# Definition:Congruence (Geometry)

*This page is about congruence in the context of geometry. For other uses, see congruence.*

## 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"

- 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

- Triangle Side-Angle-Side Congruence
- Triangle Side-Side-Side Congruence
- Triangle Angle-Angle-Side Congruence
- Triangle Angle-Side-Angle Congruence

## 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 $\cong$.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**congruent**:**1.** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**congruent**:**1.** - 2018: John M. Lee:
*Introduction to Riemannian Manifolds*(2nd ed.) ... (next): $\S 1$: What Is Curvature? The Euclidean Plane