Proof of Triangle Congruence Theorems

We want to prove that triangles ABC and DEF are congruent. Let us construct ABC onto DEF such that BC overlaps EF. The constructed triangle ABC would be called GEF.

If GEF is not congruent to DEF, there would exist another triangle congruent to GEF on the opposite side, as shown in red. That is invalid if GEF were congruent to DEF, so we know that ABC must be congruent to DEF.

We want to prove that triangles ABC and DEF are congruent. Since <BAC = <EDF, we can rotate the triangles on top of each other such that lines BE and CF intersect to form vertical angles.

We construct line EC and HK such that HK is a perpendicular bisector. We will focus on triangles BCE and FEC. We have:

BH = HE and EK and KC

--> HK = 1/2 * BC (midpoint theorem)

FH = HC and EK and KC

--> HK = 1/2 * FE (midpoint theorem)

Since HK = 1/2 * BC = 1/2 * FE --> BC = FE. Because all corresponding sides of ABC and DEF are congruent, ABC and DEF are congruent triangles.