Question

3. In parallelogram PQRS, the bisector of ZP
intersects RS at M and bisector of ZR
intersects PQ at N. Prove that PNRM is a
parallelogram.​

Answer 1

Answer:

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

o

[ Sum of adjacent angles are supplementary ]

⇒ ∠SPO+∠OPQ+∠OQP+∠OQR=180

o

⇒ 2∠OPQ+2∠OQP=180

o

[ From ( 1 ) and ( 2 ) ]

⇒ ∠OPQ+∠OQP=90

o

---- ( 3 )

Now, in △POQ,

⇒ ∠OPQ+∠OQP+∠POQ=180

o

.

⇒ 90

o

+∠POQ=180

o

[ From ( 3 ) ]

⇒ ∠POQ=90

o

.