Question

If the bisector of the exterior vertical angle of a triangle be parallel to the base. Show that the triangle is isosce

Answer 1

Given that the bisector of the 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) and∠ACB = (4)

(1) = (2)        [   ∵ AD is bisector of , ∠EAC ] 

(1)=(3)       [Corresponding angles]

and (2) = (4)    [alternative angle] 

   ⇒ (3) = (4) ⇒AB = AC

Since, in ΔABC, two sides AB and AC are equal we can say that

DABC is isosceles