{"id":229600,"date":"2025-06-08T09:59:15","date_gmt":"2025-06-08T09:59:15","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=229600"},"modified":"2025-06-08T09:59:18","modified_gmt":"2025-06-08T09:59:18","slug":"three-lines-are-shown","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/08\/three-lines-are-shown\/","title":{"rendered":"Three lines are shown"},"content":{"rendered":"\n<p>Three lines are shown. A line with points C, A, and F intersects a line with points E, A, and B at point A. A line extends from point A to point D in between angle EAF. Angle CAE is 61 degrees, angle DAF is 90 degrees, and angle FAB is 61 degrees. Find the following angle measures. mAngleEAD = \u00c2\u00b0 mAngleCAB = \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>We are given the following geometric configuration:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Line <strong>CAF<\/strong> (with points C, A, and F) intersects line <strong>EAB<\/strong> (with points E, A, and B) at point <strong>A<\/strong>.<\/li>\n\n\n\n<li>A third line extends from <strong>A to D<\/strong>, lying <strong>between angle EAF<\/strong>.<\/li>\n\n\n\n<li>\u2220<strong>CAE<\/strong> = 61\u00b0<\/li>\n\n\n\n<li>\u2220<strong>DAF<\/strong> = 90\u00b0<\/li>\n\n\n\n<li>\u2220<strong>FAB<\/strong> = 61\u00b0<\/li>\n<\/ul>\n\n\n\n<p>We are asked to find:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>m\u2220EAD = ?<\/strong><\/li>\n\n\n\n<li><strong>m\u2220CAB = ?<\/strong><\/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\">Step-by-Step Solution:<\/h3>\n\n\n\n<p>We know:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2220CAE = 61\u00b0<\/li>\n\n\n\n<li>\u2220DAF = 90\u00b0<\/li>\n\n\n\n<li>\u2220FAB = 61\u00b0<\/li>\n\n\n\n<li>Line AD lies between the rays AE and AF, meaning angle \u2220EAF is split by ray AD into \u2220EAD and \u2220DAF.<\/li>\n\n\n\n<li>Similarly, line AB lies on the opposite side of ray AF, such that \u2220FAB is adjacent to \u2220DAF.<\/li>\n<\/ul>\n\n\n\n<p>Let\u2019s break this down:<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">1. Find <strong>m\u2220EAD<\/strong><\/h4>\n\n\n\n<p>From the given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2220EAF is made up of \u2220EAD + \u2220DAF.<\/li>\n\n\n\n<li>\u2220DAF = 90\u00b0<\/li>\n\n\n\n<li>\u2220CAE = 61\u00b0, and since point C is on the same side of A as E, this tells us \u2220CAE = \u2220EAF = 61\u00b0.<\/li>\n<\/ul>\n\n\n\n<p>So,<br>\u2220EAD = \u2220EAF \u2212 \u2220DAF<br>= 61\u00b0 \u2212 90\u00b0<br>= <strong>Not possible.<\/strong><\/p>\n\n\n\n<p>This tells us that \u2220CAE must actually refer to a different angle\u2014not \u2220EAF.<\/p>\n\n\n\n<p>Let\u2019s re-interpret the configuration more carefully.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Better Interpretation of Given Angles:<\/h3>\n\n\n\n<p>Let\u2019s draw a mental image:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>C\u2014\u2014A\u2014\u2014F is a straight line.<\/li>\n\n\n\n<li>E\u2014\u2014A\u2014\u2014B is a second line crossing at A.<\/li>\n\n\n\n<li>D lies <strong>between<\/strong> \u2220EAF, forming two smaller angles: \u2220EAD and \u2220DAF = 90\u00b0.<\/li>\n<\/ul>\n\n\n\n<p>Given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2220CAE = 61\u00b0<br>\u2192 This is the angle between <strong>ray AC<\/strong> and <strong>ray AE<\/strong>.<\/li>\n\n\n\n<li>\u2220FAB = 61\u00b0<br>\u2192 This is the angle between <strong>ray AF<\/strong> and <strong>ray AB<\/strong>.<\/li>\n\n\n\n<li>\u2220DAF = 90\u00b0<br>\u2192 So ray AD is <strong>perpendicular<\/strong> to ray AF.<\/li>\n<\/ul>\n\n\n\n<p>We are to find:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2220EAD<\/li>\n\n\n\n<li>\u2220CAB<\/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\">Find <strong>m\u2220EAD<\/strong>:<\/h3>\n\n\n\n<p>We know:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2220CAE = 61\u00b0, and line CAF is straight, so angle between <strong>AE and AD<\/strong> (\u2220EAD) can be found from:<\/li>\n<\/ul>\n\n\n\n<p>\u2220EAD = \u2220EAF \u2212 \u2220DAF<br>\u2220EAF = \u2220CAE (since AE and AC are on same side) = 61\u00b0<br>\u2220DAF = 90\u00b0<br>So:<br><strong>\u2220EAD = 61\u00b0 \u2212 90\u00b0 = \u221229\u00b0<\/strong>, which is impossible.<\/p>\n\n\n\n<p>So instead, use this logic:<\/p>\n\n\n\n<p>\u2220EAD = \u2220DAF \u2212 \u2220FAE<br>But that requires knowing \u2220FAE, which we don\u2019t.<\/p>\n\n\n\n<p>However, we can treat \u2220EAD and \u2220DAF as two angles that <strong>sum to<\/strong> \u2220EAF. So:<\/p>\n\n\n\n<p><strong>\u2220EAD = \u2220EAF \u2212 \u2220DAF = 61\u00b0 \u2212 90\u00b0 = again \u221229\u00b0<\/strong><\/p>\n\n\n\n<p>Again, no good.<\/p>\n\n\n\n<p>But if <strong>\u2220DAF = 90\u00b0<\/strong>, and <strong>\u2220FAB = 61\u00b0<\/strong>, then \u2220DAB = \u2220DAF + \u2220FAB = 90\u00b0 + 61\u00b0 = <strong>151\u00b0<\/strong><\/p>\n\n\n\n<p>Similarly, \u2220EAD = \u2220CAE \u2212 \u2220CAD<br>But without \u2220CAD, we can\u2019t use that.<\/p>\n\n\n\n<p>Let\u2019s instead focus on full straight angle:<\/p>\n\n\n\n<p>Since CAF is a line, angle \u2220CAB is a straight angle:<br>\u2220CAB = \u2220CAE + \u2220EAD + \u2220DAF + \u2220FAB<\/p>\n\n\n\n<p>So:<br>\u2220CAB = 61\u00b0 (CAE) + \u2220EAD + 90\u00b0 + 61\u00b0 (FAB)<\/p>\n\n\n\n<p>But this is incorrect\u2014\u2220CAE and \u2220FAB are both parts of \u2220CAB already.<\/p>\n\n\n\n<p>Better approach:<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Use Linear Angle:<\/h3>\n\n\n\n<p>CAF is a straight line \u2192 angle between ray AC and ray AF = 180\u00b0<\/p>\n\n\n\n<p>So the three adjacent angles \u2220CAE, \u2220EAD, and \u2220DAF must <strong>sum to 180\u00b0<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2220CAE = 61\u00b0<\/li>\n\n\n\n<li>\u2220DAF = 90\u00b0<\/li>\n\n\n\n<li>So \u2220EAD = 180\u00b0 \u2212 61\u00b0 \u2212 90\u00b0 = <strong>29\u00b0<\/strong><\/li>\n<\/ul>\n\n\n\n<p>\u2705 <strong>m\u2220EAD = 29\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\">Now, find <strong>m\u2220CAB<\/strong><\/h3>\n\n\n\n<p>This is the angle between <strong>ray AC<\/strong> and <strong>ray AB<\/strong>.<\/p>\n\n\n\n<p>Given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2220DAF = 90\u00b0<\/li>\n\n\n\n<li>\u2220FAB = 61\u00b0<\/li>\n<\/ul>\n\n\n\n<p>So:<\/p>\n\n\n\n<p>\u2220CAB = \u2220DAF + \u2220FAB = 90\u00b0 + 61\u00b0 = <strong>151\u00b0<\/strong><\/p>\n\n\n\n<p>\u2705 <strong>m\u2220CAB = 151\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\">Final Answers:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>m\u2220EAD = 29\u00b0<\/strong><\/li>\n\n\n\n<li><strong>m\u2220CAB = 151\u00b0<\/strong><\/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\">Explanation<\/h3>\n\n\n\n<p>This geometry problem presents three intersecting lines with given angle measures. The key to solving it lies in understanding how these angles are arranged around point A. The line CAF is straight, which means that the total angle along this line is 180\u00b0. Points C, A, and F lie on this straight line. We are also told that \u2220CAE = 61\u00b0, \u2220DAF = 90\u00b0, and \u2220FAB = 61\u00b0.<\/p>\n\n\n\n<p>To find m\u2220EAD, we notice that angle \u2220CAE, \u2220EAD, and \u2220DAF are adjacent angles along the straight line CAF. Since a straight line measures 180\u00b0, the sum of these three angles must equal 180\u00b0. Using the given values, we calculate:<\/p>\n\n\n\n<p>\u2220EAD = 180\u00b0 \u2212 \u2220CAE \u2212 \u2220DAF = 180\u00b0 \u2212 61\u00b0 \u2212 90\u00b0 = 29\u00b0.<\/p>\n\n\n\n<p>Next, to find m\u2220CAB, we observe that it spans from ray AC to ray AB, passing through ray AD and ray AF. The angle \u2220CAB is the sum of the angles \u2220DAF (between AD and AF) and \u2220FAB (between AF and AB). Since both values are provided (90\u00b0 and 61\u00b0), we simply add:<\/p>\n\n\n\n<p>\u2220CAB = \u2220DAF + \u2220FAB = 90\u00b0 + 61\u00b0 = 151\u00b0.<\/p>\n\n\n\n<p>This solution highlights how adjacent angles along a straight line or sharing a vertex can be used to determine unknown angle measures. By recognizing the relationship between the angles, especially how they compose a straight angle or form larger composite angles, we can apply basic angle addition and subtraction to solve for unknowns accurately.<\/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-banner7-323.jpeg\" alt=\"\" class=\"wp-image-229601\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Three lines are shown. A line with points C, A, and F intersects a line with points E, A, and B at point A. A line extends from point A to point D in between angle EAF. Angle CAE is 61 degrees, angle DAF is 90 degrees, and angle FAB is 61 degrees. Find the [&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-229600","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/229600","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=229600"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/229600\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=229600"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=229600"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=229600"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}