{"id":135,"date":"2017-08-23T09:05:14","date_gmt":"2017-08-23T09:05:14","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=135"},"modified":"2017-09-18T17:47:26","modified_gmt":"2017-09-18T17:47:26","slug":"similarity-ratio","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/similarity-ratio\/","title":{"rendered":"Similarity &#038; Ratio"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/triangles\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/geometry-spatial-reasoning\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/convex-polygons\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Similarity &amp; Ratio<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, we will investigate the relationship of ratio and similarity in convex polygons, focusing on similar\u00a0triangles. We will also cover how to use ratios in order to solve problems relating to similarity and proportional\u00a0segments.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>A polygon is <strong><em>convex\u00a0<\/em><\/strong>if no two points of it lie on opposite sides of a\u00a0line containing a side of the polygon or if a line containing the\u00a0side of the polygon does not cross into the interior of the\u00a0polygon.<\/li>\n<li>Two triangles are <strong><em>congruent\u00a0<\/em><\/strong>if there is a correspondence between them such that\u00a0every pair of corresponding sides is congruent and every pair of\u00a0angles is congruent.<\/li>\n<li>A <em><strong>transversal\u00a0<\/strong><\/em>intersects two lines in distinct points.\u00a0Congruence between triangles is guaranteed by the <strong><em>SSS<\/em>,\u00a0<em>SAS<\/em>, <\/strong>and <em><strong>ASA <\/strong><\/em>congruence\u00a0postulates.<\/li>\n<\/ul>\n<section>\n<h3><strong>What are congruence and similarity?<\/strong><\/h3>\n<p>From our experience with congruent triangles in the previous\u00a0section, we know that congruence is an extremely strong condition.\u00a0Congruence essentially means that two triangles are identical.<\/p>\n<p>Similarity is a weaker concept. It means that two figures have\u00a0the same shape, but are not necessarily the same size. In other\u00a0words, one is a scaled version of the figure to which it is\u00a0similar.<\/p>\n<h4><strong>When are two convex polygons congruent or similar?<\/strong><\/h4>\n<p>Two polygons are <abbr title=\"equal in shape and size \">congruent<\/abbr> if there is a correspondence between them such that every\u00a0pair of corresponding sides is congruent and every pair of angles\u00a0is congruent.<\/p>\n<p>Two convex polygons are <abbr title=\"any pair of convex polygons with a correspondence between their vertices such that the corresponding angles are congruent, and such that the ratios of the lengths of their corresponding sides are equal \">similar<\/abbr> if there is a correspondence between their vertices such\u00a0that the corresponding angles are congruent, and such that the\u00a0ratios of the lengths of their corresponding sides are equal.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.2%20Art%20001.JPG\" alt=\"Similar quadrilaterals\" width=\"340\" height=\"122\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The polygons <em>ABCD<\/em> and <em>EFGH<\/em> shown above are\u00a0similar, and we write polygon <em>ABCD<\/em> ~ polygon <em>EFGH<\/em> to\u00a0denote the similarity.<\/p>\n<h4><strong>Where do ratios fit into the picture?<\/strong><\/h4>\n<p>In the similar quadrilaterals shown above, the ratio of the\u00a0lengths of corresponding sides is 1:2, which can be directly\u00a0verified. We can express this relationship using a series of\u00a0equations called proportions.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s5_p2_html_6b08c98d.gif\" width=\"185\" height=\"41\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>A <abbr title=\" a sequence of equations that states that two or more ratios are equal. \">proportion<\/abbr> is a sequence of equations stating that two or more ratios are\u00a0equal.<\/p>\n<p>Similarity, like congruence, is an equivalence relation. This\u00a0means that similarity is reflexive, symmetric, and transitive. Any\u00a0two of the ratios in a proportion are equal.<\/p>\n<p>Use proportions to solve problems with similar figures.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Suppose that triangle<em> ABC\u00a0<\/em>~ triangle<em> DBE\u00a0<\/em>as shown below. <em>BD\u00a0<\/em>= 3, <em>AD\u00a0<\/em>= 9, and <em>DE\u00a0<\/em>= 6. Which choice\u00a0shows the length of <em>AC<\/em>?<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.2%20Art%20002.gif\" width=\"200\" height=\"159\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<ol>\n<li>12<\/li>\n<li>15<\/li>\n<li>18<\/li>\n<li>24<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is D . The proportion we would like to\u00a0solve is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s5_p2_clip_image004.gif\" width=\"65\" height=\"34\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0. We are not given <em>AB\u00a0<\/em>directly, but can determine its value using the\u00a0information provided.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s5_p2_clip_image006.gif\" width=\"170\" height=\"11\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Let <em>x <\/em>=\u00a0<em>AC.<\/em><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s5_p2_clip_image008.gif\" width=\"66\" height=\"144\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<h3><strong>When can we show that two figures are similar?<\/strong><\/h3>\n<p>For the moment, we will focus on similar triangles because\u00a0these are the simplest polygons. As in the case of congruent\u00a0triangles, there are a number of situations in which we can give\u00a0sufficient conditions for the similarity of two triangles.<\/p>\n<h4><strong>What are the sufficient conditions for two triangles to be similar?<\/strong><\/h4>\n<h4><strong>AA similarity postulate <\/strong><\/h4>\n<ul>\n<li><em>If two angles\u00a0of one triangle are congruent to two angles of another triangle,\u00a0then the triangles are similar.<\/em><\/li>\n<\/ul>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>Note that we could have called this the <strong>AAA postulate<\/strong>, but by a\u00a0previous theorem on angles we know that if we have two congruent angles, the third is also congruent.<\/p>\n<\/div>\n<p>We will not prove the AA postulate here, but we will deduce an\u00a0important corollary from it.<\/p>\n<p><em>If a line\u00a0parallel to one side of a triangle intersects the other two sides\u00a0in distinct points, then it cuts off a triangle similar to the\u00a0original triangle. <\/em><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.2%20Art%20002.JPG\" width=\"200\" height=\"159\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>In our example, because <em>DE\u00a0<\/em>is parallel to <em>AC<\/em>,\u00a0we can view <em>AB <\/em>as\u00a0transversal. We can then apply the theorem on transversal lines of\u00a0corresponding angles, to conclude that angle\u00a0<em>BDE<\/em> is congruent\u00a0to angle<em> BAC<\/em>.\u00a0Then we can apply the AA postulate to conclude that the triangles\u00a0are similar since they share angle B.<\/p>\n<p>This corollary is extremely useful in solving word problems\u00a0that involve estimating distances. But before practicing with an\u00a0exercise, here is another version of the corollary above that may\u00a0also be useful in solving word problems.<\/p>\n<p><em>If a line\u00a0parallel to one side of a triangle intersects the other two sides\u00a0in distinct points, then it cuts off line segments which are\u00a0proportional to the sides. <\/em>The following are the two\u00a0analogues of the SAS and SSS congruence postulates.<\/p>\n<h4><em><strong>SAS similarity postulate<\/strong><\/em><\/h4>\n<ul>If two pairs of corresponding sides of triangles are proportional and the included angles are congruent, then the triangles are similar.<\/ul>\n<h4><strong><em>SSS similarity <\/em>postulate<\/strong><\/h4>\n<ul>If the corresponding sides of triangles are proportional, then the triangles are similar.<\/ul>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Determine whether the pair of triangles is similar . If they\u00a0are similar, which is the necessary postulate justifying this\u00a0fact?<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.2%20Art%20003.JPG\" alt=\"Two triangles with one angle marked congruent\" width=\"109\" height=\"188\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<ol>\n<li>Similar by the AA similarity postulate<\/li>\n<li>Not similar<\/li>\n<li>Similar by the SSS similarity postulate<\/li>\n<li>Similarity cannot be determined<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is A . In the triangles, angle\u00a0<em>DCE\u00a0<\/em>and angle<em> ACB\u00a0<\/em>are vertical angles, so they are congruent. It is also given that\u00a0angle <em>DEC\u00a0<\/em>and angle<em> CAB\u00a0<\/em>are congruent. The triangles are therefore similar by AA\u00a0similarity postulate.<\/p>\n<\/div>\n<\/section>\n<h3><strong>How do we solve word problems using ratios?<\/strong><\/h3>\n<p>Similar triangles have many applications in the real world. As\u00a0is often the case with word problems, setting the problem up\u00a0correctly is the hardest part. The following two exercises are a\u00a0few of the many ways we can use similar triangles to estimate\u00a0distances that may be impractical or impossible to measure.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>A woman is running up a hill with a constant angle of incline.\u00a0After running 10 miles, her altitude is 2 miles. Which choice\u00a0shows her altitude after she has run 15 miles?<\/p>\n<ol>\n<li>3<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s5_p4_clip_image002.gif\" width=\"9\" height=\"34\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li>4<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s5_p4_clip_image004.gif\" width=\"9\" height=\"34\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is A. We can use the following diagram and\u00a0the fact that the triangles are similar to solve the equation.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Matl%20Mod%204.2%20Art%20004.JPG\" alt=\"similar triangles with ends 10 and 15 miles\" width=\"310\" height=\"109\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Set up a ratio based on the drawing, and then cross multiply\u00a0to solve for <em>x.<\/em><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s5_p4_clip_image006.gif\" width=\"73\" height=\"144\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>The map below shows a portion of New York City. If 21st, 22nd,\u00a0and 23rd Streets are all parallel, how far is it along Broadway\u00a0from 23rd Street to 21st Street?<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.2%20Art%20005.JPG\" alt=\"streets of NYC\" width=\"223\" height=\"214\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<ol>\n<li>60 meters<\/li>\n<li>75 meters<\/li>\n<li>110 meters<\/li>\n<li>125 meters<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is D. Since the streets are parallel, the\u00a0triangle formed by Broadway, 22nd, and 5th Avenue is similar to\u00a0that formed by Broadway, 21st, and 5th Avenue. Denote the\u00a0distance along Broadway from 23rd to 21st as <em>x\u00a0<\/em>and set up a proportion.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s5_p4_html_m671b01bb.gif\" width=\"77\" height=\"87\" name=\"graphics10\" align=\"MIDDLE\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<h3><strong>How do we use ratios for line segments on transversals<br \/>\ncut by three parallel lines?<\/strong><\/h3>\n<p>The last application of ratios we will cover in this section\u00a0involves transversals and parallel lines .<\/p>\n<blockquote><p><em>If two\u00a0transversals intersect three parallel lines, then the ratio of\u00a0line segments cut off on one transversal is equal to the ratio of\u00a0corresponding line segments cut off on the second transversal. <\/em><\/p><\/blockquote>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.2%20Art%20006.JPG\" alt=\" Three parallel lines cut by transversals\" width=\"140\" height=\"121\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is the length of the segment\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s5_p5_html_m32f64e80.gif\" width=\"28\" height=\"21\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/>\u00a0depicted in the figure below?<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.2%20Art%20007.JPG\" alt=\"Three parallel lines cut by transversals with parts labeled\" width=\"140\" height=\"131\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s5_p5_clip_image002.gif\" width=\"17\" height=\"34\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s5_p5_clip_image004.gif\" width=\"17\" height=\"34\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s5_p5_clip_image006.gif\" width=\"17\" height=\"34\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s5_p5_clip_image008.gif\" width=\"17\" height=\"34\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p align=\"LEFT\">The correct answer is D . Using the postulate\u00a0above, we know that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s5_p5_clip_image010.gif\" width=\"66\" height=\"34\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0. Substituting the information given in the figure, we get that\u00a0<em>AC <\/em>=\u00a05 + 7 = 12, and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s5_p5_clip_image012.gif\" width=\"75\" height=\"37\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0where <em>x <\/em>=\u00a0<em>DE\u00a0<\/em>. Solve this equation.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s5_p5_clip_image014.gif\" width=\"86\" height=\"106\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p align=\"LEFT\">The ratio could also\u00a0be setup as:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s5_p5_html_2f38ef94.gif\" width=\"53\" height=\"109\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Therefore, if we know that <em>DE<\/em> = <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s5_p5_clip_image008.gif\" align=\"absmiddle\" \/>, then\u00a0<em>DF<\/em> = <em>DE<\/em> + 6, or <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s5_p5_html_3.gif\" align=\"absmiddle\" \/>.<\/p>\n<\/div>\n<\/section>\n<h3>Review of New Vocabulary and Concepts<\/h3>\n<ul>\n<li>Two convex polygons are <em><strong>similar\u00a0<\/strong><\/em>if there is a correspondence between their vertices\u00a0such that the corresponding angles are congruent and such that\u00a0the ratios of the lengths of their corresponding sides are equal.<\/li>\n<li>A <em><strong>proportion\u00a0<\/strong><\/em>is a sequence of equations that states that two or more ratios\u00a0are equal.<\/li>\n<li>One can use the <strong><em>AA\u00a0similarity <\/em><\/strong>postulate, <strong>SAS similarity\u00a0<\/strong>postulate, and <strong><em>SSS similarity <\/em><\/strong>postulate\u00a0to show that two triangles are similar.<\/li>\n<li>We can also solve problems\u00a0using ratios in the following two situations involving\u00a0proportional line segments:<\/li>\n<li>If a line parallel to one side of a triangle intersects the other\u00a0two sides in distinct points, then it cuts off line segments that\u00a0are proportional to the sides.<\/li>\n<li>If two transversals intersect three parallel lines, then the ratio of\u00a0line segments cut off on one transversal is equal to the ratio of\u00a0corresponding line segments cut off on the second transversal.<\/li>\n<\/ul>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/triangles\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/geometry-spatial-reasoning\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/convex-polygons\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Similarity &amp; Ratio Objective In this lesson, we will investigate the relationship of ratio and similarity in convex polygons, focusing on similar\u00a0triangles. We will also cover how to use ratios in order to solve problems relating to similarity and proportional\u00a0segments. Previously Covered: A polygon is convex\u00a0if no two points [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-135","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/135","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=135"}],"version-history":[{"count":9,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/135\/revisions"}],"predecessor-version":[{"id":661,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/135\/revisions\/661"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=135"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}