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)

∴\(\frac{1}{2}\) ∠ACB= \(\frac{1}{2}\) ∠ABC

∴ ∠OCB =∠OBC

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

---

(ii) In ∆OAB and ∆OAC,

 AO =AO (Common) 

AB = AC (Given) 

OB = OC (Proved above) 

Therefore, ∆OAB ∆OAC (By SSS congruence rule) 

∠BAO = ∠CAO (CPCT) 

∴ AO bisects A.