In δabc shown below, ∠bac is congruent to ∠bca:
triangle abc, where angles a and c are congruent

given: base ∠bac and ∠acb are congruent.

prove: δabc is an isosceles triangle.

when completed (fill in the blanks), the following paragraph proves that line segment ab is congruent to line segment bc making δabc 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 of a perpendicular bisector.

δbad is congruent to δbcd
line segment ab is congruent to line segment bc

consequently, δabc is isosceles by definition of an isosceles triangle.