Q. 14.3( 533 Votes )

# In an isosceles triangle ABC, with AB = AC, the bisectors of ∠ B and ∠ C intersect each other at O. Join A to O. Show that:

(i) OB = OC

(ii) AO bisects ∠ A

Answer :

**Given:** AB = AC

The bisectors of ∠B and ∠C intersect each other at O

**(i)** **To Prove** = OB = OC**Proof:**

ABC is an isosceles with AB = AC

Therefore,

∠B = ∠C

Multiplying both sides by 1/2, we get,

∠B = ∠C (Bisectors of angles are also be equal)

∠OBC = ∠OCB (Angles bisectors)

OB = OC (Side opposite to the equal angles are equal)

**(ii)**

To Prove: AO bisects angle A

Proof:

In

AB = AC (Given)

AO = AO (Common)

OB = OC (Proved above)

Therefore,

By SSS congruence rule

∠BAO = ∠CAO (By c.p.c.t)

Thus,

**AO bisects ∠A**

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