Reflexivity of Congruence

The distance from x to y is the same as that from y to x. This axiom asserts a property very similar to symmetry for binary relations.

Identity of Congruence

If xy is congruent with a segment that begins and ends at the same point, x and y are the same point. This is closely related to the notion of reflexivity for binary relations.

Transitivity of Congruence

Two line segments both congruent to a third segment are congruent to each other; all three segments have the same length. This axiom asserts that congruence is Euclidean, in that it respects the first of Euclid's "common notions." Hence this axiom could have been named "Congruence is Euclidean." The transitivity of congruence is an easy consequence of this axiom and Reflexivity.