{"id":258,"date":"2017-08-23T04:33:36","date_gmt":"2017-08-23T04:33:36","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/elementary-education\/?page_id=258"},"modified":"2017-09-25T20:52:38","modified_gmt":"2017-09-25T20:52:38","slug":"properties-of-polygons","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/elementary-education\/properties-of-polygons\/","title":{"rendered":"Properties of 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\/elementary-education\/polygons\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/geometry-measurement\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/transformations\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Properties of Polygons<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, we&#8217;ll explore a few properties of polygons.<\/p>\n<h4>Previously Covered<\/h4>\n<ul>\n<li>In the previous lesson we looked at examples of triangles and quadrilaterals.<\/li>\n<\/ul>\n<section><strong>Polygon<\/strong> literally means &#8220;many knees,&#8221; but mathematicians use it to mean &#8220;many sides.&#8221; In the table below you will find the names of the most common polygons. See if you can find a relationship between the number of angles a polygon has and the number of sides it has. This relationship will explain why polygons have come to be classified by the number of sides they have, rather than the number of angles they have. While you\u2019re at it, see if you can find a relationship between the numbers of sides, angles, and vertices.<center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/polygons1.gif\" alt=\"Polygons\" \/><\/center><\/p>\n<p class=\"figcaption\">Polygons and their sides<\/p>\n<p>The polygons above are the ones you&#8217;re most likely to see. The computations that you\u2019ll do will most likely be with those above and those that are named in the table below.<\/p>\n<table>\n<thead>\n<tr>\n<th>number of sides (n)<\/th>\n<th>name<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>7<\/td>\n<td>heptagon<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>nonagon<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>decagon<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Regular Polygons<\/h3>\n<p>Remember the equilateral, equiangular triangle from earlier? All of the sides and angles of that triangle are congruent. Clearly, this is a special case of a triangle, because scalene triangles do not have this property. Nor do obtuse triangles.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>Which of the following figures must be equilateral and equiangular?<\/p>\n<ol>\n<li>Circle<\/li>\n<li>Triangle<\/li>\n<li>Square<\/li>\n<li>Hexagon<\/li>\n<\/ol>\n<p><a class=\"q-answer button button-primary\">Reveal Answer<\/a><\/p>\n<p class=\"q-reveal\">The correct answer is C because all of the sides and angles of a square are congruent. Choice A is incorrect, because, as perfect as they are, circles have neither sides nor angles. Choices B and D are incorrect. Both triangles and hexagons can have unequal sides and angles. For example, here\u2019s a hexagon that is neither equilateral nor equiangular: <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/hexagon3z.gif\" alt=\"An unusual hexagon\" \/><\/p>\n<\/section>\n<p>A <strong>regular polygon<\/strong> is a polygon that is both equilateral and equiangular.<\/p>\n<p>What do you call a regular quadrilateral?<\/p>\n<p>Yes, it\u2019s a square! By the way, have you found a relationship between the number of sides, angles, and vertices in a polygon? They are all equal. A triangle has three vertices, a pentagon has five sides, and a decagon has ten angles.<\/p>\n<h3>Interior and Exterior Angles of Polygons<\/h3>\n<p>When you reviewed the definition of an angle earlier, you learned that an angle has an interior and an exterior. So it shouldn\u2019t surprise you that polygons, which are composed of angles, have interiors and exteriors, as well.<\/p>\n<p>An exterior angle of a polygon is the angle between one side of the polygon and the extension of an adjacent side. \u2220ACB is an exterior angle of \u25b3ACD.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>Assuming that \u25b3ACD is an equilateral triangle, what is the measure of \u2220ACB?<\/p>\n<ol>\n<li>60\u00b0<\/li>\n<li>90\u00b0<\/li>\n<li>120\u00b0<\/li>\n<li>135\u00b0<\/li>\n<\/ol>\n<p><a class=\"q-answer button button-primary\">Reveal Answer<\/a><\/p>\n<p class=\"q-reveal\">The correct answer is C. Since \u25b3ACD is an equilateral triangle, it is also equiangular. Thus, each of its angles measures 60\u00b0 and m\u2220ACD = 60\u00b0. \u2220BCD is a straight angle, so it measures 180\u00b0. According to the figure, m\u2220ACB + m\u2220ACD = m\u2220BCD (this relationship is actually referred to as the angle addition postulate). Now substitute: <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/equation_s2p3.jpg\" alt=\"Angle equations\" \/><\/p>\n<\/section>\n<p>What can be learned from this one example? Well, this relationship is true for all polygons: <em>The sum of the measures of a pair of interior and exterior angles of a polygon is 180\u00b0. That is, they are supplementary.<\/em><\/p>\n<h3>Review<\/h3>\n<ul>\n<li>Polygon means &#8220;many sided.&#8221;<\/li>\n<li>Regular polygons are both equilateral and equiangular.<\/li>\n<li>An exterior angle of a polygon is the angle between one side of the polygon and the extension of an adjacent side.<\/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\/elementary-education\/polygons\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/geometry-measurement\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/transformations\">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 Properties of Polygons Objective In this lesson, we&#8217;ll explore a few properties of polygons. Previously Covered In the previous lesson we looked at examples of triangles and quadrilaterals. Polygon literally means &#8220;many knees,&#8221; but mathematicians use it to mean &#8220;many sides.&#8221; In the table below you will find the [&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-258","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/258","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/comments?post=258"}],"version-history":[{"count":7,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/258\/revisions"}],"predecessor-version":[{"id":1389,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/258\/revisions\/1389"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/media?parent=258"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}