# Congruence

## Congruence Revision

**Congruence**

For two shapes to be **congruent**, they must be the **same** **shape** and the **same** **size**. **Congruent** shapes may be **mirror images** of each other.

It is important to note that this is **different from shapes being similar**, as similar shapes can be **different sizes**, but must be the same **shape**.

**Basic Congruence**

If two or more shapes have the** same** **length sides** and the **same angles**, they are **congruent** to each other, or they have **congruence**.

For example, all of these rectangles are **congruent** as the all have** sides** 12\text{ cm} and 3\text{ cm}, and **right angles**.

**Congruence** can also remain when a shape is **transformed** by either a **rotation**, a **reflection**, or a** translation**. This is because these transformations preserve **length** and **angle**, so the shapes still have the same **shape** and **size**.

Note: **congruence** does not remain when a shape is enlarged, as the **size** of the shape changes.

All of these shapes are** congruent**:

**Congruence of Triangles**

You can tell if triangles are **congruent** without knowing the **length of every side** or the **size of every angle**.

You can decide if two triangles are **congruent** if…

**1. SSS (side side side)**: 3 **sides** are the same length

In all cases, if two triangles have 3** sides the same length**, in any order, the triangles are **congruent**.

For example, these triangles are **congruent**,

**2. AAS (angle angle side**): 2 **angles** and a **side** are the same

In all cases, if two triangles have two **angles** and any one **side** the same, they are **congruent**.

For example, these triangles are **congruent**,

**3. SAS (side angle side)**: 2** sides** and the **angle** between them are the same

In all cases, if two triangles have two **sides** of equal length with an **angle** of the same size between them, they are **congruent**.

For example, these triangles are **congruent**,

**4. RHS (right-angle, hypotenuse, side):** in two **right angle** triangles, the **hypotenuse** and one other **side** will be equal

In all cases, the right **angle**, the longest **side** (the **hypotenuse**) and either of the other **sides** must be equal in two triangles for them to be **congruent**.

For example, these triangles are **congruent**,

**Example 1: Basic Congruence**

Which two of the following shapes are **congruent**?

**[2 marks]**

As these shapes all look quite similar, it is important to look at the **side lengths** to decide which are **congruent**, rather than just guessing by observation. We do not need to worry about **angles** for this as they all contain only right angles.

Let’s look side by side.

Longest side: they each have a** longest side** of length 12\text{ cm}

Top of L shape: for a), we can work out the top of the L is **length** 3\text{ cm}, which is the same as b) and d). We can ignore c) going forward as it definitely is not **congruent**.

So we now need to decide which of a), b) and d) are **congruent.**

It is clear c) and d) are not **congruent** as they have equivalent **sides** with measurements 3 and 4.

By doing 12-8=4 on a), it is clear a) and d) are **congruent**.

**Example 2: Congruence of Triangles**

Which option shows a triangle **congruent** to the one above?

**[2 marks]**

Let’s look at each option:

a) Is not **congruent** as for **SAS** to work, the **angle** must be between the two** sides**.

b) Could be congruent but we do not have enough information so cannot prove **congruence** by having three of the same **angles**.

c) This is **congruent** by **AAS**.

## Congruence Example Questions

**Question 1**: Identify the pair of congruent shapes from the options below.

**[2 marks]**

If B was rotated 90\degree clockwise, it would be shape D. Hence, B and D are congruent.

**Question 2**: All of the following triangles are congruent.

Work out the values of A, B, x and y.

**[2 marks]**

As the triangles are all congruent, the angles and sides will be the same.

By looking at the triangles we can identify:

A=8\\ B=3\\ x=40\\ y=125\\**Question 3**: Identify which pair of these triangles are congruent:

**[3 marks]**

All of these triangles have different lengths, so it is impossible to tell which are congruent. This is because the RHS rule relies on having two sides the same length and a right angle.

In order to decide, we will use Pythagoras’ Theorem to find the missing sides of each shape:

a) 13^2-5^2=144\\

\sqrt{144}=12

b) 13^2-11^2=48\\

\sqrt{48}=4\sqrt{3}

c) 13^2-12^2=25\\

\sqrt{25}=5

Therefore, a) and c) are congruent by the RHS rule.