Definition:
Polygons
are congruent when they have the same number of sides, and all corresponding sides and
interior angles
are congruent.
The polygons will have the same shape and size, but one may be a rotated, or be the mirror image of the other.

Note: This entry deals with the congruence of polygons in general. Congruent triangles are discussed in more depth in
Congruent Triangles.

Polygons are
congruent if they are equal in all respects:

Same number of sides

All corresponding sides are the same length,

All corresponding interior angles are the same measure.

But they can be rotated on the page and one can be a mirror image of the other. In the figure below, all the
irregularpentagons
shown are congruent.
Some are mirror images of the others, but are still congruent.
(See the page on congruent triangles where these ideas are illustrated in greater depth.)

One way to think about this is to imagine the polygons are made of cardboard.
If you can move them, turn them over and stack them exactly on top of each other, then they are congruent.
To see this, click on any polygon below. It will be flipped over, rotated and stacked on another as needed to demonstrate that they are congruent.

Try this
Click on 'Next' or 'Run'. Each polygon in turn will be flipped over, rotated and stacked on another as needed to show that it is congruent to it.

Mathematically speaking, each operation being done on the polygons is one of three types:

Rotation

This is where the polygon is rotated about a given point by a certain amount. In the applet above, the rotations are around a point inside the polygon, but any point can be chosen. While rotations are being done, this point is shown. See
Rotation.

Reflection

When the polygon is 'flipped over' above, this operation is called reflection. In essence the polygon is 'reflected' over a given line.
It's as if the points on each side of the line are mirror imaged, thinking of the line as the mirror. In the above applet, the line of reflection is shown while the operation is going on.
Reflection.

Translation

When the polygon is moved from one point to another, this is called 'translation'. When the polygon is translated, it is moved, but without any rotation.
Translate.

Testing for Congruence

There are four ways to test for congruence of polygons, depending on what you are given to start.
See Testing Polygons for congruence.

The three types of operation above are called 'transforms'.
In effect, they transform a shape to another by changing it in some way - rotation, reflection and translation.

What does this mean?

If you have shown that two polygons are congruent, then you know that every property of the polygons is also identical.
For example they will have the same area, perimeter, exterior angles, apothem etc.