Question

If the bisector of the pair of vertical angle of a triangle bisects the base prove that the triangle is isoceles

Answer 1

extend AD to E
AD=DE
Join E to C

In ΔABD and ΔECB 
BD=DC
AD=DE
∠ADB=∠EDC
so, ΔABD  is congurent to ΔECB

AB=EC
and ∠1=∠3
also ∠1=∠2
so, ∠2=∠3

and EC= AC
as AB=AC

so, ΔABC is isosceles