InterviewSolution
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InterviewSolution
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State AAA similarty creteripn |
| Answer» AAA similarity theorem or criterion:If the corresponding angles of two triangles are equal, then their corresponding sides are proportional and the triangles are similarIn ΔABC and ΔPQR,\xa0∠A = ∠P , ∠B = ∠Q , and\xa0∠C =\xa0∠R then AB PQ\xa0= BC QR\xa0= ACPRand ΔABC ∼ ΔPQR.Given: In\xa0ΔABC and\xa0ΔPQR,\xa0∠A =\xa0∠P,\xa0∠B =\xa0∠Q,\xa0∠C =\xa0∠R.To prove: AB PQ\xa0= BC QR\xa0= ACPRConstruction : Draw LM such that PL AB\xa0=\xa0PM AC\xa0.Proof: In\xa0ΔABC and\xa0ΔPLM,AB = PL and AC = PM (By Contruction)∠BAC =\xa0∠LPM (Given)∴\xa0ΔABC\xa0≅\xa0ΔPLM (SAS congruence rule)∠B =\xa0∠L (Corresponding angles of congruent triangles)Hence\xa0∠B =\xa0∠Q (Given)∴\xa0∠L = ∠Q\xa0LQ is a transversal to LM and QR.Hence ∠L = ∠Q\xa0(Proved)∴ LM\xa0∥ QR PL LQ\xa0= PM MR\xa0 LQ PL\xa0= MR PM (Taking reciprocals) LQ PL\xa0+ 1 = MR PM\xa0+ 1 (Adding 1 to both sides) LQ+PL PL\xa0= MR+PM PM\xa0 PQ PL\xa0= PR PM PQ AB\xa0= PR AC (AB = PL and AC =PM) AB PQ\xa0= AC PR (Taking Reciprocals) ............... (1) AB PQ\xa0= BC QR AB PQ\xa0= AC PR\xa0= BC QR\xa0∴\xa0ΔABC\xa0~\xa0ΔPQR | |