{"id":233905,"date":"2025-06-13T14:31:05","date_gmt":"2025-06-13T14:31:05","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=233905"},"modified":"2025-06-13T14:31:07","modified_gmt":"2025-06-13T14:31:07","slug":"the-perpendicular-bisector-of-side-ab-of-a%cb%86%e2%80%a0abc-intersects-the-extension-of-side-ac-at-d","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/13\/the-perpendicular-bisector-of-side-ab-of-a%cb%86%e2%80%a0abc-intersects-the-extension-of-side-ac-at-d\/","title":{"rendered":"The perpendicular bisector of side AB of \u00e2\u02c6\u2020ABC intersects the extension of side AC at D."},"content":{"rendered":"\n<p><br>The perpendicular bisector of side AB of \u00e2\u02c6\u2020ABC intersects the extension of side AC at D. Find the measure of \u00e2\u02c6 ABC if m\u00e2\u02c6 CBD = 16\u00c2\u00b0 and m\u00e2\u02c6 ACB = 118\u00c2\u00b0.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To find the measure of angle \u2220ABC, let\u2019s analyze the given information step by step:<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Given:<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2220CBD = 16\u00b0<\/li>\n\n\n\n<li>\u2220ACB = 118\u00b0<\/li>\n\n\n\n<li>D is the point where the perpendicular bisector of side AB intersects the extension of side AC.<\/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>Step-by-step Solution:<\/strong><\/h3>\n\n\n\n<p>In triangle <strong>ABC<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Point <strong>D<\/strong> lies on the extension of <strong>AC<\/strong>, and line <strong>CD<\/strong> bisects angle <strong>ABC<\/strong>, since <strong>\u2220CBD = 16\u00b0<\/strong>.<\/li>\n\n\n\n<li>Thus, <strong>\u2220ABC = \u2220CBD + \u2220ABD = 2 \u00d7 \u2220CBD = 2 \u00d7 16\u00b0 = 32\u00b0<\/strong><\/li>\n<\/ul>\n\n\n\n<p>So, <strong>m\u2220ABC = 32\u00b0<\/strong><\/p>\n\n\n\n<p>We can also double-check with the triangle angle sum rule:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The sum of interior angles of triangle <strong>ABC<\/strong> is always 180\u00b0.<\/li>\n\n\n\n<li>Given:<br>\u2220ACB = 118\u00b0,<br>\u2220ABC = 32\u00b0<br>Then:<br>\u2220BAC = 180\u00b0 \u2212 (\u2220ABC + \u2220ACB) = 180\u00b0 \u2212 (32\u00b0 + 118\u00b0) = <strong>30\u00b0<\/strong><\/li>\n<\/ul>\n\n\n\n<p>Thus, all angles in triangle ABC are:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2220BAC = 30\u00b0<\/li>\n\n\n\n<li>\u2220ABC = 32\u00b0<\/li>\n\n\n\n<li>\u2220ACB = 118\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>Final Answer:<\/strong><\/h3>\n\n\n\n<p><strong>The measure of \u2220ABC is 32\u00b0.<\/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>Book-Style Explanation (300 Words):<\/strong><\/h3>\n\n\n\n<p>In any triangle, the sum of the interior angles is always 180\u00b0. In triangle ABC, we are given the following information: angle CBD is 16\u00b0, and angle ACB is 118\u00b0. Furthermore, the perpendicular bisector of side AB intersects the extension of side AC at point D.<\/p>\n\n\n\n<p>Because CD is a perpendicular bisector and intersects the extension of AC, and because angle CBD is given as 16\u00b0, we can deduce that CD is also an <strong>angle bisector<\/strong> of \u2220ABC. Since \u2220CBD is one half of angle \u2220ABC, this means that:\u2220ABC=2\u00d7\u2220CBD=2\u00d716\u2218=32\u2218\\angle ABC = 2 \\times \\angle CBD = 2 \\times 16^\\circ = 32^\\circ\u2220ABC=2\u00d7\u2220CBD=2\u00d716\u2218=32\u2218<\/p>\n\n\n\n<p>Now, using the triangle angle sum property:\u2220ABC+\u2220BAC+\u2220ACB=180\u2218\\angle ABC + \\angle BAC + \\angle ACB = 180^\\circ\u2220ABC+\u2220BAC+\u2220ACB=180\u2218<\/p>\n\n\n\n<p>We are given that:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2220ABC = 32\u00b0<\/li>\n\n\n\n<li>\u2220ACB = 118\u00b0<\/li>\n<\/ul>\n\n\n\n<p>Then,\u2220BAC=180\u2218\u2212(32\u2218+118\u2218)=180\u2218\u2212150\u2218=30\u2218\\angle BAC = 180^\\circ &#8211; (32^\\circ + 118^\\circ) = 180^\\circ &#8211; 150^\\circ = 30^\\circ\u2220BAC=180\u2218\u2212(32\u2218+118\u2218)=180\u2218\u2212150\u2218=30\u2218<\/p>\n\n\n\n<p>All three interior angles of triangle ABC now make sense and follow the triangle sum rule:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2220ABC = 32\u00b0<\/li>\n\n\n\n<li>\u2220ACB = 118\u00b0<\/li>\n\n\n\n<li>\u2220BAC = 30\u00b0<\/li>\n<\/ul>\n\n\n\n<p>Therefore, the correct and final measure of angle \u2220ABC is <strong>32 degrees<\/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-banner8-360.jpeg\" alt=\"\" class=\"wp-image-233906\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>The perpendicular bisector of side AB of \u00e2\u02c6\u2020ABC intersects the extension of side AC at D. Find the measure of \u00e2\u02c6 ABC if m\u00e2\u02c6 CBD = 16\u00c2\u00b0 and m\u00e2\u02c6 ACB = 118\u00c2\u00b0. The Correct Answer and Explanation is: To find the measure of angle \u2220ABC, let\u2019s analyze the given information step by step: Given: Step-by-step [&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-233905","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/233905","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=233905"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/233905\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=233905"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=233905"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=233905"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}