{"id":64387,"date":"2020-11-06T22:37:32","date_gmt":"2020-11-06T17:07:32","guid":{"rendered":"https:\/\/www.oliveboard.in\/blog\/?p=64387"},"modified":"2021-02-25T09:47:26","modified_gmt":"2021-02-25T04:17:26","slug":"congruence-and-similarity-of-triangles-for-ssc","status":"publish","type":"post","link":"https:\/\/www.oliveboard.in\/blog\/congruence-and-similarity-of-triangles-for-ssc\/","title":{"rendered":"Quantitative Aptitude for SSC: Congruence and similarity of triangles"},"content":{"rendered":"<p>Geometry is an important topic of Quantitative Aptitude in SSC. To become proficient in this topic, a candidate should develop a good command over the basics, theorems, and applications. In this blog, we will be discussing about the congruence and similarity of triangles for SSC which is a subtopic of Geometry for SSC.<\/p>\n<h2><strong>Congruence and similarity of triangles for SSC<\/strong><\/h2>\n<ul>\n<li>Two triangles are said to be congruent if they have the same shape and size. In other words, two triangles are congruent if their corresponding sides and corresponding angles are equal.<\/li>\n<li>Two triangles are said to be similar if they have same shape. In other words, two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional.<\/li>\n<\/ul>\n<p><strong>\u00a0<\/strong><strong>CPCTC: <\/strong>Corresponding Parts of Congruent Triangles are Congruent<\/p>\n<p><a href=\"https:\/\/www.oliveboard.in\/ssc-cgl\/?ref=b-06-11-cmm2&amp;next= https:\/\/practice.oliveboard.in\/banking2\/ssctopictests.php\"><strong>Ace the Numerical Ability section with the help of Oliveboard<\/strong><\/a><\/p>\n<h3><strong>Conditions for Congruence of Triangles<\/strong><\/h3>\n<h4><strong style=\"font-size: 16px;\">1. SSS (Side-Side-Side): <\/strong><\/h4>\n<p><span style=\"font-size: 16px;\">Corresponding sides are equal.<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-64389\" src=\"https:\/\/www.oliveboard.in\/blog\/wp-content\/uploads\/2020\/11\/sss.png\" alt=\"\" width=\"291\" height=\"174\" \/><\/p>\n<p>In \u25b3ABC and \u25b3DEF,<\/p>\n<p>AB = FE, BC = ED and DF = CA<\/p>\n<p>Hence \u25b3ABC <strong>\u2245<\/strong> \u25b3FED<\/p>\n<p><a href=\"https:\/\/www.oliveboard.in\/ssc-cgl\/?ref=b-06-11-cmm2\"><strong>Attempt a FREE SSC mock test here.<\/strong><\/a><\/p>\n<h4><strong>2. SAS (Side-Angle-Side)<\/strong>:<\/h4>\n<p>Two corresponding sides and the corresponding angle between them are equal.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-64390\" src=\"https:\/\/www.oliveboard.in\/blog\/wp-content\/uploads\/2020\/11\/sas.png\" alt=\"\" width=\"286\" height=\"173\" \/><\/p>\n<p>In \u25b3ABC and \u25b3DEF,<\/p>\n<p>AB = FE, DF = CA and \u2220A = \u2220F<\/p>\n<p>Hence \u25b3ABC <strong>\u2245<\/strong> \u25b3FED<\/p>\n<h4><strong>3. AAS(Angle-Angle-Side):<\/strong><\/h4>\n<p>Two corresponding angles and the corresponding side which is not between them are equal.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-64391\" src=\"https:\/\/www.oliveboard.in\/blog\/wp-content\/uploads\/2020\/11\/aas.png\" alt=\"\" width=\"285\" height=\"180\" \/><\/p>\n<p>In \u25b3ABC and \u25b3DEF,<\/p>\n<p>AB = FE, \u2220C = \u2220D and \u2220A = \u2220F<\/p>\n<p>Hence \u25b3ABC <strong>\u2245<\/strong> \u25b3FED<\/p>\n<p><a href=\"https:\/\/www.oliveboard.in\/ssc-chsl\/?ref=b-06-11-cmm2\"><strong>Attempt a FREE SSC CHSL mock teste here<\/strong><\/a><\/p>\n<h4><strong>4. ASA (Angle-Side-Angle):<\/strong><\/h4>\n<p>Two corresponding angles and the corresponding side between them are equal.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-64392\" src=\"https:\/\/www.oliveboard.in\/blog\/wp-content\/uploads\/2020\/11\/asa.png\" alt=\"\" width=\"279\" height=\"197\" \/><\/p>\n<p>In \u25b3ABC and \u25b3DEF,<\/p>\n<p>AC = FD, \u2220C = \u2220D and \u2220A = \u2220F<\/p>\n<p>Hence \u25b3ABC <strong>\u2245<\/strong> \u25b3FED<\/p>\n<h4><strong>5. RHS (Right-Hypotenuse-Side):<\/strong><\/h4>\n<p>Hypotenuse and a pair of corresponding angles are equal in two right triangles.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-64393\" src=\"https:\/\/www.oliveboard.in\/blog\/wp-content\/uploads\/2020\/11\/rhs.png\" alt=\"\" width=\"297\" height=\"188\" \/><\/p>\n<p>In \u25b3ABC and \u25b3DEF,<\/p>\n<p>\u2220B = \u2220E = 90<sup>0<\/sup>, AC = DF and BC = EF<\/p>\n<p>Hence \u25b3ABC <strong>\u2245<\/strong> \u25b3DEF<\/p>\n<p><a href=\"https:\/\/www.oliveboard.in\/ssc-cgl\/?ref=b-06-11-cmm2&amp;next= https:\/\/practice.oliveboard.in\/banking2\/ssctopictests.php\"><strong>Practice questions of Geometry for SSC exams, click here.<\/strong><\/a><\/p>\n<h3><strong>Conditions for Similarity of Triangles<\/strong><\/h3>\n<h4><strong style=\"font-size: 16px;\">1. AA (Angle-Angle):<\/strong><\/h4>\n<p>Two pairs of corresponding angles are equal.<\/p>\n<h4><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-64394\" src=\"https:\/\/www.oliveboard.in\/blog\/wp-content\/uploads\/2020\/11\/aa.png\" alt=\"\" width=\"262\" height=\"190\" \/><\/h4>\n<p>\u25b3ABC and \u25b3FDE are similar<\/p>\n<h4><strong>2. SAS (Side-Angle-Side):<\/strong><\/h4>\n<p>Two pairs of corresponding sides are proportional and the angle between them is equal.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-64395\" src=\"https:\/\/www.oliveboard.in\/blog\/wp-content\/uploads\/2020\/11\/sas2.png\" alt=\"\" width=\"253\" height=\"180\" \/><\/p>\n<p>\u25b3ABC and \u25b3DEF are similar<\/p>\n<h4><strong>3. SSS (Side-Side-Side):<\/strong><\/h4>\n<p>Corresponding sides are proportional.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-64396\" src=\"https:\/\/www.oliveboard.in\/blog\/wp-content\/uploads\/2020\/11\/sss2.png\" alt=\"\" width=\"268\" height=\"185\" \/><\/p>\n<p>AB\/FD = BC\/DE = AC\/FE<\/p>\n<p>\u25b3ABC and \u25b3FDE are similar<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Note:<\/strong> If two triangles are similar then,<\/p>\n<p>1. Ratio of area of the two triangles = Ratio of squares of corresponding sides<\/p>\n<p>2. Ratio of sides of two triangles<\/p>\n<p>= Ratio of their heights\/altitudes<\/p>\n<p>= Ratio of their medians<\/p>\n<p>= Ratio of their angle bisectors<\/p>\n<p>= Ratio of their in-radii\/circum-radii<\/p>\n<p>= Ratio of their perimeters<\/p>\n<p>The conditions given above form the backbone for the preparation Congruence and similarity of triangles for SSC exams. Learning them is a must! Let us look at some theorems based on Congruence and similarity of triangles for SSC exams.<\/p>\n<p><a href=\"https:\/\/www.oliveboard.in\/ssc-cgl\/?ref=b-06-11-cmm2&amp;next= https:\/\/practice.oliveboard.in\/banking2\/ssctopictests.php\"><strong>Ace the Numerical Ability section with the help of Oliveboard<\/strong><\/a><\/p>\n<h3><strong>Congruence and similarity of triangles for SSC: Some Important Theorems<\/strong><\/h3>\n<h4><strong style=\"font-size: 16px;\">1. Basic Proportionality Theorem:<\/strong><\/h4>\n<p>A line parallel to a side of a triangle divides the other two sides in the same ratio.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-64398\" src=\"https:\/\/www.oliveboard.in\/blog\/wp-content\/uploads\/2020\/11\/bpt.png\" alt=\"\" width=\"203\" height=\"166\" \/><\/p>\n<p>If DE \u2502\u2502 BC then,<\/p>\n<p>AD\/DB = AE\/EC<strong>\u00a0<\/strong><\/p>\n<p><strong>Example:<\/strong> Area of quadrilateral DECB is 180 cm<sup>2 <\/sup>and DE divides AC in the ratio 2:5. If DE \u2502\u2502 BC, what is the area of \u25b3ADE?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-64398\" src=\"https:\/\/www.oliveboard.in\/blog\/wp-content\/uploads\/2020\/11\/bpt.png\" alt=\"\" width=\"203\" height=\"166\" \/><\/p>\n<p>a) 16 cm<sup>2<\/sup><\/p>\n<p>b) 32 cm<sup>2<\/sup><\/p>\n<p>c) 40 cm<sup>2<\/sup><\/p>\n<p>d) 20 cm<sup>2<\/sup><\/p>\n<p>Answer key: a<\/p>\n<p>Solution:<\/p>\n<p>AE\/EC = 2\/5 <strong>(<\/strong>DE divides AC in the ratio 2:5)<\/p>\n<p>AE\/EC = AD\/DB (Basic Proportionality Theorem)<\/p>\n<p>=&gt; AE\/EC = AD\/AB = 2\/7 and \u2220DAE =\u2220BAC<\/p>\n<p>=&gt; \u25b3ADE and \u25b3ABC are similar. (By SAS Similarity Rule)<\/p>\n<p>Area of \u25b3ADE \/ Area of \u25b3ABC = (2\/7)<sup>2 <\/sup>= 4\/49<\/p>\n<p>Let area of \u25b3ADE = 4x then, area of \u25b3ABC = 49 x<\/p>\n<p>Area of \u25b3ABC = Area of \u25b3ADE + Area of quadrilateral DCEB<\/p>\n<p>49 x = 4x + 180<\/p>\n<p>45 x = 180<\/p>\n<p>x = 4<\/p>\n<p>Area of \u25b3ADE = 4 x = 16 cm<sup>2<\/sup><\/p>\n<p><a href=\"https:\/\/www.oliveboard.in\/ssc-cgl\/?ref=b-06-11-cmm2\"><strong>Attempt a FREE SSC CGL mock test here.<\/strong><\/a><\/p>\n<h4><strong>2. Angle Bisector Theorem<\/strong>:<\/h4>\n<p>The angle bisector divides the side in the ratio of remaining two sides.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-64399\" src=\"https:\/\/www.oliveboard.in\/blog\/wp-content\/uploads\/2020\/11\/abt.png\" alt=\"\" width=\"221\" height=\"183\" \/><\/p>\n<p>If AD bisects \u2220A then, BD\/DC = AB\/AC<\/p>\n<p><strong>Example:<\/strong> If AD bisects \u2220BAC and AD divides BC in the ratio 1:1 then, the ratio of area of \u25b3ABC to the area of \u25b3ABD is:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-64399\" src=\"https:\/\/www.oliveboard.in\/blog\/wp-content\/uploads\/2020\/11\/abt.png\" alt=\"\" width=\"221\" height=\"183\" \/><\/p>\n<p>a) 3:1<\/p>\n<p>b) 4:1<\/p>\n<p>c) 3:2<\/p>\n<p>d) 2:1<\/p>\n<p>Answer key: d<\/p>\n<p>Solution:<\/p>\n<p>BD\/DC = AB\/AC = 1<\/p>\n<p>=&gt; AB = AC and BD = DC<\/p>\n<p>Also, AD = AD (common side)<\/p>\n<p>\u25b3ABD <strong>\u2245<\/strong> \u25b3ACD (By SSS Congruence Rule)<\/p>\n<p>Area of \u25b3ABC = Area of \u25b3ABD + Area of \u25b3ACD<\/p>\n<p>= 2 Area of \u25b3ABD (Since, area of congruent triangles are equal)<\/p>\n<p>Required ratio = 2:1<\/p>\n<p><a href=\"https:\/\/www.oliveboard.in\/ssc-cgl\/?ref=b-06-11-cmm2&amp;next= https:\/\/practice.oliveboard.in\/banking2\/ssctopictests.php\">Practice more questions on Congruence and similarity of triangles for SSC<\/a> to become proficient in this topic. Also, don\u2019t forget to solve previous year question papers.<\/p>\n<p>All the best for your examination!<\/p>\n<p><a href=\"https:\/\/www.oliveboard.in\/bank-exams\/?ref=b-06-11-cmm2\"><strong>Crack any SSC exam or Railways exam- Register with Oliveboard now!<\/strong><\/a><\/p>\n<p><a href=\"https:\/\/play.google.com\/store\/apps\/details?id=in.oliveboard.prep\"><strong>Study at your own convenience anywhere. Download the Oliveboard app now!<\/strong><\/a><\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Geometry is an important topic of Quantitative Aptitude in SSC. To become proficient in this topic, a candidate should develop<\/p>\n","protected":false},"author":2,"featured_media":64400,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,2151,5217,10231,63],"tags":[],"class_list":["post-64387","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-articles","category-quiz_test","category-ssc-prep","category-ssc-railways","category-ssc-exams","generate-columns","tablet-grid-50","mobile-grid-100","grid-parent","grid-50"],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v26.6 (Yoast SEO v26.6) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Congruence and similarity of triangles for SSC Quantitative Aptitude<\/title>\n<meta name=\"description\" content=\"In this blog, we will be look at the congruence and similarity of triangles for SSC along with few important theorems and some examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.oliveboard.in\/blog\/congruence-and-similarity-of-triangles-for-ssc\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Quantitative Aptitude for SSC: Congruence and similarity of triangles\" \/>\n<meta property=\"og:description\" content=\"Geometry is an important topic of Quantitative Aptitude in SSC. 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