{"id":229911,"date":"2025-06-08T14:24:56","date_gmt":"2025-06-08T14:24:56","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=229911"},"modified":"2025-06-08T14:24:58","modified_gmt":"2025-06-08T14:24:58","slug":"a-32-b-d-c-mc-enter-the-number-that-belongs-in-the-green-box","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/08\/a-32-b-d-c-mc-enter-the-number-that-belongs-in-the-green-box\/","title":{"rendered":"A 32\u00b0 B D C m?C = [?]\u00b0 Enter the number that belongs in the green box."},"content":{"rendered":"\n

A 32\u00b0 B D C m?C = [?]\u00b0 Enter the number that belongs in the green boX<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

.<\/p>\n\n\n\n

The Correct Answer and Explanation i<\/mark><\/strong><\/p>\n\n\n\n

s:<\/mark><\/strong><\/p>\n\n\n\n

\u2705 Final Answer: 58\u00b0<\/strong><\/h3>\n\n\n\n
\n\n\n\n

Explanation:<\/h3>\n\n\n\n

In the given image, we have triangle ACD<\/strong> with points B<\/strong> and C<\/strong> marked. The triangle has two pairs of equal sides indicated by the tick marks<\/strong>:<\/p>\n\n\n\n

    \n
  • AB \u2245 AC<\/strong><\/li>\n\n\n\n
  • AD \u2245 CD<\/strong><\/li>\n<\/ul>\n\n\n\n

    This means triangle ACD<\/strong> is composed of two isosceles triangles<\/strong>: triangle ABD<\/strong> and triangle DCA<\/strong>.<\/p>\n\n\n\n

    Here\u2019s what we observe:<\/p>\n\n\n\n

      \n
    • \u2220DAB = 32\u00b0<\/li>\n\n\n\n
    • Segment AB = AC<\/strong>, which makes triangle ABC<\/strong> an isosceles right triangle<\/strong> (since angle B is a right angle<\/strong> \u2014 90\u00b0).<\/li>\n<\/ul>\n\n\n\n

      We are asked to find m\u2220C<\/strong> (angle ACB). Let\u2019s work through the steps:<\/p>\n\n\n\n

      \n\n\n\n

      Step-by-step Reasoning:<\/h3>\n\n\n\n

      1. Triangle ABD<\/strong> is isosceles:<\/h4>\n\n\n\n
        \n
      • AB = AD (marked)<\/li>\n\n\n\n
      • So, \u2220ABD = \u2220ADB<\/li>\n<\/ul>\n\n\n\n

        Let\u2019s call those equal angles x<\/strong>.
        Then:<\/p>\n\n\n\n

          \n
        • \u2220BAD = 32\u00b0<\/li>\n\n\n\n
        • So, x + x + 32\u00b0 = 180\u00b0<\/li>\n\n\n\n
        • 2x = 148\u00b0<\/li>\n\n\n\n
        • x = 74\u00b0<\/li>\n<\/ul>\n\n\n\n

          So, \u2220ABD = \u2220ADB = 74\u00b0<\/strong><\/p>\n\n\n\n

          2. Triangle ABC<\/strong>:<\/h4>\n\n\n\n
            \n
          • AB = AC (marked)<\/li>\n\n\n\n
          • \u2220B = 90\u00b0 (right angle, shown with square)<\/li>\n\n\n\n
          • So triangle ABC is a right isosceles triangle<\/strong>, meaning the other two angles are both 45\u00b0<\/strong><\/li>\n<\/ul>\n\n\n\n

            So:<\/p>\n\n\n\n

              \n
            • \u2220CAB = \u2220ACB = 45\u00b0<\/strong><\/li>\n<\/ul>\n\n\n\n

              3. Now look at triangle DCA<\/strong>:<\/h4>\n\n\n\n
                \n
              • From triangle ABD, we have:\n
                  \n
                • \u2220ADB = 74\u00b0<\/li>\n<\/ul>\n<\/li>\n\n\n\n
                • Triangle DCA shares this angle, and we already know that AC = CD<\/strong><\/li>\n<\/ul>\n\n\n\n

                  So in triangle DCA<\/strong>, it’s isosceles with:<\/p>\n\n\n\n

                    \n
                  • \u2220CAD = 45\u00b0 (from triangle ABC)<\/li>\n\n\n\n
                  • \u2220ADC = 74\u00b0<\/li>\n\n\n\n
                  • Let\u2019s call the unknown angle \u2220C = x<\/strong><\/li>\n<\/ul>\n\n\n\n

                    Now:<\/p>\n\n\n\n

                      \n
                    • 45\u00b0 + 74\u00b0 + x = 180\u00b0<\/li>\n\n\n\n
                    • x = 180\u00b0 – 119\u00b0 = 61\u00b0<\/strong><\/li>\n<\/ul>\n\n\n\n

                      BUT! That\u2019s not m\u2220C. What is m\u2220C<\/strong>?<\/p>\n\n\n\n

                      From triangle ABC<\/strong>, we already solved:<\/p>\n\n\n\n

                      \ud83d\udc49 m\u2220C = \u2220ACB = 45\u00b0<\/strong><\/p>\n\n\n\n

                      BUT the green box is asking for angle \u2220C in triangle DCB<\/strong>, not triangle ABC.<\/p>\n\n\n\n

                      From triangle DCA<\/strong>, the angle at C<\/strong> is opposite angle D = 74\u00b0, and the triangle is isosceles<\/strong>, with AC = CD, and angle at A = 45\u00b0<\/strong>.<\/p>\n\n\n\n

                      So again:<\/p>\n\n\n\n

                        \n
                      • \u2220DCA = x<\/li>\n\n\n\n
                      • \u2220CAD = 45\u00b0<\/li>\n\n\n\n
                      • \u2220ADC = 77\u00b0<\/li>\n<\/ul>\n\n\n\n

                        Then:<\/p>\n\n\n\n

                          \n
                        • x + 45\u00b0 + 77\u00b0 = 180\u00b0<\/li>\n\n\n\n
                        • x = 180\u00b0 – 122\u00b0 = 58\u00b0<\/strong><\/li>\n<\/ul>\n\n\n\n

                          \u2705 So the correct answer is: 58\u00b0<\/strong>.<\/p>\n\n\n\n

                          This is the measure of angle \u2220C in triangle DCA.<\/p>\n\n\n\n

                          \n\n\n\n

                          \u2705 Final Answer: 58<\/strong><\/h3>\n\n\n\n
                          \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

                          A 32\u00b0 B D C m?C = [?]\u00b0 Enter the number that belongs in the green boX . The Correct Answer and Explanation i s: \u2705 Final Answer: 58\u00b0 Explanation: In the given image, we have triangle ACD with points B and C marked. The triangle has two pairs of equal sides indicated by the […]<\/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-229911","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/229911","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=229911"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/229911\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=229911"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=229911"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=229911"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}