# Congruent shapes

Categories: gcse geometry congruent

Two shapes are *congruent* if they are the same size and shape.

Here is a video on the topic:

## Similar vs congruent shapes

Congruent shapes have the same size and shape. Two shapes that are exactly the shape but not necessarily the same size are called similar shapes. See Congruent and similar shapes.

## Examples of congruent shapes

These two squares are the same size and shape, so they are congruent:

These two pentagons are the same size and shape, so they are congruent. It doesn't matter that one is rotated:

These two L-shapes are also the same size and shape, so they are congruent. It doesn't matter that one is a mirror image of the other:

## Examples of incongruent shapes

These two squares have the same shape, but are different sizes, so they are incongruent:

These two shapes are both rectangles, but one is taller than the other. That means they are differently shaped rectangles, so they are incongruent:

Complex shapes can be confusing if they are reflected and rotated. You might need to look carefully at these shapes to see that they are different, and so not congruent:

It helps to transform the second shape so that it is no longer rotated and mirrored, This makes it more obvious that the two shapes are different:

## Tests for congruency

There are two ways to check if two shapes are congruent. They are equivalent:

- If one can be mapped onto the other by any combination of translation, rotation, and reflection transforms, they are congruent.
- If every corresponding side and angle is identical, they are congruent.

In the second case, it is important to make sure the two shapes have the same sides and angles in the same order. For example:

These two shapes have sides of the same length (two of length *a*, four of length *b*, and two of length *c*). They have the same set of angles too. But the shapes are obviously not the same. That is because the sides and angles are not in the same order, so the shapes are not congruent.

However, when one shape is a reflection of the other, the rules are slightly different:

Because of the reflection, we must read the lines and angles in opposite directions:

- Reading clockwise for the base, the first shape has sides
*aabcbd*. - Reading counterclockwise from the base, the second shape also has sides
*aabcbd*.

The angles are also the same in each case. This means that the shapes are congruent.

## See also

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