Given that the bisector of exterior vertical angle of a triangle is parallel to the base and we have to prove that the triangle is isosceles.

Let, ABC be a triangle such that AD is the angular bisector of exterior vertical angle EAC and AD ‖ BC

Let, ∠EAD = 1

∠DAC = 2

∠ABC = 3

∠ACB = 4

We have,

1 = 2 (Therefore, AD is the bisector of ∠EAC)

1 = 3 (Corresponding angles)

And,

2 = 4 (Alternate angles)

3 = 4 = AB = AC

Since, in two sides AB and AC are equal we can say that is isosceles.