Question

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

Answer 1

Solution:-
In Δ ABC,
∠ BAD = ∠ CAD
BD = CD
AD bisects AC
Extend AD to point E such that CE is parallel to AB. Join CE.
Taking transversal AE intersecting AB and CE.
∠ BAD = ∠ DEC (interior alternate angles)
We know, ∠ BAD = ∠ CAD =  ∠ DEC
In triangle ACE,
∠ CAD = ∠ DEC 
Therefore,
AC = CE
In triangles ABD and ECD,
∠ ADB = ∠ EDC (vertically opposite angles)
∠ BAD = ∠ CED (interior alternate angles)
BD = CD (Given)
By AAS, triangle ABD is congruent to triangle ECD
Therefore, AB = EC (CPCT)
We know CE = AC (already proven)
Therefore, AB = AC and triangle ABC is an isosceles triangle.
Hence proved.