Question

24. In the figure, PQRS is square in which RM and SM are the angle bisectors of ZR and ZS. Prove that ZRMS = ZP = ZQ = ZR = ZS.
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Answer 1

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Answer 2

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PQRS is a parallelogram.

PO is angle bisector of ∠P

∴ ∠SPO=∠OPQ --- ( 1 )

QO is an angle bisector of ∠Q

∴ ∠RQO=∠OQP ---- ( 2 )

∴ PS∥QR

⇒ ∠SPQ+∠PQR=180°

[ Sum of adjacent angles are supplementary ]

⇒ ∠SPO+∠OPQ+∠OQP+∠OQR=180°

⇒ 2∠OPQ+2∠OQP=180° [ From ( 1 ) and ( 2 ) ]

⇒ ∠OPQ+∠OQP=90° ---- ( 3 )

Now, in △POQ,

⇒ ∠OPQ+∠OQP+∠POQ=180°

⇒ 90° +∠POQ=180° [ From ( 3 ) ]

⇒ ∠POQ=90°