Refer to the image attached.
Given: and are congruent.
To Prove: is an isosceles triangle.
Construction: Construct a perpendicular bisector from point B to line segment AC. Label the point of intersection between this perpendicular bisector and line segment AC as point D.
Proof:
Consider
(By the definition of perpendicular bisector)
(By the definition of perpendicular bisector)
So, Line segment AD is congruent to DC by the definition of perpendicular bisector.
= (given)
So, by ASA congruence postulate.
∆BAD is congruent to ∆BCD by the ASA congruence Postulate.
Line segment AB is congruent to line segment BC because corresponding parts of congruent triangles are congruent (CPCTC).
So, Option C is the correct answer.