Question

In∆APQ.AO=OP=OQ.Prove that anglePOQ=2×anglePAQ.​

Answer 1

Given : ∆APQ.AO=OP=OQ.

To Find : Prove that anglePOQ=2×anglePAQ.​

Solution:

In∆APQ.AO=OP=OQ.

In∆APO AO=OP

=> ∠OAP = ∠OPA  = α

In∆AQO AO=OQ

=> ∠OAQ = ∠OQA  = β

In∆POQ PO=QO

=> ∠OPQ = ∠OQP  = γ

∠APQ = α + γ

∠AQP = γ + β

∠PAQ =β +  α

α + γ+  γ + β  + β + α = 180°

=> α + β + γ  =  90°

∠PAQ =β +  α     = 90° - γ  

∠POQ + ∠OPQ +  ∠OQP = 180°

=> ∠POQ + γ + γ  = 180°

=> ∠POQ  = 180° - 2γ

=> ∠POQ  = 2(90° - γ  )

=> ∠POQ  =  2∠PAQ

QED

Hence Proved

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