In △ ABC, AB = AC and the bisectors of angles B and C intersect at point O. Prove that :

(i) BO = CO

(ii) AO bisects angle BAC.

(i) In Δ ABC,

AB = AC (Given)

⇒ ∠B = ∠C [Angles opposite to equal sides are equal]

Also OB and OC are bisectors of angles B and C.

⇒ ∠OBC = ∠OCB

∴ OB = OC [Sides opposite to equal angles are equal]

Hence, proved that BO = CO.

(ii) In Δ AOB and Δ AOC,

⇒ OA = OA (Common side)

⇒ AB = AC (Given)

⇒ OB = OC (Proved above)

∴ Δ AOB ≅ Δ AOC (By S.S.S. axiom)

We know that,

Corresponding parts of congruent triangles are equal.

∴ ∠OAB = ∠OAC

Hence, proved that OA is bisector ∠A.