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:

1. OB = OC
2. AO bisects ∠A

Answer 1

(i) ABC is an isosceles triangle in which AB = AC

∠C = ∠B    ...[Angles opposite to equal sides in a triangle are equal.]

⇒ ∠OCA + ∠OCB = ∠OBA + ∠OBC

⇒ ∠OCB + ∠OCB = ∠OBC + ∠OBC

∵ OB bisects ∠B.

∴ ∠OBA = ∠OBC

And OC bisects ∠C.

∴ ∠OCA = ∠OCB

⇒ 2∠OCB = 2∠OBC

⇒ ∠OCB = ∠OBC

Now, in △OBC,

∠OCB = ∠OBC       ...[Proved above]

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

(ii) Now, in △AOB and △AOC,

AB = AC      ...[Given]

∠OBA = ∠OCA

∠B = ∠C

BO bisects ∠B and CO bisects ∠C.

∠OBA = ∠OCA

OB = OC

∴ △AOB ≌ △AOC      ...[By SAS congruence rule]

⇒ ∠OAB = ∠OAC       ...[Corresponding parts of congruent triangles]

So, AO bisects ∠A.