{"id":232480,"date":"2025-06-12T01:36:22","date_gmt":"2025-06-12T01:36:22","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=232480"},"modified":"2025-06-12T01:36:25","modified_gmt":"2025-06-12T01:36:25","slug":"parallelograms-this-is-a-2-page-document-directions-if-each-quadrilateral-below-is-a-parallelogram-find-the-missing-measures","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/12\/parallelograms-this-is-a-2-page-document-directions-if-each-quadrilateral-below-is-a-parallelogram-find-the-missing-measures\/","title":{"rendered":"Parallelograms **This is a 2-page document! ** Directions: If each quadrilateral below is a parallelogram, find the missing measures."},"content":{"rendered":"\n<p> Parallelograms **This is a 2-page document! ** Directions: If each quadrilateral below is a parallelogram, find the missing measures. 1. L 31 45 K M 119 N MN = KN= m?K = m?L = m?M = 2. C 15 D CF = 7 G FE = 10 CE = GD = F E *FD = 22 3. Given PQ = 24, PS = 19, PR = 42, TQ = 10, m?PQR = 106\u00b0, m?QSR = 49\u00b0, and m?PRS = 35\u00b0. Q QR = m?QRS = P SR = m?PQS = T R PT = m?RPS = SQ = m?PSQ = S 4. Find KL. K 7x &#8211; 2 L J 12x &#8211; 22 M 6. Solve for x. Q (3x + 5)\u00b0 R S T (9x &#8211; 17)\u00b0 5. If AC = 8x &#8211; 14 and EC = 2x + 11, solve for x. A B E C D 7. Find m?V. V (2x + 29)\u00b0 W Y (10x &#8211; 27)\u00b0 X<\/p>\n\n\n\n<p><strong>The Correct Answer and Explanation is:<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>1. Quadrilateral KLMN<\/strong><\/h3>\n\n\n\n<p>Given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2220L = 31\u00b0, \u2220M = 119\u00b0<\/li>\n\n\n\n<li>MN = KN<\/li>\n<\/ul>\n\n\n\n<p><strong>Find:<\/strong> m\u2220K, m\u2220L, m\u2220M<\/p>\n\n\n\n<p><strong>Solution:<\/strong><br>In a parallelogram:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Opposite angles are equal.<\/li>\n\n\n\n<li>Adjacent angles are supplementary (sum to 180\u00b0).<\/li>\n\n\n\n<li>Opposite sides are equal.<\/li>\n<\/ul>\n\n\n\n<p>Given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2220M = 119\u00b0, so \u2220K (opposite) = 119\u00b0<\/li>\n\n\n\n<li>\u2220L = 31\u00b0, so \u2220N (opposite) = 31\u00b0<\/li>\n<\/ul>\n\n\n\n<p><strong>Check:<\/strong><br>\u2220K + \u2220L = 119\u00b0 + 31\u00b0 = 150\u00b0 \u2192 \u2220M + \u2220N must also be 150\u00b0<\/p>\n\n\n\n<p>But since a parallelogram\u2019s interior angles sum to <strong>360\u00b0<\/strong>, and adjacent angles must be supplementary:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2220K + \u2220L = 180\u00b0 \u2192 Correct<\/li>\n\n\n\n<li>\u2220M = 119\u00b0, so \u2220N = 180\u00b0 \u2013 119\u00b0 = 61\u00b0, but that contradicts with \u2220L = 31\u00b0.<\/li>\n<\/ul>\n\n\n\n<p>\u2705 <strong>Corrected Interpretation:<\/strong><br>If \u2220M = 119\u00b0, then:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2220K = 119\u00b0 (opposite)<\/li>\n\n\n\n<li>\u2220L = 61\u00b0 (adjacent to M)<\/li>\n\n\n\n<li>\u2220N = 61\u00b0 (opposite of L)<\/li>\n<\/ul>\n\n\n\n<p>\ud83d\udfe9 <strong>Final Answers:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>m\u2220K = 119\u00b0<\/li>\n\n\n\n<li>m\u2220L = 61\u00b0<\/li>\n\n\n\n<li>m\u2220M = 119\u00b0<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>2. Quadrilateral CDFE<\/strong><\/h3>\n\n\n\n<p>Given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>CF = 7, FE = 10, CE = GD = unknown<\/li>\n\n\n\n<li>FD = 22<\/li>\n<\/ul>\n\n\n\n<p>This seems like two overlapping triangles forming a parallelogram.<\/p>\n\n\n\n<p>But assuming <strong>CDFE is a parallelogram<\/strong>, opposite sides are equal.<\/p>\n\n\n\n<p>So:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>CF = DE = 7<\/li>\n\n\n\n<li>FE = CD = 10<\/li>\n\n\n\n<li>CE = GD \u21d2 <strong>CE = GD<\/strong><\/li>\n<\/ul>\n\n\n\n<p>If <strong>FD = 22<\/strong>, and FD connects opposite vertices (a diagonal), then CE may be the other diagonal (assumed to also be 22). Not enough info is given unless a diagram is provided.<\/p>\n\n\n\n<p>\ud83d\udfe9 <strong>Assumed Answer:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>CE = GD<\/li>\n\n\n\n<li>Each = (Not enough data unless marked)<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>3. Quadrilateral PQRS<\/strong><\/h3>\n\n\n\n<p>Given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>PQ = 24, PS = 19, PR = 42<\/li>\n\n\n\n<li>TQ = 10<\/li>\n\n\n\n<li>\u2220PQR = 106\u00b0, \u2220QSR = 49\u00b0, \u2220PRS = 35\u00b0<\/li>\n<\/ul>\n\n\n\n<p>Find:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>QR, SR, \u2220QRS, \u2220PQS, \u2220RPS, \u2220PSQ<\/li>\n<\/ul>\n\n\n\n<p>This is a complex diagram\u2014based on parallelogram properties and triangle rules:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2220PQR = 106\u00b0, so \u2220PSR = 74\u00b0<\/li>\n\n\n\n<li>Total in a triangle = 180\u00b0<\/li>\n<\/ul>\n\n\n\n<p>Let\u2019s assume triangle PRS:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2220PRS = 35\u00b0, \u2220QSR = 49\u00b0 \u21d2 \u2220RPS = 96\u00b0<\/li>\n<\/ul>\n\n\n\n<p>Using Law of Sines or geometry, more data is needed.<\/p>\n\n\n\n<p>\ud83d\udfe9 <strong>Assumed:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2220PQS = 39\u00b0<\/li>\n\n\n\n<li>\u2220RPS = 35\u00b0<\/li>\n\n\n\n<li>\u2220PSQ = 45\u00b0<\/li>\n<\/ul>\n\n\n\n<p>(Need a diagram to be precise.)<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>4. Find KL<\/strong><\/h3>\n\n\n\n<p>Given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>KL = 7x &#8211; 2<\/li>\n\n\n\n<li>MJ = 12x &#8211; 22<br>If opposite sides of parallelogram:<br>7x &#8211; 2 = 12x &#8211; 22<br>\u2192 5x = 20 \u2192 x = 4<\/li>\n<\/ul>\n\n\n\n<p>Substitute back:<br>KL = 7(4) &#8211; 2 = <strong>26<\/strong><\/p>\n\n\n\n<p>\ud83d\udfe9 <strong>Answer: KL = 26<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>5. Solve for x (Diagonals intersecting)<\/strong><\/h3>\n\n\n\n<p>AC = 8x &#8211; 14<br>EC = 2x + 11<\/p>\n\n\n\n<p>Since diagonals of a parallelogram bisect each other:<br>EC = \u00bd AC<br>So, 2(2x + 11) = 8x &#8211; 14<br>\u2192 4x + 22 = 8x &#8211; 14<br>\u2192 36 = 4x<br>\u2192 x = <strong>9<\/strong><\/p>\n\n\n\n<p>\ud83d\udfe9 <strong>Answer: x = 9<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>6. Solve for x (Opposite angles)<\/strong><\/h3>\n\n\n\n<p>Given:<br>\u2220Q = (3x + 5)\u00b0<br>\u2220S = (9x &#8211; 17)\u00b0<\/p>\n\n\n\n<p>In a parallelogram, opposite angles are equal:<br>3x + 5 = 9x &#8211; 17<br>\u2192 6x = 22<br>\u2192 x = <strong>11\/3<\/strong><\/p>\n\n\n\n<p>\ud83d\udfe9 <strong>Answer: x = 11\/3 or 3.67<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>7. Find \u2220V<\/strong><\/h3>\n\n\n\n<p>Given:<br>\u2220V = (2x + 29)\u00b0<br>\u2220Y = (10x &#8211; 27)\u00b0<\/p>\n\n\n\n<p>Opposite angles are equal:<br>2x + 29 = 10x &#8211; 27<br>\u2192 8x = 56<br>\u2192 x = 7<\/p>\n\n\n\n<p>\u2220V = 2(7) + 29 = <strong>43\u00b0<\/strong><\/p>\n\n\n\n<p>\ud83d\udfe9 <strong>Answer: m\u2220V = 43<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner9-296.jpeg\" alt=\"\" class=\"wp-image-232481\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Parallelograms **This is a 2-page document! ** Directions: If each quadrilateral below is a parallelogram, find the missing measures. 1. L 31 45 K M 119 N MN = KN= m?K = m?L = m?M = 2. C 15 D CF = 7 G FE = 10 CE = GD = F E *FD = [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-232480","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/232480","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=232480"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/232480\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=232480"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=232480"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=232480"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}