Parallelogram | Brilliant Math & Science Wiki

<!DOCTYPE html> <html lang="en-us"> <head> <meta http-equiv="Content-type" content="text/html; charset=utf-8"> <meta name="google" content="notranslate" /> <meta http-equiv="Content-Language" content="en_US" /> <title>Parallelogram | Brilliant Math & Science Wiki</title> <meta name="viewport" content="width=device-width,initial-scale=1.0,maximum-scale=1.0"> <meta property="fb:app_id" content="326254770799145" /> <meta property="al:ios:url" content="com.brilliant.Brilliant://wiki/parallelogram/" /> <meta property="al:ios:app_store_id" content="913335252" /> <meta property="al:ios:app_name" content="Brilliant.org" /> <meta property="al:android:url" content="https://brilliant.org/wiki/parallelogram/"> <meta property="al:android:package" content="org.brilliant.android"> <meta property="al:android:app_name" content="Brilliant"> <link rel="canonical" href="https://brilliant.org/wiki/parallelogram/"> <link rel="alternate" href="android-app://org.brilliant.android/http/brilliant.org/wiki/parallelogram/"> <link rel="apple-touch-icon" href="/apple-touch-icon-precomposed.png"> <link rel="apple-touch-icon" href="/apple-touch-icon-72x72-precomposed.png"> <link rel="apple-touch-icon" href="/apple-touch-icon-76x76-precomposed.png"> <link rel="apple-touch-icon" href="/apple-touch-icon-114x114-precomposed.png"> <link rel="apple-touch-icon" href="/apple-touch-icon-120x120-precomposed.png"> <link rel="apple-touch-icon" href="/apple-touch-icon-144x144-precomposed.png"> <link rel="apple-touch-icon" href="/apple-touch-icon-152x152-precomposed.png"> <link rel="apple-touch-icon" href="/apple-touch-icon-180x180-precomposed.png"> <meta name="description" content="A parallelogram is a quadrilateral whose opposite sides are parallel. A parallelogram, in its most general form, looks something like this: Note that the arrowheads are used to indicate which pair of sides are parallel. A parallelogram is a quadrilateral with two pairs of parallel sides. The fundamental definition of a parallelogram is as follows: In the diagram of a general parallelogram above, ..." > <meta property="og:type" content="website"> <meta property="og:title" content="Parallelogram | Brilliant Math & Science Wiki"> <meta property="og:description" content="A parallelogram is a quadrilateral whose opposite sides are parallel. A parallelogram, in its most general form, looks something like this: Note that the arrowheads are used to indicate which pair of sides are parallel. A parallelogram is a quadrilateral with two pairs of parallel sides. The fundamental definition of a parallelogram is as follows: In the diagram of a general parallelogram above, ..."> <meta property="og:image" content="https://brilliant.org/site_media/version-1a4dfc4f1b/images/open-graph/default.png"> <meta property="og:image:alt" content="https://brilliant.org/site_media/version-1a4dfc4f1b/images/open-graph/default.png"> <meta property="og:url" content="https://brilliant.org/wiki/parallelogram/"> <meta name="twitter:site" content="@brilliantorg"> <meta name="twitter:card" content="summary_large_image"> <link href="/site_media/build/dist/brilliant_desktop.b75ee11fd0804510d4a6.css" rel="stylesheet" media="all" type="text/css"> <link href="/site_media/build/dist/wiki.a788b031a6461c585e3d.css" rel="stylesheet" media="all" type="text/css"> <link href="/site_media/build/dist/feed_page.367e4e839de43cf7373b.css" rel="stylesheet" media="all" type="text/css">      <script type="text/javascript"> (function (w, loc) { var u = 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href="https://brilliant.org/account/signup/?signup=true&next=/wiki/parallelogram/" class="btn signup-email ax-click" data-ax-id="clicked_signup_modal_email" data-ax-type="button"> Sign up manually</a> </div> <div class="alternative"> <p> Already have an account? <a href="https://brilliant.org/account/login/?next=/wiki/parallelogram/" class="ax-click" data-ax-id="clicked_signup_modal_login" data-ax-type="link"> Log in here. </a> </p> </div> </div> </div> <div class="wiki-top-editors" id="cmp_wiki_top_editors_id"> <b>Niranjan Khanderia</b>, <b>Lawrence Chiou</b>, <b>Karthik .</b>, and <div class="dropdown tipsy"> <button class="btn-link dropdown-toggle" data-toggle="dropdown"> 4 others </button> <ul class="dropdown-menu"> <li> <b>Hamza Waseem</b> </li> <li> <b>Andrew Ellinor</b> </li> <li> <b>ZhiJie Goh</b> </li> <li> <b>Jimin Khim</b> </li> </ul> </div> contributed </div> <div id="wiki-main" data-controller="app/newsfeed:feed"> <div class="summary-container" id="cmp_wiki_canonical_page_id"> <div class="summary wiki-content" data-controller="app/wiki:summary,app/zoomable:images" data-cmp-url="/wiki/parallelogram/" data-page-key="wiki_canonical_page" data-cmp-key="wiki_canonical_page"> <div class="section collapsed" id="section-pre-header-section"><div class="section-container"><p>A <strong>parallelogram</strong> is a <a href="/wiki/quadrilateral-classification/" class="wiki_link" title="quadrilateral"target="_blank">quadrilateral</a> whose opposite sides are parallel. A parallelogram, in its most general form, looks something like this: <span class="image-caption center"> <img src="http://syllabus.bostes.nsw.edu.au/assets/mathematicsk10/images/parallelogram.png" srcset="http://syllabus.bostes.nsw.edu.au/assets/mathematicsk10/images/parallelogram.png 1x" alt="" /> </span> Note that the arrowheads are used to indicate which pair of sides are parallel. </p> </div> </div> <div class="toc wiki-toc"> <h4>Contents</h4> <ul class="unstyled"> <li> <a href="#special-cases">Special Cases</a> </li> <li> <a href="#basic-properties">Basic Properties</a> </li> <li> <a href="#area">Area</a> </li> <li> <a href="#some-cooler-properties">Some Cooler Properties</a> </li> <li> <a href="#properties-of-parallelograms">Properties of Parallelograms</a> </li> <li> <a href="#properties">Properties</a> </li> <li> <a href="#special-parallelograms">Special Parallelograms</a> </li> </ul> </div> <div id="special-cases" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Special Cases</h2> </header> </div> <div class="section collapsed" id="section-special-cases"><div class="section-container"> </div> </div> <div id="basic-properties" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Basic Properties</h2> </header> </div> <div class="section collapsed" id="section-basic-properties"><div class="section-container"> </div> </div> <div id="area" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Area</h2> </header> </div> <div class="section collapsed" id="section-area"><div class="section-container"> </div> </div> <div id="some-cooler-properties" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Some Cooler Properties</h2> </header> </div> <div class="section collapsed" id="section-some-cooler-properties"><div class="section-container"> </div> </div> <div id="properties-of-parallelograms" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Properties of Parallelograms</h2> </header> </div> <div class="section collapsed" id="section-properties-of-parallelograms"><div class="section-container"> <p>A <strong>parallelogram</strong> is a <a href="/wiki/quadrilateral/" class="wiki_link" title="quadrilateral"target="_blank">quadrilateral</a> with two pairs of <a href="/wiki/parallel/?wiki_title=parallel" class="wiki_link new" title="page not yet created"target="_blank" rel="nofollow">parallel</a> sides.</p> </div> </div> <div id="properties" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Properties</h2> </header> </div> <div class="section collapsed" id="section-properties"><div class="section-container"> <p>The fundamental definition of a <strong>parallelogram</strong> is as follows:</p> <blockquote class="definition"><p> A <strong>parallelogram</strong> is a quadrilateral whose <a href="/wiki/opposite-sides/?wiki_title=opposite sides" class="wiki_link new" title="page not yet created"target="_blank" rel="nofollow">opposite sides</a> are parallel. </p></blockquote> <p><span class="image-caption center"> <img src="https://upload.wikimedia.org/wikipedia/commons/4/41/Parallelogram.svg" srcset="http://upload.wikimedia.org/wikipedia/commons/4/41/Parallelogram.svg 1x" alt="" /> </span></p> <p>In the diagram of a general parallelogram above, \( AB || DC \) and \( AD || BC \). Several important properties then follow. First, it is clear that the opposite angles must be equal \(( \angle A = \angle C \) and \( \angle B = \angle D) \) since they make corresponding angles with the two sets of parallel lines. Meanwhile, consecutive angles are supplementary.</p> <blockquote class="property"><p> <strong>Property 1.</strong> The opposite angles of a parallelogram are equal.</p> <p><strong>Property 2.</strong> Consecutive angles of a parallelogram are supplementary. </p></blockquote> <p>One can also show that the opposite sides are equal \(( AB = DC \) and \( AD = BC): \) the two triangles formed by drawing in a diagonal of the parallelogram \((\)i.e. either the segment \( AC \) or \( BD \) above\()\) must be congruent by angle-side-angle, so the corresponding sides of the two triangles must be congruent as well.</p> <blockquote class="property"><p> <strong>Property 3.</strong> The lengths of the opposite sides of a parallelogram are equal. </p></blockquote> <p>Now that the lengths of all of the sides are known, it is easy to compute the <a href="/wiki/perimeter/" class="wiki_link" title="perimeter"target="_blank">perimeter</a> of a parallelogram.</p> <blockquote class="property"><p> <strong>Property 4.</strong> A parallelogram whose side lengths are \( a \) and \( b \) has perimeter \( 2a + 2 b \). </p></blockquote> <p>Meanwhile, the area of the parallelogram can be found by computing the sum of the area of the two triangles formed.</p> <blockquote class="property"><p> <strong>Property 5.</strong> The area of a parallelogram with side lengths \( a \) and \( b \), with the acute angle formed between them \( \theta \), is given by \( ab \sin{\theta} \). </p></blockquote> <p>Drawing in both diagonals simultaneously produces four congruent triangles. Therefore, the intersection of the two diagonals must be the midpoint of each diagonal.</p> <blockquote class="property"><p> <strong>Property 6.</strong> The diagonals of a parallelogram bisect each other. </p></blockquote> <p>One last result is left as an exercise for the reader.</p> <blockquote class="property"><p> <strong>Property 7.</strong> Lines that connect midpoints of opposite sides with opposite vertices trisect the diagonal. </blockquote> </p> <p><div class="problem-modal-container anchor" id="problem-paralleogramania"> <div class="wiki-problem" data-controller="app/wiki:wikiShowProblemAnswer"> <div class="problem-container" > <div class="answer-container solv-details"> <button class="btn reveal-solution-btn">Reveal the answer</button> </div> <div class="question-container"> <p><span class="image-caption right"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/uploads/Lh9vHdoqDf-24448.svg?width=450" srcset="https://ds055uzetaobb.cloudfront.net/brioche/uploads/Lh9vHdoqDf-24448.svg?width=450 1x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/Lh9vHdoqDf-24448.svg?width=900 2x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/Lh9vHdoqDf-24448.svg?width=1350 3x" alt="" style="width:450px;max-width:100%;" /> </span></p> <p>In a parallelogram \(ABCD\), the length of \(AB\) is twice the length of \(AD\). \(E\) is the midpoint of \(AB\). </p> <p>Find the measure of \(\angle DEC\) in degrees.</p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>90</strong> </div> </div> </div> </div> </p> <p><div class="problem-modal-container anchor" id="problem-whats-the-shaded-area"> <div class="wiki-problem" data-controller="app/wiki:wikiShowProblemAnswer"> <div class="problem-container" > <div class="answer-container solv-details"> <div class="solv-mcq-wrapper row"> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> \[\frac{1}{3}\] </span> </span> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> \[ \frac{3}{8}\] </span> </span> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> \[\frac{1}{2}\] </span> </span> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> \[\frac{3}{5}\] </span> </span> </div> <button class="btn reveal-solution-btn">Reveal the answer</button> </div> <div class="question-container"> <p><span class="image-caption right"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/solvable/d62c06b750.bb31c936a7.BIVoa3.jpg?width=250" srcset="https://ds055uzetaobb.cloudfront.net/brioche/solvable/d62c06b750.bb31c936a7.BIVoa3.jpg?width=250 1x,https://ds055uzetaobb.cloudfront.net/brioche/solvable/d62c06b750.bb31c936a7.BIVoa3.jpg?width=500 2x,https://ds055uzetaobb.cloudfront.net/brioche/solvable/d62c06b750.bb31c936a7.BIVoa3.jpg?width=750 3x" alt="" style="width:250px;max-width:100%;" /> </span></p> <p>In the given parallelogram, the midpoints of two adjacent sides of the parallelogram are joined and then connected to the opposite vertex to form a triangle. </p> <p>What fraction of the total parallelogram is the shaded area? </p> <hr /> <p>Click here for more from this set.</p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>\[ \frac{3}{8}\]</strong> </div> </div> </div> </div> </p> <p><div class="problem-modal-container anchor" id="problem-a-geometrical-enigma-ii"> <div class="wiki-problem" data-controller="app/wiki:wikiShowProblemAnswer"> <div class="problem-container" > <div class="answer-container solv-details"> <button class="btn reveal-solution-btn">Reveal the answer</button> </div> <div class="question-container"> <p>A parallelogram has sides measuring 7 and 9. Its shorter diagonal has a length of 8. </p> <p>Find the measure of the longer diagonal.</p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>14</strong> </div> </div> </div> </div> </p> <p><div class="problem-modal-container anchor" id="problem-interesting-parallelogram-2"> <div class="wiki-problem" data-controller="app/wiki:wikiShowProblemAnswer"> <div class="problem-container" > <div class="answer-container solv-details"> <button class="btn reveal-solution-btn">Reveal the answer</button> </div> <div class="question-container"> <p><span class="image-caption right"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/solvable/d264a378bd.b41c73effd.u2yN5j.png?width=250" srcset="https://ds055uzetaobb.cloudfront.net/brioche/solvable/d264a378bd.b41c73effd.u2yN5j.png?width=250 1x,https://ds055uzetaobb.cloudfront.net/brioche/solvable/d264a378bd.b41c73effd.u2yN5j.png?width=500 2x,https://ds055uzetaobb.cloudfront.net/brioche/solvable/d264a378bd.b41c73effd.u2yN5j.png?width=750 3x" alt="" style="width:250px;max-width:100%;" /> </span></p> <p>A point \(X\) is drawn on the diagonal of a parallelogram \(ABCD\). The lines parallel to the sides through \(X\) are constructed, and quadrilaterals \(P\) and \(Q\) are formed, as shown in the diagram. </p> <p>Given that \(Q\) has base \(12\) and perpendicular height \(4\) and that the area of \(ABCD\) is \(200\), find the sum of all possible values of the base of \(P\).</p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>26</strong> </div> </div> </div> </div> </p> </div> </div> <div id="special-parallelograms" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Special Parallelograms</h2> </header> </div> <div class="section collapsed" id="section-special-parallelograms"><div class="section-container"> <p>Commonly encountered special cases of parallelograms include</p> <ul> <li><a href="/wiki/properties-of-rectangles/" class="wiki_link" title="rectangles"target="_blank">rectangles</a>, all of whose angles are equal;</li> <li><a href="/wiki/rhombus/" class="wiki_link" title="rhombuses"target="_blank">rhombuses</a>, all of whose sides are equal;</li> <li><a href="/wiki/properties-of-squares/" class="wiki_link" title="squares"target="_blank">squares</a>, all of whose sides and angles are equal.</li> </ul> <p>Because opposite angles of a <a href="/wiki/cyclic-quadrilaterals/" class="wiki_link" title="cyclic quadrilateral"target="_blank">cyclic quadrilateral</a> are supplementary, all cyclic parallelograms are rectangles. Furthermore, the only rectangles with an <a href="/wiki/incircle/?wiki_title=incircle" class="wiki_link new" title="page not yet created"target="_blank" rel="nofollow">incircle</a> are squares, so the only bicyclic parallelogram is a square.</p> </div> </div> </div> </div> </div> <div class="wiki-self-citation" data-controller="app/wiki:getCitationTime"> <strong>Cite as:</strong> Parallelogram. <em>Brilliant.org</em>. 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