Proof: Isosceles Triangle - Congruent Base Angles

Given Hilbert's axioms, show that the base angles of an isosceles triangle are congruent. Proof: Consider the isosceles triangles

ABC and

. Assume that the full SAS condition has been established. Thus, we know that AB

, AC

, and

BAC =

. Note that since the triangles we are considering are isosceles, we can also conclude that BC

. By Hilbert's Axiom (III-5), we know that

ABC =

. We can then conclude that

ACB =

. Therefore, in triangle

ABC, the base angles

BAC and

ACB are congruent. Similarly, we know that in triangle

that base angles

and

are congruent.

The picture does not tie directly to the proof. Labels are different. It is only used for reference.