Math resources Geometry

Congruence and similarity

# Congruence and similarity

Here you will learn about congruence and similarity, including what congruent shapes are and what similar shapes are. This will extend to congruent triangles and scaling.

Students will first learn about congruence and similarity as a part of geometry in 8 th grade and will expand that knowledge into high school.

## What is congruence and similarity?

Congruence and similarity are mathematical terms used to describe how two or more shapes are related to each other.

### Congruent shapes

Congruent shapes are exactly the same shape and the same size.

The corresponding sides are the same and the corresponding angles are the same. Shapes can be congruent if they have been reflected or rotated.

For example, these polygons are congruent.

Step-by-step guide: Congruent shapes

### Congruent triangles

Congruent shapes can be extended to congruent triangles. There are 4 special conditions to help us recognise congruent triangles. These conditions help us prove triangle congruence.

The 4 conditions are:

SSS (side-side-side),

RHS (right-angled triangle, hypotenuse and a side),

ASA or AAS (angle-side-angle or angle-angle-side),

SAS (side-angle-side, two sides and the included angle).

Step-by-step guide: Congruent triangles

### Similar shapes

Similar shapes (similar figures) are the same shape but they have different sizes. The corresponding sides are in the same ratio and the corresponding angles are the same.

If you know that 2 shapes are mathematically similar, you can find any missing sides using the ratio, or scale factor.

For example, here are 2 similar triangles.

The ratio of the sides is 4\text{:}12=1\text{:}3, the scale factor is 12\div4=3.

You can use either of these to work out the size of the required length.

Therefore x=3\times 7=21.

Step-by-step guide: Similar shapes

## Common Core State Standards

How does this relate to 8 th grade math?

Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

## How to use congruence and similarity

There are a lot of ways to use congruence and similarity. For more specific step-by-step guides, check out the individual pages linked in the “What is congruence and similarity?” section above or read through the examples below.

## Congruence and similarity examples

### Example 1: congruent shapes

Are these 2D shapes congruent?

1. Check the type of 2D shape.

Both shapes are rectangles.

2Check the corresponding angles and corresponding sides.

All the angles are 90^{\circ}.

The short sides on both rectangles are 3.

The long sides on both rectangles are 5.

3State if the shapes are congruent or not.

The shapes are the same shape and the same size – they are congruent shapes.

### Example 2: recognize congruent triangles

Decide whether this pair of triangles are congruent. If they are congruent, state why.

Check the corresponding angles and corresponding sides.

Decide if the shapes are congruent or not.

If the triangles are congruent, which congruence condition fits the pair of triangles.

### Example 3: prove congruent triangles (higher)

Prove that triangle ABC is congruent to triangle XYZ.

Pair up the corresponding sides.

Pair up the corresponding angles.

State which congruence condition fits the pair of triangles.

### Example 4: finding a missing length using similar shapes

Here are two similar shapes. Find the length PQ.

Decide which sides are pairs of corresponding sides.

Find the scale factor.

Use the scale factor to find the missing length.

### Example 5: finding a missing area using similar shapes

These two figures are similar.

The area of shape A is 10 \, cm^{2}.

Find the area of shape B.

Find the scale factor.

Use the scale factor to find the missing value.

### Example 6: finding a missing volume using similar shapes

These two shapes are similar.

The volume of shape A is 120 \, cm^{3}.

Find the volume of shape B.

Find the scale factor.

Use the scale factor to find the missing value.

### Teaching tips for congruence and similarity

• When first introducing the concept of congruence and similarity, use visual aids, including diagrams and models, that are clearly marked and demonstrate the difference between congruent shapes and similar shapes.

• There are a lot of interactive apps that are created by fellow educators that allow students to manipulate shapes to see how the transformations affect the shapes. There is also access to interactive games and quizzes that give students plenty of practice and immediate feedback.

### Easy mistakes to make

• Shapes can be congruent but in different orientations
The second shape may be in a different orientation to the first shape. The shapes can still be congruent. The use of tracing paper can help you check.

• In most diagrams the diagrams are NOT drawn to scale
Questions about congruent shapes are often on grids. But sometimes diagrams may have shapes which are NOT drawn to scale. So use the measurements given, rather than measuring for yourself.

• Scaling up or down
If you are finding a missing length in the larger shape, you can multiply by the scale factor. The scale factor will be a number greater than 1.

If you are finding a missing length in the smaller shape, you can multiply by the scale factor, but the scale factor will be a number between 0 and 1.

### Practice congruence and similarity questions

1. Which of these shapes is congruent to shape X?

A

B

C

D

Shape B is the same as the original shape, but is upside down.

2. Here is a pair of congruent triangles. Which congruence condition is satisfied?

ASA

SAS

SSS

RHS

The triangles are right angle triangles. They have the same hypotenuse and the same short side.

3. Prove that triangle ABD is congruent to triangle DEF.

SSS

Because

\begin{aligned}AB &= PQ \\\\ AC &= RQ \\\\ BC&=PR \end{aligned}

SSS

Because

\begin{aligned}AB &= RQ \\\\ AC &= PQ \\\\ BC&=PR \end{aligned}

SSS

Because

\begin{aligned}AB &= PR \\\\ AC &= RQ \\\\ BC&=PQ \end{aligned}

SSS

Because

\begin{aligned}AB &= PR \\\\ BC&=PQ \\\\ AC &= RQ \end{aligned}

The corresponding sides need to be paired up. The lengths of sides are all equal.

4. These parallelograms are similar. Find the value of x.

x=3.5

x=3

x=4

x=4.5

The ratio of the bases is \quad 5\text{:}20

The scale factor of enlargement is 4. But you will need to divide by 4 as you are finding a length on the smaller shape.

x=12\div4=3

5. Here are two similar shapes. The area of shape A is 60 \, cm^{2}. Find the area of shape B.

240 \, cm^{2}

120 \, cm^{2}

180 \, cm^{2}

200 \, cm^{2}

The ratio of the bases is \quad 10\text{:}20

The scale factor of enlargement is 2. But you are finding the area, so you need to multiply the area of shape A by 2^{2} or 4.

60\times 2^2=60\times 4=240

The area of shape B is 240 \, cm^{2}.

6. Here are two similar shapes. The volume of shape B is 35,840 \, cm^{3}. Find the volume of shape A.

4,480 \, cm^{3}

560 \, cm^{3}

8,960 \, cm^{3}

2,240 \, cm^{3}

The ratio of the bases is \quad 10\text{:}40

The scale factor of enlargement is 4. But you are finding the volume, so you need to use 4^{3} or 64.

Since you are finding the volume of the smaller shape, divide by the scale factor.

35840\div 4^3=35840\div 64=560

The volume of shape A is 560 \, cm^{3}.

## Congruence and similarity FAQs

What’s the difference between congruent shapes and similar shapes?

Congruent shapes are identical in both size and shape. Similar shapes have the same shape but differ in size. The corresponding angles are equal, and the corresponding sides are proportional.

What transformation can be used to prove two shapes are similar?

You can prove two shapes are similar by using scaling combined with rotation, reflection and translation. Scaling may change the size of the shape, but will keep the shape proportional.

What is a scale factor and how does it relate to similar shapes?

The scale factor is the ratio of the lengths of corresponding sides in similar shapes. It tells you how much one shape has been enlarged or reduced compared to the other. A scale factor that is larger than 1 will result in a larger shape, and a scale factor less than 1 will result in a smaller shape.

Step-by-step guide: Scale math

Step-by-step guide: Scale drawing

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