{"id":262725,"date":"2025-07-20T18:43:53","date_gmt":"2025-07-20T18:43:53","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=262725"},"modified":"2025-07-20T18:43:55","modified_gmt":"2025-07-20T18:43:55","slug":"prove-that-opposite-angles-of-a-parallelogram-are-equal","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/20\/prove-that-opposite-angles-of-a-parallelogram-are-equal\/","title":{"rendered":"Prove that opposite angles of a parallelogram are equal"},"content":{"rendered":"\n<p>Prove that opposite angles of a parallelogram are equal.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\">The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To prove that opposite angles of a parallelogram are equal, consider a parallelogram ABCDABCD, where AB\u2225CDAB \\parallel CD and AD\u2225BCAD \\parallel BC. We need to show that \u2220DAB=\u2220BCD\\angle DAB = \\angle BCD and \u2220ABC=\u2220CDA\\angle ABC = \\angle CDA.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Label the Angles<\/h3>\n\n\n\n<p>Let the angles at the vertices be:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2220DAB=\u03b8\\angle DAB = \\theta<\/li>\n\n\n\n<li>\u2220ABC=\u03d5\\angle ABC = \\phi<\/li>\n\n\n\n<li>\u2220BCD=\u03b3\\angle BCD = \\gamma<\/li>\n\n\n\n<li>\u2220CDA=\u03b4\\angle CDA = \\delta<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Use Parallel Line Properties<\/h3>\n\n\n\n<p>Since AB\u2225CDAB \\parallel CD, the angle \u2220DAB\\angle DAB is an alternate interior angle with \u2220BCD\\angle BCD (because ADAD is a transversal line between the two parallel lines). By the Alternate Interior Angle Theorem, we conclude that: \u2220DAB=\u2220BCD.\\angle DAB = \\angle BCD.<\/p>\n\n\n\n<p>Therefore, \u03b8=\u03b3\\theta = \\gamma.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Apply the Same Reasoning to the Other Pair of Opposite Angles<\/h3>\n\n\n\n<p>Similarly, since AD\u2225BCAD \\parallel BC, the angle \u2220ABC\\angle ABC is an alternate interior angle with \u2220CDA\\angle CDA. Again, applying the Alternate Interior Angle Theorem, we get: \u2220ABC=\u2220CDA.\\angle ABC = \\angle CDA.<\/p>\n\n\n\n<p>Thus, \u03d5=\u03b4\\phi = \\delta.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 4: Conclusion<\/h3>\n\n\n\n<p>From the two observations above, we have shown that opposite angles in a parallelogram are congruent: \u2220DAB=\u2220BCDand\u2220ABC=\u2220CDA.\\angle DAB = \\angle BCD \\quad \\text{and} \\quad \\angle ABC = \\angle CDA.<\/p>\n\n\n\n<p>This completes the proof.<\/p>\n\n\n\n<p>The proof relies on the basic properties of parallel lines and transversals, specifically the Alternate Interior Angle Theorem, which ensures that alternate interior angles formed by parallel lines are equal.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Prove that opposite angles of a parallelogram are equal. The correct answer and explanation is: To prove that opposite angles of a parallelogram are equal, consider a parallelogram ABCDABCD, where AB\u2225CDAB \\parallel CD and AD\u2225BCAD \\parallel BC. We need to show that \u2220DAB=\u2220BCD\\angle DAB = \\angle BCD and \u2220ABC=\u2220CDA\\angle ABC = \\angle CDA. Step 1: [&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-262725","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/262725","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=262725"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/262725\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=262725"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=262725"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=262725"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}