Question

Bisectors of the angles B and C of an isosceles triangle with AB = AC intersect each other at O. BO is produced to a point M. Prove that ∠MOC = ∠ABC.

Answer 1

Given in the question, bisectors of the angles B and C of an isosceles triangle ABC with AB = AC intersect each other at O. Now BO is produced to a point M.

In triangle ABC,

AB = AC  

∠ABC = ∠ACB   ...[Angle opposite to equal sides of a triangle are equal]

`1/2 ∠ABC = 1/2 ∠ACB`

That is ∠1 = ∠2  ...[Since, BO and CO are bisectors of ∠B and ∠C]

In triangle OBC,

Exterior ∠MOC = ∠1 + ∠2  ...[Exterior angle of a triangle is equal to the sum of interior opposite angles]

Exterior ∠MOC = 2∠1   ...[∠1 = ∠2]

Hence, ∠MOC = ∠ABC.