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<h2 class="wp-block-heading">Key Concepts</h2>



<ul class="wp-block-list"><li>Prove that a quadrilateral is a parallelogram.</li><li>Find the relation between the angles of a parallelogram.</li><li>Use properties of sides, and diagonals to identify a parallelogram.</li></ul>



<h3 class="wp-block-heading">Parallelogram </h3>



<p>A simple quadrilateral in which the opposite sides are of equal length and parallel is called a <strong>parallelogram.</strong>&nbsp;</p>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="259" height="173" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1788.png" alt="Parallelogram " class="wp-image-16981"/></figure></div>


<h3 class="wp-block-heading">Real-life examples of a parallelogram&nbsp;</h3>



<ol class="wp-block-list"><li><strong>Dockland office building in Hamburg, Germany.</strong> </li></ol>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="391" height="288" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1790.png" alt="Dockland office building in Hamburg, Germany. " class="wp-image-16983" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1790.png 391w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1790-300x221.png 300w" sizes="auto, (max-width: 391px) 100vw, 391px" /></figure></div>


<ol class="wp-block-list" start="2"><li><strong>An eraser </strong></li></ol>


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1792.png" alt="eraser" class="wp-image-16986" width="277" height="207" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1792.png 598w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1792-300x224.png 300w" sizes="auto, (max-width: 277px) 100vw, 277px" /></figure></div>


<ol class="wp-block-list" start="3"><li><strong>A striped pole </strong></li></ol>


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1790.png" alt="A striped pole " class="wp-image-16984" width="333" height="245" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1790.png 391w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1790-300x221.png 300w" sizes="auto, (max-width: 333px) 100vw, 333px" /></figure></div>


<ol class="wp-block-list" start="4"><li><strong>A solar panel </strong></li></ol>


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1791.png" alt="A solar panel " class="wp-image-16985" width="357" height="349" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1791.png 491w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1791-300x293.png 300w" sizes="auto, (max-width: 357px) 100vw, 357px" /></figure></div>


<h4 class="wp-block-heading">Theorem&nbsp;</h4>



<p>If one pair of opposite sides of a quadrilateral is parallel and congruent, then the quadrilateral is a parallelogram.&nbsp;</p>


<div class="wp-block-image">
<figure class="aligncenter size-large is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1789-1024x526.png" alt="Theorem 1" class="wp-image-16982" width="506" height="260" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1789-1024x526.png 1024w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1789-300x154.png 300w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1789-768x395.png 768w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1789.png 1530w" sizes="auto, (max-width: 506px) 100vw, 506px" /></figure></div>


<p><strong>Given:</strong> WX=ZY</p>



<p><strong>To prove:</strong> WXYZ is a parallelogram.&nbsp;</p>



<p><strong>Proof: </strong> Now, in △XZY and △ZXW, </p><div class="fillWidthInMob"><a loading='lazy' class="showAdInAllRes"  data-toggle="modal" data-target=#homeLeadForm data-heading="Get Expert Guidance" href="#"><img class="img-fluid" alt="parallel" src=https://a.storyblok.com/f/128066/1100x230/00b28b4876/11-applying-for-collages_leaderboard-1100x230.jpg /></a></div>



<p>WX = ZY </p>



<p>∠WXZ = ∠XZY [Alternate interior angles]</p>



<p>XZ = ZX [Reflexive property] </p>



<p>So, △XZY ≅ △ZX by Side-Angle-Side congruence criterion </p>



<p>If two triangles are congruent, their corresponding sides are equal.&nbsp;</p><div class="fillWidthInMob"><a loading='lazy' class="showAdInAllRes"  data-toggle="modal" data-target=#homeLeadForm data-heading="Get Expert Guidance" href="#"><img class="img-fluid" alt="parallel" src=https://a.storyblok.com/f/128066/1100x230/00b28b4876/11-applying-for-collages_leaderboard-1100x230.jpg /></a></div>



<p>Hence, WZ=XY</p>



<p>Therefore,</p>



<p><strong>WXYZ is a parallelogram. </strong></p>



<h4 class="wp-block-heading">Theorem&nbsp;</h4>



<p>If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a <strong>parallelogram</strong>.&nbsp;</p>


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1795.png" alt="Theorem 2" class="wp-image-16992" width="366" height="217"/></figure></div>


<p><strong>Given:</strong> ∠A+∠B = 180° and ∠A+∠D = 180°</p>



<p><strong>To prove: </strong>ABCD is a parallelogram.&nbsp;</p>



<p><strong>Proof: </strong></p>



<p>Given: ∠A and ∠B are supplementary. </p>



<p>If two lines are cut by a transversal, then the consecutive interior angles are supplementary, and the lines are parallel.&nbsp;</p>



<p>So, AD ∥ BC …(1) </p>



<p>Given: ∠A and ∠D are supplementary.&nbsp;</p>



<p>We know that if the consecutive interior angles are supplementary, then the lines are parallel.&nbsp;</p>



<p>So, AB ∥ CD …(2) </p>



<p>From (1) and (2), we get AB∥CD and AD∥BC</p>



<p>Hence,</p>



<p><strong>ABCD is a parallelogram. </strong></p>



<h4 class="wp-block-heading">Theorem&nbsp;</h4>



<p>If both the pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a <strong>parallelogram</strong>.&nbsp;</p>


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1793.png" alt="Theorem 3" class="wp-image-16991" width="284" height="263"/></figure></div>


<p><strong>Given:</strong> PQ=RS and PS=QR</p>



<p><strong>To prove:</strong> PQRS is a parallelogram.&nbsp;</p>



<p><strong>Proof:&nbsp;</strong>&nbsp;</p>



<p>Now, in △PQS and △RSQ, </p>



<p>PQ = RS </p>



<p>PS = RQ</p>



<p>QS = SQ [Reflexive property] </p>



<p>So, △PQS ≅  △RSQ by Side-Side-Side congruence criterion </p>



<p>If two triangles are congruent, their corresponding angles are equal.&nbsp;</p>



<p>Hence, ∠PQS = ∠RSQ and ∠QSP = ∠SQR</p>



<p>We know that if two lines are cut by a transversal, then the alternate interior angles are congruent, and the lines are parallel.&nbsp;</p>



<p>So, PQ ∥ SR and PS ∥ QR</p>



<p>Therefore,</p>



<p><strong>PQRS is a parallelogram. </strong></p>



<h4 class="wp-block-heading">Theorem&nbsp;</h4>



<p>If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a <strong>parallelogram</strong>.&nbsp;</p>


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1794.png" alt="Theorem 4" class="wp-image-16990" width="322" height="286"/></figure></div>


<p><strong>Given: </strong>LP=PN and OP=PM&nbsp;</p>



<p><strong>To prove: </strong>LMNO is a parallelogram.&nbsp;</p>



<p><strong>Proof:&nbsp;</strong>&nbsp;</p>



<p>Now, in</p>



<p>△OPN and △MPL, OP=MP</p>



<p>∠OPN=∠MPL [Vertically opposite angles]&nbsp;</p>



<p>PN=PL</p>



<p>So,</p>



<p>△OPN ≅  △MPL by Side-Angle-Side congruence criterion. </p>



<p>If two triangles are congruent, their corresponding sides are equal.&nbsp;</p>



<p>Hence,</p>



<p>ON = LM  …(1) </p>



<p>And, in △LPO and △NPM,</p>



<p>LP = NP</p>



<p>∠LPO = ∠NPM [Vertically opposite angles] </p>



<p>PO = PM</p>



<p>So, △LPO ≅  △NPM by Side-Angle-Side congruence criterion. </p>



<p>If two triangles are congruent, their corresponding sides are equal.&nbsp;</p>



<p>Hence,</p>



<p>LO = MN …(2) </p>



<p>From (1) and (2), we get</p>



<p>ON = LM and LO = MN. </p>



<p><strong>∴LMNO is a parallelogram.</strong> </p>



<h4 class="wp-block-heading">Theorem&nbsp;</h4>



<p>If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a <strong>parallelogram</strong>.&nbsp;</p>


<div class="wp-block-image">
<figure class="aligncenter size-large is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1796-1024x744.png" alt="Theorem 5" class="wp-image-16993" width="356" height="259" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1796-1024x744.png 1024w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1796-300x218.png 300w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1796-768x558.png 768w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1796.png 1375w" sizes="auto, (max-width: 356px) 100vw, 356px" /></figure></div>


<p><strong>Given:</strong> ∠A=∠C; ∠B=∠D</p>



<p><strong>To prove: </strong>ABCD is a parallelogram.&nbsp;</p>



<p><strong>Proof:&nbsp;</strong>&nbsp;</p>



<p>We know that the sum of angles of a quadrilateral is 360°.&nbsp;</p>



<p>So,</p>



<p>∠A+∠B+∠C+∠D=360°</p>



<p>∠A+∠B+∠A+∠B=360°</p>



<p>2(∠A+∠B)&nbsp;=360°</p>



<p>∠A+∠B=180°</p>



<p>If two lines are cut by a transversal, then the consecutive interior angles are supplementary, and the lines are parallel.&nbsp;</p>



<p>Since ∠A and ∠B are supplementary,</p>



<p>so, AD ∥ BC …(1) </p>



<p>Or we can write&nbsp;</p>



<p>∠A+∠B+∠C+∠D=360°</p>



<p>∠A+∠D+∠A+∠D=360°</p>



<p>2(∠A+∠D)&nbsp;=360°</p>



<p>∠A+∠D=180°</p>



<p>Since the angles ∠A and ∠D are supplementary, so,</p>



<p>AB ∥ CD       …(2) </p>



<p>From (1) and (2), we get AB ∥ CD and AD ∥ BC</p>



<p><strong>So, ABCD is a parallelogram. </strong></p>



<h2 class="wp-block-heading">Exercise</h2>



<ul class="wp-block-list" type="1"><li>For what values of x and y is the given quadrilateral a parallelogram?</li></ul>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="334" height="135" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1799.png" alt="For what values of x and y is the given quadrilateral a parallelogram?" class="wp-image-17005" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1799.png 334w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1799-300x121.png 300w" sizes="auto, (max-width: 334px) 100vw, 334px" /></figure></div>


<ul class="wp-block-list"><li>For what values of <em>w </em>and <em>z</em> is the given figure a parallelogram?</li></ul>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="285" height="188" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1800.png" alt="For what values of w and z is the given figure a parallelogram?" class="wp-image-17006"/></figure></div>


<ul class="wp-block-list"><li>Is the figure below a parallelogram?</li></ul>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="363" height="135" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1801.png" alt="Is the figure below a parallelogram?" class="wp-image-17007" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1801.png 363w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1801-300x112.png 300w" sizes="auto, (max-width: 363px) 100vw, 363px" /></figure></div>


<ul class="wp-block-list"><li>For what values of <em>x</em> and <em>y</em> is the given quadrilateral a parallelogram?</li></ul>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="307" height="284" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1802.png" alt="For what values of x and y is the given quadrilateral a parallelogram?" class="wp-image-17008" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1802.png 307w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1802-300x278.png 300w" sizes="auto, (max-width: 307px) 100vw, 307px" /></figure></div>


<ul class="wp-block-list"><li>Is the figure below a parallelogram?</li></ul>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="291" height="150" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1803.png" alt="Is the figure below a parallelogram?" class="wp-image-17009"/></figure></div>


<h3 class="wp-block-heading">Concept Map</h3>



<figure class="wp-block-image size-large"><img loading="lazy" decoding="async" width="1024" height="322" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1798-1024x322.png" alt="Concept Map" class="wp-image-17004" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1798-1024x322.png 1024w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1798-300x94.png 300w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1798-768x241.png 768w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1798.png 1051w" sizes="auto, (max-width: 1024px) 100vw, 1024px" /></figure>



<h3 class="wp-block-heading">What we have learned</h3>



<ul class="wp-block-list"><li>A quadrilateral whose two pairs of opposite sides are parallel is called a parallelogram.</li><li>If a quadrilateral is a parallelogram, then its opposite sides are congruent.</li><li>If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.</li><li>If a quadrilateral is a parallelogram, then opposite angles are congruent.</li><li>If a quadrilateral is a parallelogram, then its diagonals bisect each other.</li></ul>
