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

**R***n* are called congruent if there exists an isometry

*f*:

**R***n* →

**R***n* (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.