In ∆abc shown below, ∡bac is congruent to ∡bca. given: base ∡bac and ∡acb are congruent.
prove: ∆abc is an isosceles triangle.

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.
m∡bda and m∡bdc is 90° by the definition of a perpendicular bisector.
∡bda is congruent to ∡bdc by the definition of congruent angles. line segment ad is congruent to line segment dc by
∆bad is congruent to ∆bcd by the line segment ab is congruent to line segment bc because corresponding parts of congruent triangles are congruent (cpctc).
consequently, ∆abc is isosceles by definition of an isosceles triangle.

options are:
a)1. angle-side-angle (asa) postulate
2. corresponding parts of congruent triangles are congruent (cpctc)

b) 1. corresponding parts of congruent triangles are congruent (cpctc)
2. angle-side-angle (asa) postulate

c) 1. the definition of a perpendicular bisector
2. angle-side-angle (asa) postulate

d) 1. corresponding parts of congruent triangles are congruent (cpctc)
2. the definition of a perpendicular bisector