Prove congruent angles have congruent supplements.

Question

Prove congruent angles have congruent supplements.

I do not yet have degrees. Could I somehow use the base angles of isosceles triangles are congruent?

Answer 1

Let $\angle$BAC $\cong$ $\angle$EDF where AB $\cong$ DE and AC $\cong$ DF. We then have congruent triangles ABC and DEF by connecting B to C and E to F (they are congruent due to side-angle-side). Extend BA to a point P, and extend ED to a point Q such that AP $\cong$ DQ. Then PBC $\cong$ QEF (again using side-angle-side), meaining $\angle$APC $\cong$ $\angle$DQF. Since AC $\cong$ DF and PC $\cong$ QF, APC $\cong$ DQF so that $\angle$PAC $\cong$ $\angle$QDF, which is the desired result.