Question

If a triangle is isosceles, then prove that two angle bisectors drawn from vertices at the base to the sides are of equal length.

Answer 1

Considering your question to be:
If a triangle is isosceles, then the two angle bisectors drawn from vertices at the base to the sides are of equal length.

Proof
Let ABC be an isosceles triangle with sides AC and BC of equal length (Figure).
We need to prove that the angle bisectors AD and BE are of equal length. Note that the angles ABC and BAC are congruent as the angles at the base of the isosceles triangle ABC.
This fact was proved in the lesson Isosceles triangles under the topic Triangles in the section Geometry in this site.
This implies that the angles ABE and BAD are congruent as halves of the angles ABC and BAC.
Now consider the triangles ABD and BAE. They have the congruent angles ABD and BAE, as well as the congruent
angles ABE and BAD.
These congruent angles include the common side AB.Hence, the triangles ABD and BAE are congruent in accordance to ASA Congruence condition.
Therefore, the angle bisectors AD and BE are of equal length as the corresponding sides of these triangles.