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

Given that in ΔABC AB = AC

Bisectors of ∠B and ∠C meet at ‘O’.

To prove 

i) OB = OC 

∠B = ∠C (Angles opposite to equal, sides)

1/2 ∠B =1/2 ∠C (Dividing both sides by 2)

∠OBC = ∠OCB

⇒ OB = OC (∵ Sides opposite to equal angles in ΔOBC)

ii) AO bisects ∠A. 

In ΔAOB and ΔAOC 

AB = AC (given) 

BO = CO (already proved) 

∠ABO = ∠ACO (∵ ∠B =∠C) 

∴ ΔAOB ≅ ΔAOC 

⇒ ∠BAO = ∠CAO [ ∵ CPCT of ΔAOB and ΔAOC] 

∴ AO is bisector of ∠A.