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Lines and Angles
Question
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In figure, OP bisects ∠AOC, OQ bisects ∠BOC and OP ⊥ OQ. Show that the points A, O and B are collinear.

Answer

Given: OP bisects ∠AOC, OQ bisects ∠BOC and OP ⊥ OQ.
To Prove: The points A, O and B are collinear.
Proof: ∵ OP bisects ∠AOC
∴ ∠AOP = ∠COP    ...(1)
∵ OQ bisects ∠BOC
∠BOQ = ∠COQ    ...(2)
Now, ∠AOB
= ∠AOP + ∠COP + ∠COQ + ∠BOQ
= ∠COP + ∠COP + ∠COQ + ∠COQ
| From (1) and (2)
= 2(∠COP + ∠COQ)
= 2 ∠POQ
= 2(90°)    | ∵ OP ⊥ OQ
= 180°
∴ The points A, O and B are collinear.
| By converse of Linear Pair Axiom

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