Question

In the parallelogram PQRS the bisector of angle p and angle q meet SR at O Show that

angle POQ =900​

Answer 1

Step-by-step explanation:

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 degree    [Sum of adjacent angles are supplementary]

⇒  ∠SPO+∠OPQ+∠OQP+∠OQR=180 degree

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

⇒  ∠OPQ+∠OQP=90 degree          ---- ( 3 )

Now, in △POQ,

⇒  ∠OPQ+∠OQP+∠POQ=180 degree

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

⇒  ∠POQ=90 degree

∴ LHS = RHS

Therefore, Proved

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