Question

If a ray OC stands on line AB such that angleAOC =angleCOB ,show that angleAOC=90

Answer 1

Answer:

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

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.