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

Data: In an isosceles triangle ABC, with AB = AC, the bisectorts of ∠B and ∠C intersect each other at O. Join A to O. 

To Prove: 

(i) OB = OC 

(ii) AO bisects ∠A. 

Proof: 

(i) In ∆ABC, 

AB = AC 

∴ ∠ABC = ∠ACB 

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

∠OBC = ∠OCB. 

In ∆OBC 

Now, ∠OBC = ∠OCB is proved. 

∴ ∆OBC is an isosceles triangle. 

∴ OB = OC. 

(ii) In ∆AOB and ∆AOC, 

AB = AC (Data) 

OB = OC (proved) 

AO is common. 

Side, Side, Side postulate. 

∴ ∆AOB ≅ ∆AOC 

∴ ∠OAB = ∠OAC 

∴ AO bisects ∠A.