{"id":136,"date":"2017-08-23T09:05:38","date_gmt":"2017-08-23T09:05:38","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=136"},"modified":"2017-09-18T17:49:50","modified_gmt":"2017-09-18T17:49:50","slug":"convex-polygons","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/convex-polygons\/","title":{"rendered":"Convex Polygons"},"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\/similarity-ratio\">\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\/quadrilaterals\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Convex Polygons<\/h1>\n<h4>Objective<\/h4>\n<p>We will cover the formulas for the sum of the interior angles and the sum of the exterior angles for convex polygons.\u00a0We will use these formulas to solve related problems, such as determining when two convex polygons are similar or\u00a0congruent .<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>A polygon is <em><strong>convex <\/strong><\/em>if no two points of it lie on opposite sides of a line containing a side of the polygon or that no side containing a side of the polygon crosses the interior of the polygon.<\/li>\n<li>A polygon is called <strong><em>regular<\/em><\/strong> if it is convex, all of its sides are congruent, and all of its angles are congruent.<\/li>\n<li>Two polygons are <strong><em>congruent <\/em><\/strong>if there is a correspondence between them such that every pair of corresponding sides is congruent and every pair of angles is congruent.<\/li>\n<li>Two convex polygons are <strong><em>similar <\/em><\/strong>if there is 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.<\/li>\n<li>The sum of the angles of a triangle is 180\u00ba.<\/li>\n<\/ul>\n<section>\n<h3>Convex Polygons<\/h3>\n<p>At this point, we know how to show that two polygons (that are not triangles) are congruent or similar is by verifying the respective definitions directly.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which of the figures below are congruent?<\/p>\n<p align=\"CENTER\"><a name=\"#second\"><\/a><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/MAth%20Mod%204.2%20Art%20008.JPG\" alt=\"8 polygons\" width=\"369\" height=\"164\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<ol>\n<li>Figures C and E<\/li>\n<li>Figures B and H<\/li>\n<li>Figures A and F<\/li>\n<li>No figures are similar<\/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. Figures C and E are congruent, or identical. The pairs of figures A and F and figures B and H are both similar. In fact, any two regular <em>n<\/em>-gons are similar.<\/p>\n<\/div>\n<\/section>\n<h4><strong>What other ways can we verify that two polygons are\u00a0congruent or similar?<\/strong><\/h4>\n<p>Let\u2019s first answer the question for regular polygons.\u00a0Although it may seem obvious to you, we have not yet proved the\u00a0following fact.<\/p>\n<p class=\"lesson_subheadt\"><em><strong>Two regular polygons with the same number of sides are\u00a0similar.<\/strong><\/em><\/p>\n<p>By definition, a regular polygon has congruent sides and\u00a0congruent angles. Therefore, a pair of regular polygons with <em>n\u00a0<\/em>sides automatically satisfies the definition of similarity.\u00a0If the side length of the first polygon is <em>d<\/em> and that of\u00a0the second polygon is <em>e<\/em>, then the ratio of similarity is\u00a0<em>d<\/em>:<em>e<\/em>.<\/p>\n<h4><strong>What if the polygons are not regular?<\/strong><\/h4>\n<p>One approach to studying complex geometric figures is to break\u00a0them into simpler pieces. The figure below shows this division for\u00a0polygons with <em>n <\/em><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/greaterthanorequalto.gif\" width=\"13\" height=\"16\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>4, using diagonals.<\/p>\n<p>A line segment whose endpoints lie on two nonconsecutive\u00a0vertices of the polygon is called a <abbr title=\"a line segment whose endpoints lie on two nonconsecutive vertices of a polygon \">diagonal<\/abbr>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.2%20Art%20009.JPG\" alt=\"Two polygons with n greater than or equal to 4\" width=\"209\" height=\"87\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>In the figures above some of the diagonals are shown in blue.<\/p>\n<h4><strong>What is the sum of the measures of the interior angles\u00a0of a convex polygon with n sides?<\/strong><\/h4>\n<p>We can determine the answer to this question by a clever use of\u00a0diagonals. We divide a convex polygon with <em>n<\/em> sides into (<em>n\u00a0\u2013 <\/em>2) triangles. Divided in this manner, the sum of the\u00a0measures of the polygon is the sum of the (<em>n \u2013 <\/em>2)\u00a0triangles This gives a total of 180 (<em>n \u2013 <\/em>2). We have\u00a0proved the following:<\/p>\n<p><em>The sum of the measures of\u00a0the interior angles of a convex polygon with n sides is\u00a0180(n \u2013 2). <\/em><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which choice shows the measure of an interior angle of a\u00a0regular octagon?<\/p>\n<ol>\n<li>135\u00b0<\/li>\n<li>180\u00b0<\/li>\n<li>200\u00b0<\/li>\n<li>240\u00b0<\/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. By the formula above, the sum of the\u00a0interior angles of a octagon is 180(6)=1,080\u00b0. Since a\u00a0regular octagon has equal angles, an interior angle has measure\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s6_p2_clip_image002.gif\" width=\"77\" height=\"34\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<\/div>\n<\/section>\n<p>Although the idea of a polygon&#8217;s interior angle is an obvious\u00a0one, the idea of an exterior angle may not be so obvious . The\u00a0<abbr title=\"angle formed externally between two adjacent sides or by extending a side of the triangle.\">exterior\u00a0angle<\/abbr> of a polygon is the angle formed externally\u00a0between two adjacent sides or the angle formed on the outside of\u00a0the polygon by the extension of a side of the polygon.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.2%20Art%20010.JPG\" alt=\" Exterior angle\" width=\"194\" height=\"147\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<h3><strong>What is the sum of the measures of the exterior angles\u00a0of a convex polygon with <em>n<\/em> sides?<\/strong><\/h3>\n<p>We can find a formula for the sum of exterior angles using the\u00a0sum of the measures of interior angles for a convex polygon with <em><span style=\"text-decoration: none;\">n\u00a0<\/span><\/em><span style=\"text-decoration: none;\">s<\/span>ides. In the diagram shown below, the interior and\u00a0exterior angles at each vertex are supplementary, which means they\u00a0add up to equal 180\u00b0.<\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math Mod 4.2 Art 010.jpg\" alt=\"A polygon showing that the interior and exterior angles are supplementary at each vertex.\" width=\"194\" height=\"147\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s6_p3_clip_image002.gif\" width=\"366\" height=\"85\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><em><span style=\"text-decoration: none;\">The sum of the\u00a0measures of the exterior angles of a convex polygon with n sides\u00a0is 360. <\/span><\/em>Strangely enough, this result\u00a0does not depend on the number of sides of the polygon.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>If the measure of an interior angle of a regular <em><span style=\"text-decoration: none;\">n<\/span><\/em><span style=\"text-decoration: none;\">&#8211;<\/span>gon\u00a0is 156\u00b0, which choice shows the value of <em>n? <\/em><\/p>\n<ol>\n<li>10<\/li>\n<li>12<\/li>\n<li>15<\/li>\n<li>16<\/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 C. There are two ways to solve this\u00a0problem. Since the <em><span style=\"text-decoration: none;\">n<\/span><\/em>-gon\u00a0is regular, the interior angles all have the same measure.\u00a0Therefore, all exterior angles have an equal measure. An\u00a0exterior angle has measure 180-156=24\u00b0. So, using the formula\u00a0for the sum of exterior angles, we get\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s6_p3_clip_image004.gif\" width=\"57\" height=\"34\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0, or\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s6_p3_clip_image006.gif\" width=\"80\" height=\"34\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0We can also use the formula for the interior sum.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s6_p3_clip_image008.gif\" width=\"202\" height=\"169\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<p>We can use the formulas on the sum of the measures of interior\u00a0and exterior angles to solve problems relating to congruence and\u00a0similarity. If we are given the measures of <span style=\"text-decoration: none;\">(<\/span><em><span style=\"text-decoration: none;\">n\u00a0<\/span><\/em><span style=\"text-decoration: none;\">\u2013 1) of the angles in a polygon with <\/span><em><span style=\"text-decoration: none;\">n\u00a0<\/span><\/em><span style=\"text-decoration: none;\">sides, we can find the last angle using the formula. <\/span><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which of the following statements is true?<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.2%20Art%20011.JPG\" alt=\"Two similar quadrilaterals\" width=\"250\" height=\"94\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<ol>\n<li>The two figures shown above are congruent.<\/li>\n<li>The two figures shown above are similar.<\/li>\n<li>The two figures shown above are neither congruent nor similar.<\/li>\n<li>There is not enough information given to determine similarity<br \/>\nor congruence.<\/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 B. Each pair of corresponding sides has\u00a0ratio 3:2, and because three pairs of the angles are congruent,\u00a0the last pair is congruent by the interior angle sum formula.<\/p>\n<\/div>\n<\/section>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/similarity-ratio\">\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\/quadrilaterals\">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 Convex Polygons Objective We will cover the formulas for the sum of the interior angles and the sum of the exterior angles for convex polygons.\u00a0We will use these formulas to solve related problems, such as determining when two convex polygons are similar or\u00a0congruent . Previously Covered: A polygon is [&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-136","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/136","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=136"}],"version-history":[{"count":9,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/136\/revisions"}],"predecessor-version":[{"id":754,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/136\/revisions\/754"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=136"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}