In the given picture,  is an isosceles triangle with   and AD bisects the exterior angle  A.  Prove that

The correct answer is: Hence proved

In  ABC,
AB =  AE (given)

Then ,∠ B = ∠ C (angles opposite to equal sides are equal )
We  know that in a triangle, the exterior angle is always equal to the sum of the interior opposite angles.

∠ B  +  ∠ C = ∠ EAC
Substitute ∠B  =  ∠C we get,

2∠C =  ∠ EAC

But 2 ∠ ADC  =  ∠ EAC (AD is the angular bisector)

2 ∠C =  2 ∠ ADC

∠ C = ∠ ADC
As we know , ∠ C and  ∠ ADC are alternate interior angles .
If alternate interior angles are equal then the lines are parallel.

So, AD || BC
Hence proved