{"id":257,"date":"2017-08-23T04:33:27","date_gmt":"2017-08-23T04:33:27","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/elementary-education\/?page_id=257"},"modified":"2017-09-25T20:45:59","modified_gmt":"2017-09-25T20:45:59","slug":"polygons","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/elementary-education\/polygons\/","title":{"rendered":"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\/two-and-three-dimensional-figures\">\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\/properties-of-polygons\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Polygons<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, we&#8217;ll begin to define, identify, and classify polygons.<\/p>\n<h4>Previously Covered<\/h4>\n<ul>\n<li>So far, we&#8217;ve covered the basics of two-dimensional figures: points, lines, rays, planes, angles, and how to construct them.<\/li>\n<\/ul>\n<section>A <abbr title=\"A closed plane figure with three or more sides.\">polygon<\/abbr> is a closed figure made by joining line segments, called sides, so that the line segments intersect exactly two other segments. The point where each pair of segments intersects is called a vertex.Can you identify the figures below that are polygons and those that are not?<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/polygonquestion.jpg\" alt=\"Polygon question\" \/><\/center>Only figure A is a polygon. Notice that each of the segments that compose the figure is joined with only two others. Figure B is not a polygon, because it is not made of line segments; figure C is not a polygon, because it is not closed; and figure D is not a polygon, because two of the segments of which it is composed intersect three segments.<\/p>\n<h3>Triangles<\/h3>\n<p>Let\u2019s begin with triangles, since they are the simplest polygons. That is, they are the polygon with the least number of sides. Then we\u2019ll explore 4-sided polygons, or <abbr title=\"A four-sided polygon\">quadrilaterals<\/abbr>, and generalize to polygons with more than 4 sides.<\/p>\n<p><abbr title=\"A three-sided polygon.\">Triangles<\/abbr> may be classified by their sides or by their angles. To do so, remember that two figures are <abbr title=\"Two (or more) figures are congruent if they coincide exactly when superimposed.\">congruent<\/abbr> if they have the same size and the same shape.<\/p>\n<p>There are three classifications of a triangle according to the number of congruent sides it has: scalene, isosceles, and equilateral.<\/p>\n<table>\n<tbody>\n<tr>\n<td><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/mse6.1scalene.jpg\" alt=\"Scalene triangle\" \/><\/td>\n<td>This is a <strong>scalene triangle<\/strong>. None of its sides are congruent. That is, they all have different lengths.<\/td>\n<\/tr>\n<tr>\n<td><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/mse6.1isosceles.jpg\" alt=\"Isosceles triangle\" \/><\/td>\n<td>This is an <strong>isosceles triangle<\/strong>. Two of its sides are congruent. The two congruent sides are called the legs and the third side is called the base. In fact, according to the Isosceles Triangle Theorem, the <em>base angles<\/em> (those that are opposite the sides that are congruent) are also congruent. The third angle is called the <em>vertex angle<\/em>. The triangle is marked with congruence symbols to show which parts are congruent.<\/td>\n<\/tr>\n<tr>\n<td><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/equilateral.jpg\" alt=\"Equilateral triangle\" \/><\/td>\n<td>This is an <strong>equilateral triangle<\/strong>. All of its sides are congruent. It is also <em>equiangular<\/em>, which means that all of its angles are also congruent. Since the sum of the measures of the angles in a triangle is 180\u00b0, and all of the angles in an equilateral triangle are congruent, then each angle measures 180\u00b0\u00f7 3, or 60\u00b0. The triangle is marked with congruence symbols to show which parts are congruent.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Classifying Triangles By Their Angles<\/h3>\n<p>There are three classifications of a triangle according to the measures of its angles: acute, right, and obtuse.<\/p>\n<table>\n<tbody>\n<tr>\n<td><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/mse6.1acute.jpg\" alt=\"Acute triangle\" \/><\/td>\n<td>Each of the angles in this triangle is acute, so this is called an <strong>acute triangle<\/strong>. Notice that the sum of the measures of these angles is 180\u00b0.<\/td>\n<\/tr>\n<tr>\n<td><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/mse6.1right.jpg\" alt=\"Right triangle\" \/><\/td>\n<td>Because this triangle contains a right angle, it is called a <strong>right triangle<\/strong>. Do you see why a triangle can contain at most one right angle?<\/td>\n<\/tr>\n<tr>\n<td><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/mse6.1obtuse.jpg\" alt=\"Obtuse triangle\" \/><\/td>\n<td>Because this triangle contains an <strong>obtuse angle<\/strong>, it is called an obtuse triangle. Do you see why a triangle can contain at most one obtuse angle?<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>Which of the following describes this triangle?<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/mse6.1triquestion.jpg\" alt=\"Triangle question\" \/><\/center><\/p>\n<ol>\n<li>Isosceles, right<\/li>\n<li>Isosceles, obtuse<\/li>\n<li>Scalene, acute<\/li>\n<li>Scalene, right<\/li>\n<\/ol>\n<p><a class=\"q-answer button button-primary\">Reveal Answer<\/a><\/p>\n<p class=\"q-reveal\">The correct choice is B. This is an isosceles, obtuse triangle. The base angles are congruent, so we know that the sides opposite those angles are also congruent, which makes this an isosceles triangle. The base angles each measure 40\u00b0, for a total of 80\u00b0. The sum of the angles in a triangle is 180\u00b0, therefore, the vertex angle must measure 180\u00b0 \u2013 80\u00b0, or 100\u00b0. Thus, the triangle is obtuse. You cannot assume that the vertex angle is a right angle, although it does look like one. Objects are often not drawn to scale.<\/p>\n<\/section>\n<h3>Quadrilaterals<\/h3>\n<p>Quadrilateral literally means \u201cfour sides.\u201d A quadrilateral is a four-sided polygon. This is the <em>Quadrilateral Family Tree<\/em>.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/mse6.1quadrilaterals.jpg\" alt=\"Quadrilateral family tree\" \/><\/center><\/p>\n<table>\n<tbody>\n<tr>\n<td><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/quad_small.gif\" alt=\"Quadrilateral\" \/><\/td>\n<td>A <strong>quadrilateral<\/strong> is any four-sided polygon.<\/td>\n<\/tr>\n<tr>\n<td><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/parallelogram_small_s1_p9.gif\" alt=\"Parallelogram\" \/><\/td>\n<td>\n<h4>parallelogram<\/h4>\n<ul>\n<li>opposite sides parallel<\/li>\n<li>opposite sides congruent<\/li>\n<li>opposite angles congruent<\/li>\n<li>diagonals bisect each other<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/rectangle_small_s1_p9.gif\" alt=\"Rectangle\" \/><\/td>\n<td>\n<h4>rectangle<\/h4>\n<ul>\n<li>properties of a parallelogram<\/li>\n<li>all angles congruent and right<\/li>\n<li>diagonals congruent<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/rhombus_small_s1_p9.gif\" alt=\"Rhombus\" \/><\/td>\n<td>\n<h4>rhombus<\/h4>\n<ul>\n<li>properties of a parallelogram<\/li>\n<li>all sides congruent<\/li>\n<li>diagonals perpendicular<\/li>\n<li>bisect the angles they originate from<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/square_small_s1_p9.gif\" alt=\"Square\" \/><\/td>\n<td>\n<h4>square<\/h4>\n<ul>\n<li>properties of a parallelogram<\/li>\n<li>properties of a rectangle<\/li>\n<li>properties of a rhombus<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/trapezoid_small_s1_p9.gif\" alt=\"Trapezoid\" \/><\/td>\n<td>\n<h4>trapezoid<\/h4>\n<ul>\n<li>exactly one pair of opposite sides parallel<\/li>\n<li>consecutive angles between parallel sides are supplementary<\/li>\n<li>In an <strong>isosceles trapezoid<\/strong>, the nonparallel sides are congruent<\/li>\n<li>A <strong>right trapezoid<\/strong> contains two right angles.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>How can you use what you know about the sum of the measures of the angles in a triangle in order to find the sum of the measures of the angles in a quadrilateral?<\/p>\n<p>See if these pictures help you answer the question.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/mse6.1quadquestion.jpg\" alt=\"Angles in quadrilaterals\" \/><\/center>Notice that each quadrilateral can be cut into two triangles. Since the sum of the measures of the angles in a triangle is 180\u00b0, and each quadrilateral is composed of two triangles, then the sum of the measures of the angles in a quadrilateral is twice the sum of the measures of the angles in a triangle: 2 x 180\u00b0 = 360\u00b0.<\/p>\n<p><em>The sum of the measures of the angles in a quadrilateral is 360\u00b0.<\/em><\/p>\n<h3>Review<\/h3>\n<ul>\n<li>A polygon is a closed figure made of segments called sides.<\/li>\n<li>A triangle is the simplest polygon. It has three sides.<\/li>\n<li>Triangles may be classified by their sides or by the measures of their angles.<\/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\/two-and-three-dimensional-figures\">\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\/properties-of-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 Polygons Objective In this lesson, we&#8217;ll begin to define, identify, and classify polygons. Previously Covered So far, we&#8217;ve covered the basics of two-dimensional figures: points, lines, rays, planes, angles, and how to construct them. A polygon is a closed figure made by joining line segments, called sides, so that [&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-257","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/257","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=257"}],"version-history":[{"count":6,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/257\/revisions"}],"predecessor-version":[{"id":1388,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/257\/revisions\/1388"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/media?parent=257"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}