Question

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 1

(i) It is given that in triangle ABC, AB = AC

∴ ∠ACB = ∠ABC (Angles opposite to equal sides of a triangle are equal)

∴ 1/2 ∠ACB = 1/2 ∠ABC

∴ ∠OCB = ∠OBC

∴ OB = OC (Sides opposite to equal angles of a triangle are also equal) 

(ii) In ΔOAB and ΔOAC,

AO =AO (Common) 

AB = AC (Given)

OB = OC (Proved above)

Therefore, ΔOAB ≅ ΔOAC (By SSS congruence rule)

∴ ∠BAO = ∠CAO (CPCT)

∴ AO bisects ∠A.