{"id":137,"date":"2017-08-23T09:06:04","date_gmt":"2017-08-23T09:06:04","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=137"},"modified":"2017-09-18T18:01:15","modified_gmt":"2017-09-18T18:01:15","slug":"quadrilaterals","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/quadrilaterals\/","title":{"rendered":"Quadrilaterals"},"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\/convex-polygons\">\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\/straightedge-and-compass-constructions\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Quadrilaterals<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will study the relationship of the angles, sides, and diagonals in special quadrilaterals.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>A line segment whose endpoints lie on two nonconsecutive vertices of the polygon is called a <em><strong>diagonal<\/strong><\/em>.<\/li>\n<li>The <strong><em>parallel postulate<\/em><\/strong> states: <em><span style=\"text-decoration: none;\">Given a point not on a line, there is exactly one line parallel to the given line containing that point.<\/span><\/em><\/li>\n<li>If two parallel lines are cut by a transversal, then any pair of alternate interior angles is congruent.<\/li>\n<\/ul>\n<section>\n<h3><strong>What is a quadrilateral?<\/strong><\/h3>\n<p><abbr title=\"a four-sided figure. It can also be called a four-sided polygon.\">Quadrilaterals<\/abbr> are the simplest polygons after triangles; they are polygons with four sides. There is a special class of quadrilaterals for which we have a plethora of theorems. In fact, there are so many we will mention just a handful of them here. But first, let\u2019s get comfortable with quadrilateral terminology.<\/p>\n<h4><strong>Which quadrilaterals are special?<\/strong><\/h4>\n<p>In a quadrilateral, two sides are <abbr title=\"sides that do not share a common endpoint \">opposite<\/abbr> if they do not share a common endpoint. Two sides are <abbr title=\" sides that share a common endpoint\">consecutive<\/abbr> if they share a common endpoint.<\/p>\n<p>Special quadrilaterals have at least one pair of opposite sides parallel.<\/p>\n<p>A <abbr title=\"A trapezoid is a quadrilateral with exactly one pair of parallel sides \">trapezoid<\/abbr> is a quadrilateral with exactly one pair of parallel sides.<\/p>\n<p>A <abbr title=\"a quadrilateral that has exactly two pairs of opposite parallel sides. \">parallelogram<\/abbr> is a quadrilateral in which both pairs of opposite sides are parallel. Other properties of parallelograms include:<\/p>\n<ul>\n<li>Opposite sides are congruent<\/li>\n<li>Opposite angles are congruent.<\/li>\n<li>Consecutive angles are supplementary.<\/li>\n<li>The diagonals bisect each other.<\/li>\n<\/ul>\n<p>As you probably know, the most famous of quadrilaterals, a <abbr title=\"a quadrilateral with four right angles\">rectangle<\/abbr>, is a quadrilateral with four right angles.<\/p>\n<p>The less famous <abbr title=\"a parallelogram with four congruent sides\">rhombus<\/abbr> is a parallelogram with four congruent sides. In any rhombus, the diagonals are perpendicular to each other.<\/p>\n<p>A <abbr title=\"a rectangle with congruent sides \">square <\/abbr>is a rectangle with congruent sides.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which of the following statements is <em><span style=\"text-decoration: none;\">false<\/span><\/em>?<\/p>\n<ol>\n<li>Every rhombus is a parallelogram.<\/li>\n<li>Every square is a parallelogram.<\/li>\n<li>Every rectangle is a square.<\/li>\n<li>Every square is a rhombus.<\/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. The truth of A and B follow directly from the definitions. The truth of D requires more thought; but according to the definitions, a square can also be thought of as a rhombus with four congruent angles.<\/p>\n<\/div>\n<\/section>\n<h3><strong>What are some properties of<br \/>\ntrapezoids?<\/strong><\/h3>\n<p>A trapezoid is a quadrilateral with <span style=\"font-style: italic;\">exactly<\/span> parallel segments called the <abbr title=\" side or face of a geometrical figure from which an altitude can be constructed\">bases<\/abbr>; the nonparallel sides are called <abbr title=\" nonparallel sides of a trapezoid or the sides that make the right angle in a right triangle \">legs<\/abbr>. Each base forms two angles called, appropriately, base angles. Thus, each trapezoid contains two pairs of base angles.<\/p>\n<p>An <abbr title=\"a trapezoid with congruent legs \">isosceles<\/abbr> trapezoid is a trapezoid with congruent legs.<\/p>\n<ul>\n<li>Each pair of base angles in an isosceles trapezoid is congruent.<\/li>\n<li>The diagonals of an isosceles trapezoid are congruent.<\/li>\n<\/ul>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>You can prove these theorems about isosceles trapezoids using the following corollary of the theorem on transversals cutting parallel lines: <em><span style=\"text-decoration: none;\">If two lines are parallel, the distance between the two lines is constant<\/span><\/em><span style=\"text-decoration: none;\">. <\/span><\/p>\n<p>Remember, the distance between two lines is the distance from any one point of the line to the other.<\/p>\n<\/div>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which of the following quadrilaterals does <em><u>not<\/u><\/em> necessarily have congruent diagonals?<\/p>\n<ol>\n<li>Rectangle<\/li>\n<li>Square<\/li>\n<li>Rhombus<\/li>\n<li>Isosceles trapezoid<\/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. A rhombus that is not a square will have non-congruent diagonals.<\/p>\n<\/div>\n<\/section>\n<h3>Review of New Vocabulary and Concepts<\/h3>\n<ul>\n<li>When the endpoints of a line segment lie on two nonconsecutive vertices of a polygon, that\u00a0line segment is called a <strong><em>diagonal<\/em><\/strong>.<\/li>\n<li style=\"margin-bottom: 0in;\">The sum of the measures of the interior angles of a convex polygon with <em>n<\/em> sides is 180(<em>n<\/em> \u2013 2).<\/li>\n<li>The sum of the measures of the exterior angles of a convex polygon with <em>n<\/em> sides is 360\u00b0.<\/li>\n<li>A <em><strong>trapezoid <\/strong><\/em>is a quadrilateral with exactly one pair of parallel sides. Isosceles trapezoids have the following properties:\n<ul>\n<li>Each pair of base angles in an isosceles trapezoid is congruent.<\/li>\n<li>The diagonals of an isosceles trapezoid are congruent.<\/li>\n<\/ul>\n<\/li>\n<li>A <em><strong>parallelogram<\/strong><\/em> is a quadrilateral in which both pairs of opposite sides are\u00a0parallel.<\/li>\n<li>A <em><strong>rectangle<\/strong><\/em> is a quadrilateral with four right angles.<\/li>\n<li>A <strong><em>rhombus<\/em><\/strong> is a parallelogram with four congruent sides.<\/li>\n<li>A <strong><em>square <\/em><\/strong>is\u00a0a rectangle with congruent sides.<\/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\/convex-polygons\">\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\/straightedge-and-compass-constructions\">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 Quadrilaterals Objective In this lesson, you will study the relationship of the angles, sides, and diagonals in special quadrilaterals. Previously Covered: A line segment whose endpoints lie on two nonconsecutive vertices of the polygon is called a diagonal. The parallel postulate states: Given a point not on a line, [&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-137","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/137","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=137"}],"version-history":[{"count":6,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/137\/revisions"}],"predecessor-version":[{"id":512,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/137\/revisions\/512"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=137"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}