Congruence axiom: The side-angle-side congruence property holds for triangles.

Definitions: A triangle consists of three points A, B, C, such that no two of the points are equal, together with the three line segments AB, BC, and AC. The points A, B, C are called the vertices of the triangle (each of the points A, B, C is a vertex), and the segments AB, BC, AC are called the sides of the triangle. For short, we refer to the triangle as ABC.

The traditional way to state the side-angle-side congruence axiom uses absolute angle measure.

Two triangles are congruent if there is a bijection (one-to-one correspondence) between the vertices so that corresponding sides have the same length and corresponding angles have the same absolute angle measure. In other words, we can label the vertices ABC and A'B'C' so that AB = A'B', AC = A'C', BC = B'C', a(C,A,B) = a(C',A',B'), a(A,B,C) = a(A',B',C'), a(B,C,A) = a(B',C',A').

Congruence Axiom (side-angle-side) - absolute version

If ABC and A'B'C' are two triangles with AB = A'B', AC = A'C', and a(C,A,B) = a(C',A',B'), then ABC and A'B'C' are congruent.

Sometimes further precision is needed, and one distinguishes between positive congruence and negative congruence, using the angle measure @.

Two triangles are positively congruent (or directly congruent) if there is a bijection (one-to-one correspondence) between the vertices so that corresponding sides have the same length and corresponding angles have the same angle measure. In other words, we can label the vertices ABC and A'B'C' so that AB = A'B', AC = A'C', BC = B'C', @(C,A,B) = @(C',A',B'), @(A,B,C) = @(A',B',C'), @(B,C,A) = @(B',C',A').

Example of positively congruent triangles:

Two triangles are negatively congruent (or oppositely congruent) if there is a bijection (one-to-one correspondence) between the vertices so that corresponding sides have the same length and corresponding angles have opposite angle measure. In other words, we can label the vertices ABC and A'B'C' so that AB = A'B', AC = A'C', BC = B'C', @(C,A,B) = - @(C,'A',B'), @(A,B,C) = - @(A',B',C'), @(B,C,A) = - @(B',C',A').

Example of negatively congruent triangles:
 

Congruence Axiom (side-angle-side) - signed version

If ABC and A'B'C' are two triangles with AB = A'B', AC = A'C', and @(C,A,B) = @(C',A',B'), then ABC and A'B'C' are positively congruent.

If ABC and A'B'C' are two triangles with AB = A'B', AC = A'C', and @(C,A,B) = - @(C',A',B'), then ABC and A'B'C' are negatively congruent.