Question

prove that angle opposite to equal side of an isosceles triangle are equal

Answer 1

Since angle C is bisected, 

angle (x) = angle (y)

Segment AC = segment BC ( This one was given)

Segment CF = segment CF (Common side is the same for both triangle ACF and triangle BCF)

Triangles ACF and triangle BCF are then congruent by SAS or side-angle-side 

In other words, by 

AC-angle(x)-CF 

and 

BC-angle(y)-CF 

Since triange ACF and triangle BCF are congruent, angle A = angle B

Answer 2

consider triangle ABC

AB=AC

draw a perpendicular from A to BC

AB=AC

AD=AD

angle ADB=angle ADC=90⁰

according to hypotenuse leg axiom triangles ABD,ADC are congruent

∴ angle ABD=angle ACD