If the bisector of the exterior vertical angle of a triangle . . .

Question : If the bisector of the exterior vertical angle of a triangle is parallel to the base, show that the triangle is isosceles.

Doubt by Veer

Solution : 

Given : AD is the angle bisector of ∠ EAC
AD|| BC

To Prove : ABC is an isosceles triangle. 

Proof : 

∠1=∠2 (AD is angle bisector) — (1) 
∠1=∠3 (Corresponding Angles) — (2) 
∠2=∠4 (Alternate Interior Angles) — (3) 

From equations (1) , (2)  and (3) 

∠3=∠4

AB=AC (Sides opposite to equal angles of a triangle are equal)

⇒ABC is an isosceles triangle. 

Hence Proved.