Question

The bisectors of angle B and angle C of an isosceles triangle with AB equal to AC intersects each other at a point O.BO is produced to meet AC at a point M prove that angle MOC equal angle ABC​

Answer 1

Given:

Lines OB and OC are the bisectors of ∠B and ∠C of an isosceles ΔABC such that AB=AC which intersect each other at O and BO is produced to M.

To prove:

∠MOC=∠ABC

Consider the diagram shown below.

Proof:

In ΔABC,

AB=AC (given)

∠ACB=∠ABC (angles opposite to equal sides are equal)

∠ACB= ∠ABC (dividing both sides by 2)

Therefore,

∠OCB=∠OBC …… (1)

(Since, OB and OC are the bisector of ∠B and ∠C)

Now, from equation (1), we have

∠MOC=∠OBC+∠OBC

∠MOC=∠OBC

⇒∠MOC=2∠ABC

(Since, OB is the bisector of ∠B)

Hence, proved.