Question

In a parallelogram PQRS, the bisectors of angle P and
angle S intersect at M. What is the measure of angle PMS?​

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=180o          [ Sum of adjacent angles are supplementary ]

⇒  ∠SPO+∠OPQ+∠OQP+∠OQR=180o

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

⇒  ∠OPQ+∠OQP=90o         ---- ( 3 )

Now, in △POQ,

⇒  ∠OPQ+∠OQP+∠POQ=180o.

⇒  90o+∠POQ=180o           [ From ( 3 ) ]

⇒  ∠POQ=90o.