# 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

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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