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Vk Academy, 10th Triangles Notes, CBSE Class 10 Maths Chapter 6 Notes Triangles, , Triangles Class 10 Notes Understanding the Lesson, , In X standard we have learnt about congruent figures., , Congruent figure: Those two geometric figures having the same shape and size are known as, congruent figures., , Rules of Congruency, , 1. SAS (Side-Angle-Side): Two triangles are congruent if two sides and the included angle of one, triangle are equal to the two sides and the included angle of the other triangle., , 2. ASA (Angle-Side-Angle): Two triangles are congruent if two angles and the included side of one, triangle are equal to two angles and the included side of other triangle., , 3. AAS (Angle-Angle-Side): Two triangles are Congruent if any two pairs of angles and a pair of, corresponding sides are equal., , 4. SSS (Side-Side-Side): If three sides of one triangle are equal to the three sides of another triangle,, then the two triangles are congruent., , 5. RHS (Right angle-Hypotenuse-Side: In two right angle triangles, the hypotenuse and one side of one, triangle are equal to the hypotenuse and one side of the other triangle, then the two triangles are, congruent., , Note: All congruent figures or triangles are similar., , Similar Figure: Two figure which are of same shape (but not necessarily the same size) are called, similar figures, For example,, , All line segments are similar., , All circles are similar., , « Two or more squares are similar., , « Two or more equilateral triangles are similar., , Note:, , « All rectangles are not similar., « Alltriangles are not similar.
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Similar Polygons: Two polygons with the same number of sides are similar, if (1) their corresponding, angles are equal. (2) their corresponding sides in the same ratio., , Similarity of Triangles, , Two triangles are similar if, , ¢ Their corresponding angles are equal; and, ¢ Their corresponding sides are in the same ratio, , Famous Greek mathematician Militus Thales gives the relation to the two equiangular triangle is known, as BPT or Thales theorem., , Equiangular triangles: If corresponding angles of two triangles are equal then they are equiangular, triangles, , Theorem 6.1: If a line is drawn parallel to one side of a triangle to intersect the other two sides in, distinct points, the other two sides are divided in the same ratio., , Given: A AABC in which a line DE || BC intersects the other two sides AB and AC at D and E respectively., , AD _ AE, To Prove that 5a = =C, , Construction: Join BE and CD and draw DM -L AC and EN -L AB, , , , 1, arAADE 9 ADSEN, Proof: arABDE ~ 1., 4., arABDE * DBxEN, arAADE AD, Therefore arABDE ~ DB (1), 1, N arAADE 9 AEX DM, Ow es ea, arADEC 1lioy pm, arA ADE AE, Therefore arADEC * EG ++(2), ar A BDE = ar A DEC ++(3), , (Because both are on the same base DE and between the same parallels BC and DE) from eqn (1), (2), and (3) AD AE, , AD _ AE, DB EC, AD AE, Note: We have, DB EC’, f DB _ EC, we can write AD = AE, Adding 1 on both sides, > DE. +1l= EC +1
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DB+AD EC+AE, , , , we can write, , zis Ble ©, > >, ale ale, , Theorem 6.2: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to, the third side., , iven: AD _ AE, AD AE Given: In AABC, DB — Ec, To prove: DE || BC, , Construction: Let us suppose that DE is not parallel to BC, so we draw a line DF || BC, Proof: DF || BC, Therefore by Basic Proportionality Theorem,, , , , AD AF, DB" FC -(1), AD AE ‘, But DB" EC , (given) ...(2), : AF AE, From equation (1) and (2) FC ~ EC, AF AE, ~41=s5t1, ; Fo’. EC D E, Adding 1 on both sides AF+FC AE+EC, FC ~ EC, AC _ AC, or FC ~ EC . S, or EC =FC, , which is not possible. We come at the contradiction. So our supposition was wrong, it is only possible,, if point F will coincide the point E., Therefore DE || BC., , Criteria for Similarity of Triangles, , In previous section, we have studied that two triangles are similar, if (|) their corresponding angles are, similar (II) their corresponding sides are proportional (or are in the same ratio)., , Theorem 6.3: AAA Criterion: If in two triangles, corresponding angles are equal, then their, corresponding sides are In the same ratio (or proportion) and hence the two triangles are similar., , In AABC and ADEF,
