Question

In an isosceles triangle ABC , with AB = AC , the bisectors of angle b and C intersect each other at O. Join A to O.
Show that i) OB = OC

ii) AO bisects angle A​

Answer 1

Given :- In triangle ABC

AB = AC

BO is the bisector of angle B.

Therefore, angle 1 = angle 2

== CO is the bisector of angle C.

Therefore , angle 3 = angle 4.

To proof :- i) OB = OC

ii) AO bisects angle A.

Proof :- In triangle ABC

i) AB = AC

ii) angle B = angle C.

or angle 1 + angle 2 = angle 3 + angle 4

= (angle 1 = angle 2 and angle 3 = angle 4).

Therefore , 2 angle 2 = 2 Angle 4

=> angle 2 = angle 4

Therefore, OB = OC (sides opposite to equal angles are equal).

ii) In triangle ABO and triangle ACO.

AO = OA (common)

AB = AC (given)

OB = OC (from 3rd)

Therefore , ABO congruent ACO (SSS)

Angle 5 = angle 6 (cpct).

AO bisects angle A.