Proof. Congruent base angles
Given Hilbert's axioms, show that the base angles of an isosceles triangle are congruent.Consider two triangles 
ABC and CBA. Notice that line AB C*B* and notice that BC B*A*, and angle ABC C*B*A*. Using Hilbert's Axiom of Congruence (III-5), it implies that angles BAC B*C*A*. We can further conclude that angle ABC ACB, and the base angles of triangle ABC are congruent!
ABC and CBA. Notice that line AB C*B* and notice that BC B*A*, and angle ABC C*B*A*. Using Hilbert's Axiom of Congruence (III-5), it implies that angles BAC B*C*A*. We can further conclude that angle ABC ACB, and the base angles of triangle ABC are congruent!