# Definition:Congruence (Geometry)

## Definition

In the field of Euclidean geometry, two geometric figures are congruent if and only if:

they are, informally speaking, both "the same size and shape"
they differ only in position in space
one figure can be overlaid on the other figure with a series of rotations, translations, and reflections.

Specifically:

all corresponding angles of the congruent figures must have the same measurement
all corresponding sides of the congruent figures must be be the same length.

## Warning

Two geometric figures which are congruent are not necessarily identical.

For example, consider two scalene triangles with identical sides and identical internal angles embedded in the plane.

These scalene triangles are not identical if one is a reflection of the other.

However, they are still congruent, because they can be rotated in space $180 \degrees$ about an axis in the plane in which they are both embedded.

This has the same effect of picking one triangle up, flipping it over, and placing it down again the other way round.

In three-dimensional space, this point is important because mirror images cannot be superimposed by physically manipulating them in space.

### Direct Congruence

Let $A$ and $B$ be $3$-dimensional figures which are congruent.

$A$ and $B$ are directly congruent if and only if $A$ and $B$ can be made to coincide with rotations and translations.

### Opposite Congruence

Let $A$ and $B$ be $3$-dimensional figures which are congruent.

$A$ and $B$ are oppositely congruent if and only if $A$ and $B$ cannot be made to coincide with rotations and translations, but also need a reflection for this to happen.

## Also known as

Some sources feel the need to hammer home the fact that overlay means the same thing as superimpose.

Hence the ugly sesquipedalianism superimposable.

Whether this can be backed up by reference in a source work is dubious.

## Also see

• Results about congruence in the context of geometry can be found here.

## Historical Note

The symbol introduced by Gottfried Wilhelm von Leibniz to denote geometric congruence was $\simeq$.

This is still in use and can still be seen, but is not universal.

Also in current use is $\cong$.