In the figure, if AOB is a line, OP bisects ∠ BOC and OQ bisects ∠ AOC, show that
∠ POQ is a right angle.
Ray OP bisects ∠ BOC
... ∠ COP = ∠ POB
... ∠ COP = 1/2 ∠ BOC ------------(1)
Ray OQ bisects ∠ AOC
... ∠ AOQ = ∠ QOC
∠ QOC = 1/2 ∠ AOC -----------(2)
Adding (1) and (2)
... ∠COP + ∠ QOC = 1/2 [ ∠BOC + ∠ AOC ] (∠AOC and ∠ BOC form a linear pair.)
= 1/2 x 180° (Linear Pair Axiom)
∠QOP = 90°
... ∠POQ is a right angle.
... ∠ COP = ∠ POB
... ∠ COP = 1/2 ∠ BOC ------------(1)
Ray OQ bisects ∠ AOC
... ∠ AOQ = ∠ QOC
∠ QOC = 1/2 ∠ AOC -----------(2)
Adding (1) and (2)
... ∠COP + ∠ QOC = 1/2 [ ∠BOC + ∠ AOC ] (∠AOC and ∠ BOC form a linear pair.)
= 1/2 x 180° (Linear Pair Axiom)
∠QOP = 90°
... ∠POQ is a right angle.
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