Question

in an isosceles triangle ABC AB is equal to AC the bisector of Angle B and angle C intersect each other at O join a to show that a b is equal to AC bisects angle A​

Answer 1

Answer:  Triangle ABC is isosceles in which AB=AC also OB and OC are bisectors of angle B and angle C

To Prove: i) OB = OC ii) AO bisects ∠A

i) OB = OC

It is given that in triangle ABC,

AB = AC (given)

∠ACB = ∠ABC (Angles opposite to equal sides of an isosceles triangle are equal)

1/2 ∠ACB = 1/2 ∠ABC

⇒ ∠OCB = ∠OBC (Since OB and OC are the angle bisectors of ∠ABC and ∠ACB)

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

ii) AO bisects ∠A

In ΔOAB and ΔOAC,

AO = AO (Common)

AB = AC (Given)

OB = OC (Proved above)

Therefore,

ΔOAB ≅ ΔOAC (By SSS congruence rule)

Also, we can use an alternative approach as shown below,

∠OBA = ∠OCA (OB and OC bisects angle ∠B and ∠C)

AB = AC (Given)

OB = OC (Proved above)

ΔOAB ≅ ΔOAC (By SAS congruence rule)

⇒ ∠BAO = ∠CAO (CPCT)

∴ AO bisects ∠A or AO is the angle bisector of ∠A.