Q. In figure 10, QPR = PQR and M and N are respectively points on sides QR and PR of PQR, such that QM = PN. Prove that OP = OQ, where O is the point of intersection of PM and QN.

Dear Student,

Given: QPR =  PQR, QM = PNTo Prove: OP = OQProof: In PQR,QPR =  PQR RP = RQ                                [Sides opposite equal angles of a triangle are equal to each other]      ...i PN + NR = RM + MQ        [ QM = PN] NR = RM                                 ...iiIn RQN and RPM,QRN = PRM                         [Common]RP = RQ                                       [From i]RM = NR                                       [From ii] By SAS congruence criterion,RQN  RPM corresponding parts of congruent triangles are congruent (cpct), RPM = RQN RPQ - MPQ = RQP - NQP                      [ RPQ = RQP] MPQ = NQP                                                             ...iii In  OPQ,OPQ = OQP                                                                  [From iii] OP = OQ                                                                          [ Sides opposite equal angles of a triangle are equal]Hence, proved 
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