Introduction

Recognizing and using congruent and similar shapes can make calculations and design work easier. For instance, in the design below, only two different shapes were actually drawn. The design was put together by copying and manipulating these shapes to produce versions of them of different sizes and in different positions.

In this unit we will look in a little more depth at the mathematical meaning of the terms similar and congruent, which describe the relation between shapes like those in this design.

Recognizing and using congruent and similar shapes can make calculations and design work easier. For instance, in the design below, only two different shapes were actually drawn. The design was put together by copying and manipulating these shapes to produce versions of them of different sizes and in different positions.

In this unit we will look in a little more depth at the mathematical meaning of the terms similar and congruent, which describe the relation between shapes like those in this design.

Congruence and similarity

The two shapes below are said to be **congruent**. This means that they are the same shape and size. If you move or rotate the shape on the right below, it will still be congruent to the shape on the left. The shapes would also remain congruent if you reflected the shape on the right, producing its mirror image, because all it sides and angles retain their size. Try moving and rotating the right-hand shape in Fig.1 below.

Click on the figure below to interact with the model.

Each box in Fig.2 below contains

**congruent**shapes.

The two shapes below are said to be

**similar**. This means that they are the same shape but can be different sizes, i.e. they have the same proportions. If you move, rotate, reflect, or change the scale of the shape on the right in Fig.3 below, the two shapes will still be similar.

Click on the figure below to interact with the model.

Each box in Fig.4 below contains

**similar**shapes.

The difference between congruence and similarity is that

**similar**Think about all the ways that you might draw a square. A few examples are shown in Fig.5 above.

Fig.6 shows some different triangles.

Finally, imagine all the circles that you could draw. Again, some different circles are shown in Fig.7 above.

Congruence

If two shapes are congruent, then we know that their measurements (including lengths and angles) are the same. The two triangles
in Fig.8 below are congruent.In congruent shapes, we say that equivalent side or

**angle***e*and

*d*, are

**corresponding**.

Similarity and proportion

As you can see in the unit *Enlargement, Angle and Length*,

**enlargement**

**constant**

**scale factor**The model in Fig.10 below shows a

**triangle**

**image**Click on the figure below to interact with the model.

Although the size of a shape changes during enlargement, it remains in proportion to the original shape. That is, although the new shape is a different size from the original, it retains the same proportions.

This is a useful aspect of similarity. In fact, it means that we can use whether shapes are in proportion to one another as a test for similarity. For instance, look at the two triangles in Fig.11 below (they are not necessarily drawn to scale). Marked on them are all the measurements that we know of them. Are they similar?

To find out, we first identify what would would be the corresponding sides if the triangles were similar.

If these are indeed similar triangles, then the

**ratio***a*to

*b*is often written

*a*:

*b*. For example, if the ratio of the width to the length of a swimming pool is 1:3, the length is three times the width.ratio of the lengths in each pair will be a constant (the scale factor of enlargement). We therefore need to calculate the ratio of lengths for each of the paired sides.

So the triangles are related by an enlargement. They are therefore in proportion to one another and so they are similar triangles. |

Summary

Congruent shapes are the same shape and size as one another, but can be in different positions, orientations, or form mirror images.

Shapes are similar if they have the same proportions. They may have different positions and orientations.

Whether shapes are congruent or similar can be found by comparing their internal angles and the length of their sides.

Congruent shapes are the same shape and size as one another, but can be in different positions, orientations, or form mirror images.

Shapes are similar if they have the same proportions. They may have different positions and orientations.

Whether shapes are congruent or similar can be found by comparing their internal angles and the length of their sides.

Well done!

Try again!