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.

Solution:

Given, ABC is an isosceles triangle with AB = AC

Bisectors of the angles B and C intersect each other at O

BO is produced to a point M

We have to prove that ∠MOC = ∠ABC

Considering triangle ABC,

Given, AB = AC
We know that the angles opposite to equal sides are equal.

∠ACB = ∠ABC

Since OB is the bisector of angle B

∠ABO = ∠OBC

∠ABC = ∠ABO + ∠OBC

∠ABC = ∠OBC + ∠OBC

∠ABC = 2∠OBC ------------ (1)

Since OC is the bisector of angle C

∠ACO = ∠OCB

∠ACB = ∠ACO + ∠OCB

∠ACB = ∠OCB + ∠OCB

∠ACB = 2∠OCB ------------- (2)

So, 2∠OCB = 2∠OBC

∠OCB = ∠OBC --------------- (3)

We know that the exterior angle of a triangle is equal to the sum of two interior angles.

∠MOC = ∠OBC + ∠OCB

From (3),

∠MOC = ∠OBC + ∠OBC

∠MOC = 2∠OBC

From (1),

Therefore, ∠MOC = ∠ABC

✦ Try This: In figure, sides QP and RQ of ΔPQR are produced to points S and T respectively. If ∠SPR=135° and ∠PQT=110°, find ∠PRQ.

☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 7

---

NCERT Exemplar Class 9 Maths Exercise 7.3 Problem 9

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

Summary:

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. It is proven that ∠MOC = ∠ABC

---

☛ Related Questions:

• Bisectors of the angles B and C of an isosceles triangle ABC with AB = AC intersect each other at O. . . . .
• In Fig. 7.8, AD is the bisector of ∠BAC. Prove that AB > BD
• In Fig. 7.9, ABC is a right triangle and right angled at B such that ∠BCA = 2 ∠BAC. Show that hypote . . . .