Congruence is an equivalence relation.  In geometry, the congruence is studied as a relationship between triangles. We can define it thus:   

Definition: Two triangles are congruent if there exists a one-to-one correspondence between their vertices so that the corresponding sides and corresponding angles are congruent.

In a Euclidean Geometric system, congruence is fundamental and  is the counterpart of equality for numbers.

A more formal definition can be given as follows:
Two subsets A and B of Euclidean Space Rn are called congruent if there exists an isometry f: Rn → Rn (an element of the Euclidean Group E(n)) with f(A) = B..

If triangle ABC is congruent to triangle DEF, the relationship can be written mathematically as:



In many cases it is sufficient to establish the equality of three corresponding parts and use one of the following results to deduce the congruence of the two triangles.