 55 degrees, and vertex C labeled x\" width=\"365\" height=\"192\" \/><\/div>\n<p>&nbsp;<\/p>\n<\/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> 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+82+x=180[\/latex] [latex]137+x=180[\/latex] [latex]x=43[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check.<\/strong><\/td>\n<td>[latex]55+82+43\\stackrel{?}{=}180[\/latex] [latex]180=180\\checkmark[\/latex]<\/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<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm7023\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=7023&theme=lumen&iframe_resize_id=ohm7023&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>Right Triangles<\/h3>\n<p>Sometimes you will be asked to find the unknown angle in a right triangle. As we just learned, a right triangle has one [latex]90^\\circ[\/latex] angle, which is often marked with the symbol shown in the triangle below.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 109px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" 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.\" width=\"109\" height=\"86\" \/><figcaption class=\"wp-caption-text\">Figure 2. The angle marks a right triangle<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>If 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<section class=\"textbox example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q738782\">Show Solution<\/button> <\/p>\n<div id=\"q738782\" class=\"hidden-answer\" style=\"display: none\">\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 right triangle with vertex A labeled 90 degrees, vertex B labeled 28 degrees, and vertex C labeled x.\" 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> 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]x+90+28=180[\/latex] [latex]x+118=180[\/latex] [latex]x=62[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check.<\/strong><\/td>\n<td>[latex]180\\stackrel{?}{=}90+28+62[\/latex] [latex]180=180\\checkmark[\/latex]<\/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<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm7024\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=7024&theme=lumen&iframe_resize_id=ohm7024&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\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<section class=\"textbox example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q255227\">Show Solution<\/button> <\/p>\n<div id=\"q255227\" class=\"hidden-answer\" style=\"display: none\">\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>&nbsp;<\/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. Now draw the figure and label it with the given information.<\/td>\n<td>Let [latex]a={1}^{st}[\/latex] angle. [latex]a+20={2}^{nd}[\/latex] angle. [latex]90={3}^{rd}[\/latex] angle (the right angle).<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 right triangle with vertex A labeled a, vertex B labeled a + 20, and vertex C labeled 90 degrees.\" width=\"212\" height=\"197\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong> Write the appropriate formula and substitute into the formula.<\/td>\n<td>[latex]m\\angle A+m\\angle B+m\\angle C=180[\/latex] [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] [latex]2a=70[\/latex] [latex]a=35[\/latex] first angle [latex]a+20[\/latex] second angle [latex]\\color{red}{35}+20[\/latex] [latex]55[\/latex] second angle [latex]90[\/latex] third angle.<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check.<\/strong><\/td>\n<td>[latex]35+55+90\\stackrel{?}{=}180[\/latex] [latex]180=180\\checkmark[\/latex]<\/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<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm7025\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=7025&theme=lumen&iframe_resize_id=ohm7025&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":12,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":71,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2981"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":29,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2981\/revisions"}],"predecessor-version":[{"id":15640,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2981\/revisions\/15640"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/71"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2981\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=2981"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=2981"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=2981"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=2981"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}