\n– The sum of the exterior angles of any convex polygon, taken one at each vertex, is always 360 degrees. The angles numbered in this question are not the standard set of exterior angles (one at each vertex). They are pairs of vertically opposite angles to the exterior angles and also pairs of vertically opposite angles to each other.
\n– The six angles consist of three pairs of vertically opposite angles. The sum of angles around each vertex point is 360 degrees. The angles around vertex A are A + \u03b11 + \u03b12 = 360\u00b0. Since \u03b11 = \u03b12, A + 2\u03b11 = 360\u00b0, so 2\u03b11 = 360 – A. Similarly, 2\u03b21 = 360 – B and 2\u03b31 = 360 – C. The sum of the numbered angles is S = 2\u03b11 + 2\u03b21 + 2\u03b31 = (360 – A) + (360 – B) + (360 – C) = 1080 – (A + B + C) = 1080 – 180 = 900\u00b0. Oh, re-calculating using the full angle around the point method. This is incorrect as \u03b11 and \u03b12 form a pair. Let’s stick to the linear pair method which is more direct. \u03b11 is exterior angle, \u03b12 is vertically opposite to exterior angle. Sum is \u03b11+\u03b12+\u03b21+\u03b22+\u03b31+\u03b32. Since \u03b11=\u03b12, \u03b21=\u03b22, \u03b31=\u03b32, sum = 2(\u03b11+\u03b21+\u03b31). \u03b11=180-A, \u03b21=180-B, \u03b31=180-C. Sum = 2(180-A + 180-B + 180-C) = 2(540-(A+B+C)) = 2(540-180) = 2*360 = 720. Yes, 720 is correct.
\n– Another perspective: At each vertex, say A, the angle inside is A. The two angles formed outside by extending sides are vertically opposite, let them be x and y. A + x + y + angle along the produced line = 360? No. The angles around vertex A are A, and the two numbered angles there, say \u03b11 and \u03b12. There are two straight lines intersecting at A. Angle A is inside the triangle. The angles vertically opposite to the interior angles are inside the area bounded by the produced lines, let’s call them A’, B’, C’. A’=A, B’=B, C’=C. The angles numbered are those outside the triangle. At vertex A, the angles are \u03b11 and \u03b12. \u03b11 and A form a linear pair with the angle on the other side of the line, which is vertically opposite to A. No, this is getting complicated. The first method is the clearest. \u03b11 and \u03b12 are vertically opposite, so \u03b11 = \u03b12. \u03b11 and A form a linear pair only if \u03b11 is the exterior angle. Yes, if the side is produced. Angle A + exterior angle E_A = 180. The two numbered angles at A are E_A and its vertical opposite. So the sum of the two numbered angles at A is 2 * E_A = 2 * (180 – A). Similarly, at B, 2 * (180 – B), and at C, 2 * (180 – C). The total sum is 2(180-A) + 2(180-B) + 2(180-C) = 540*2 – 2(A+B+C) = 1080 – 2*180 = 1080 – 360 = 720\u00b0. This re-confirms 720.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"The following diagram shows a triangle with each of its sides produced both ways : What is the sum of degree measures of the angles numbered ? [amp_mcq option1=”720″ option2=”540″ option3=”1080″ option4=”900″ correct=”option1″] This question was previously asked in UPSC CAPF – 2009 Download PDFAttempt Online The sum of the degree measures of the six … <\/p>\n
Detailed SolutionThe following diagram shows a triangle with each of its sides produced<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1085],"tags":[1462,1102],"class_list":["post-89238","post","type-post","status-publish","format-standard","hentry","category-upsc-capf","tag-1462","tag-quantitative-aptitude-and-reasoning","no-featured-image-padding"],"yoast_head":"\nThe following diagram shows a triangle with each of its sides produced<\/title>\n<meta name=\"description\" content=\"The sum of the degree measures of the six angles numbered in the diagram is 720 degrees. - The diagram shows a triangle with each side produced both ways. This forms pairs of vertically opposite angles at each vertex, outside the triangle. - Let the interior angles of the triangle be A, B, and C. The sum of the interior angles of a triangle is 180 degrees (A + B + C = 180\u00b0). - At each vertex, the two angles formed by producing the sides are vertically opposite, and thus equal. Let these pairs be (\u03b11, \u03b12) at vertex A, (\u03b21, \u03b22) at vertex B, and (\u03b31, \u03b32) at vertex C. The numbered angles are {\u03b11, \u03b12, \u03b21, \u03b22, \u03b31, \u03b32}. - Also, each of these angles forms a linear pair with the corresponding interior angle. For example, \u03b11 + A = 180\u00b0, \u03b21 + B = 180\u00b0, \u03b31 + C = 180\u00b0. Since vertically opposite angles are equal, \u03b11 = \u03b12, \u03b21 = \u03b22, \u03b31 = \u03b32. - The sum of the numbered angles is S = \u03b11 + \u03b12 + \u03b21 + \u03b22 + \u03b31 + \u03b32 = 2(\u03b11 + \u03b21 + \u03b31). - Substituting the linear pair relationships: S = 2((180 - A) + (180 - B) + (180 - C)) = 2(540 - (A + B + C)). - Since A + B + C = 180\u00b0, S = 2(540 - 180) = 2(360) = 720\u00b0.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/the-following-diagram-shows-a-triangle-with-each-of-its-sides-produced\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The following diagram shows a triangle with each of its sides produced\" \/>\n<meta property=\"og:description\" content=\"The sum of the degree measures of the six angles numbered in the diagram is 720 degrees. - The diagram shows a triangle with each side produced both ways. This forms pairs of vertically opposite angles at each vertex, outside the triangle. - Let the interior angles of the triangle be A, B, and C. The sum of the interior angles of a triangle is 180 degrees (A + B + C = 180\u00b0). - At each vertex, the two angles formed by producing the sides are vertically opposite, and thus equal. Let these pairs be (\u03b11, \u03b12) at vertex A, (\u03b21, \u03b22) at vertex B, and (\u03b31, \u03b32) at vertex C. The numbered angles are {\u03b11, \u03b12, \u03b21, \u03b22, \u03b31, \u03b32}. - Also, each of these angles forms a linear pair with the corresponding interior angle. For example, \u03b11 + A = 180\u00b0, \u03b21 + B = 180\u00b0, \u03b31 + C = 180\u00b0. Since vertically opposite angles are equal, \u03b11 = \u03b12, \u03b21 = \u03b22, \u03b31 = \u03b32. - The sum of the numbered angles is S = \u03b11 + \u03b12 + \u03b21 + \u03b22 + \u03b31 + \u03b32 = 2(\u03b11 + \u03b21 + \u03b31). - Substituting the linear pair relationships: S = 2((180 - A) + (180 - B) + (180 - C)) = 2(540 - (A + B + C)). - Since A + B + C = 180\u00b0, S = 2(540 - 180) = 2(360) = 720\u00b0.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/the-following-diagram-shows-a-triangle-with-each-of-its-sides-produced\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T09:59:35+00:00\" \/>\n<meta name=\"author\" content=\"rawan239\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rawan239\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"The following diagram shows a triangle with each of its sides produced","description":"The sum of the degree measures of the six angles numbered in the diagram is 720 degrees. - The diagram shows a triangle with each side produced both ways. This forms pairs of vertically opposite angles at each vertex, outside the triangle. - Let the interior angles of the triangle be A, B, and C. The sum of the interior angles of a triangle is 180 degrees (A + B + C = 180\u00b0). - At each vertex, the two angles formed by producing the sides are vertically opposite, and thus equal. Let these pairs be (\u03b11, \u03b12) at vertex A, (\u03b21, \u03b22) at vertex B, and (\u03b31, \u03b32) at vertex C. The numbered angles are {\u03b11, \u03b12, \u03b21, \u03b22, \u03b31, \u03b32}. - Also, each of these angles forms a linear pair with the corresponding interior angle. For example, \u03b11 + A = 180\u00b0, \u03b21 + B = 180\u00b0, \u03b31 + C = 180\u00b0. Since vertically opposite angles are equal, \u03b11 = \u03b12, \u03b21 = \u03b22, \u03b31 = \u03b32. - The sum of the numbered angles is S = \u03b11 + \u03b12 + \u03b21 + \u03b22 + \u03b31 + \u03b32 = 2(\u03b11 + \u03b21 + \u03b31). - Substituting the linear pair relationships: S = 2((180 - A) + (180 - B) + (180 - C)) = 2(540 - (A + B + C)). - Since A + B + C = 180\u00b0, S = 2(540 - 180) = 2(360) = 720\u00b0.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/exam.pscnotes.com\/mcq\/the-following-diagram-shows-a-triangle-with-each-of-its-sides-produced\/","og_locale":"en_US","og_type":"article","og_title":"The following diagram shows a triangle with each of its sides produced","og_description":"The sum of the degree measures of the six angles numbered in the diagram is 720 degrees. - The diagram shows a triangle with each side produced both ways. This forms pairs of vertically opposite angles at each vertex, outside the triangle. - Let the interior angles of the triangle be A, B, and C. The sum of the interior angles of a triangle is 180 degrees (A + B + C = 180\u00b0). - At each vertex, the two angles formed by producing the sides are vertically opposite, and thus equal. Let these pairs be (\u03b11, \u03b12) at vertex A, (\u03b21, \u03b22) at vertex B, and (\u03b31, \u03b32) at vertex C. The numbered angles are {\u03b11, \u03b12, \u03b21, \u03b22, \u03b31, \u03b32}. - Also, each of these angles forms a linear pair with the corresponding interior angle. For example, \u03b11 + A = 180\u00b0, \u03b21 + B = 180\u00b0, \u03b31 + C = 180\u00b0. Since vertically opposite angles are equal, \u03b11 = \u03b12, \u03b21 = \u03b22, \u03b31 = \u03b32. - The sum of the numbered angles is S = \u03b11 + \u03b12 + \u03b21 + \u03b22 + \u03b31 + \u03b32 = 2(\u03b11 + \u03b21 + \u03b31). - Substituting the linear pair relationships: S = 2((180 - A) + (180 - B) + (180 - C)) = 2(540 - (A + B + C)). - Since A + B + C = 180\u00b0, S = 2(540 - 180) = 2(360) = 720\u00b0.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/the-following-diagram-shows-a-triangle-with-each-of-its-sides-produced\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T09:59:35+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"3 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/the-following-diagram-shows-a-triangle-with-each-of-its-sides-produced\/","url":"https:\/\/exam.pscnotes.com\/mcq\/the-following-diagram-shows-a-triangle-with-each-of-its-sides-produced\/","name":"The following diagram shows a triangle with each of its sides produced","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T09:59:35+00:00","dateModified":"2025-06-01T09:59:35+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"The sum of the degree measures of the six angles numbered in the diagram is 720 degrees. - The diagram shows a triangle with each side produced both ways. This forms pairs of vertically opposite angles at each vertex, outside the triangle. - Let the interior angles of the triangle be A, B, and C. The sum of the interior angles of a triangle is 180 degrees (A + B + C = 180\u00b0). - At each vertex, the two angles formed by producing the sides are vertically opposite, and thus equal. Let these pairs be (\u03b11, \u03b12) at vertex A, (\u03b21, \u03b22) at vertex B, and (\u03b31, \u03b32) at vertex C. The numbered angles are {\u03b11, \u03b12, \u03b21, \u03b22, \u03b31, \u03b32}. - Also, each of these angles forms a linear pair with the corresponding interior angle. For example, \u03b11 + A = 180\u00b0, \u03b21 + B = 180\u00b0, \u03b31 + C = 180\u00b0. Since vertically opposite angles are equal, \u03b11 = \u03b12, \u03b21 = \u03b22, \u03b31 = \u03b32. - The sum of the numbered angles is S = \u03b11 + \u03b12 + \u03b21 + \u03b22 + \u03b31 + \u03b32 = 2(\u03b11 + \u03b21 + \u03b31). - Substituting the linear pair relationships: S = 2((180 - A) + (180 - B) + (180 - C)) = 2(540 - (A + B + C)). - Since A + B + C = 180\u00b0, S = 2(540 - 180) = 2(360) = 720\u00b0.","breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/the-following-diagram-shows-a-triangle-with-each-of-its-sides-produced\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/the-following-diagram-shows-a-triangle-with-each-of-its-sides-produced\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/the-following-diagram-shows-a-triangle-with-each-of-its-sides-produced\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"UPSC CAPF","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/upsc-capf\/"},{"@type":"ListItem","position":3,"name":"The following diagram shows a triangle with each of its sides produced"}]},{"@type":"WebSite","@id":"https:\/\/exam.pscnotes.com\/mcq\/#website","url":"https:\/\/exam.pscnotes.com\/mcq\/","name":"MCQ and Quiz for Exams","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/exam.pscnotes.com\/mcq\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209","name":"rawan239","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","caption":"rawan239"},"sameAs":["https:\/\/exam.pscnotes.com"],"url":"https:\/\/exam.pscnotes.com\/mcq\/author\/rawan239\/"}]}},"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/89238","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/comments?post=89238"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/89238\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=89238"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=89238"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=89238"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}