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Theorem 6.5(SAS Criteria) - Class 10th - If two sides are in ratio and an angle is equal - Theorems

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  1. Chapter 6 Class 10 Triangles
  2. Serial order wise
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Theorem 6.5 (SAS Similarity) If one angle of a triangle is equal to one angle of the other triangle and sides including these angles are proportional then the triangles are similar. Given :- Two triangles ∆ABC and ∆DEF such that ∠A = ∠D AB﷮DE ﷯ = AC﷮DF﷯ To Prove : ∆ABC ~ ∆DEF Construction :- Draw P and Q on DE & DF such that DP = AB and DQ = AC respectively and join PQ. Proof :- Given AB﷮DE ﷯ = CA﷮FD﷯ And DP = AB, DQ = AC ∴ DP﷮DE ﷯ = DQ﷮DF﷯ DE﷮DP﷯ = DF﷮DQ﷯ Subtracting 1 on both sides DE﷮DP﷯ − 1 = DF﷮DQ﷯ − 1 DF − DP﷮DP﷯ = DF − DQ﷮DQ﷯ PE﷮DP ﷯ = QF﷮DQ﷯ DP﷮PE ﷯ = DQ﷮QF﷯ Using theorem 6.2 : If a line divides any two sides of a triangle in the same ratio, then the line is parallel to third side. ∴ PQ ∥ EF. Thus, ∠P = ∠E and ∠Q = ∠F In ∆ABC and ∆DPQ AB = DP AC = DQ BC = PQ ⇒ ∆ABC ≅ ∆DPQ ∴ ∠B = ∠P ∠C = ∠Q But From (1) ∠P = ∠E and ∠Q = ∠F Therefore, ∠B = ∠P = ∠E and ∠C = ∠Q = ∠F Therefore, In Δ ABC & Δ DEF ∠B = ∠E ∠C = ∠F ∴ ∆ABC ~ ∆DEF Hence Proved

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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