What transformations will maintain the congruence of shapes?

Shapes will be congruent even if they are rotated, translated or reflected.

What does it mean for two shapes to be congruent?

Two shapes are congruent if they are exactly the same.

What are the four ways to prove two triangles are congruent?

SSS, SAS, ASA and RHS are the four ways to prove two triangles are congruent.

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Chapter overview

Learning goals

Maths

Congruent shapes

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Summary

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Congruent shapes

In a nutshell

Congruent shapes are shapes which are exactly the same. There are four conditions to identify congruent triangles.

Congruent shapes

Shapes will be congruent if they are exactly the same, which means all the angles and all the side lengths between the shapes will be equal. This also means that shapes will still be congruent even if a shape is rotated, translated or reflected.

Example 1

Identify whether the two shapes on the grid below are congruent.

Maths; Properties of shapes; KS3 Year 7; Congruent shapes

At first glance, both shapes appear to be a square.

Using the grid you can count the side lengths of each shape.

The side lengths of Shape $B$  are all $2$ units long.

The side lengths of Shape $B'$ are also all $2$ units long. 

Using the grid lines identify the angles of each shape.

As the grid lines are perpendicular to each other, and the sides of both shapes follow the grid lines, all the angle in each shape must be a right angle.

Both shapes are squares, where each corresponding side is the same length and all the corresponding angles are the same size.

Therefore, shape B and shape B' are congruent.

Congruent triangles

There are four ways to identify whether two triangles are congruent.

NAME	DESCRIPTION	EXAMPLE
$SSS$	All three sides of the triangle are the same
$SAS$	Two sides and the angle between them are the same
$ASA$	Two angles and a corresponding side are the same
$RHS$	Both triangles have a right-angle, the same hypotenuse and other common side

Example 2

Are the two triangles shown congruent to one another?

At first glance, both triangles have the angles of $40^\circ$ and $80^\circ$ and they both have a side of $2.2cm$ . This fits the $ASA$ rule of congruence.

However, this is not the case because the two labelled sides are not corresponding.

The triangle to the left has a side length of $2.2cm$ which is between the two angles of $40^\circ$ and $80^\circ$ .

The triangle to the right has a side length of $2.2cm$ which is not between the two angles of $40^\circ$ and $80^\circ$ .

Therefore, the triangles are not congruent.

Congruent shapes

In a nutshell

Congruent shapes are shapes which are exactly the same. There are four conditions to identify congruent triangles.

Congruent shapes

Example 1

Identify whether the two shapes on the grid below are congruent.

At first glance, both shapes appear to be a square.

Using the grid you can count the side lengths of each shape.

The side lengths of Shape $B$  are all $2$ units long.

The side lengths of Shape $B'$ are also all $2$ units long. 

Using the grid lines identify the angles of each shape.

As the grid lines are perpendicular to each other, and the sides of both shapes follow the grid lines, all the angle in each shape must be a right angle.

Both shapes are squares, where each corresponding side is the same length and all the corresponding angles are the same size.

Therefore, shape B and shape B' are congruent.

Congruent triangles

There are four ways to identify whether two triangles are congruent.

NAME	DESCRIPTION	EXAMPLE
$SSS$	All three sides of the triangle are the same
$SAS$	Two sides and the angle between them are the same
$ASA$	Two angles and a corresponding side are the same
$RHS$	Both triangles have a right-angle, the same hypotenuse and other common side

Example 2

Are the two triangles shown congruent to one another?

At first glance, both triangles have the angles of $40^\circ$ and $80^\circ$ and they both have a side of $2.2cm$ . This fits the $ASA$ rule of congruence.

However, this is not the case because the two labelled sides are not corresponding.

The triangle to the left has a side length of $2.2cm$ which is between the two angles of $40^\circ$ and $80^\circ$ .

The triangle to the right has a side length of $2.2cm$ which is not between the two angles of $40^\circ$ and $80^\circ$ .

Therefore, the triangles are not congruent.

Frequently Asked Questions (FAQ)

FAQs

Question: What transformations will maintain the congruence of shapes?

Answer: Shapes will be congruent even if they are rotated, translated or reflected.
Question: What does it mean for two shapes to be congruent?

Answer: Two shapes are congruent if they are exactly the same.
Question: What are the four ways to prove two triangles are congruent?

Answer: SSS, SAS, ASA and RHS are the four ways to prove two triangles are congruent.

Theory

Exercises

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Congruent shapes

Congruent shapes

In a nutshell

Congruent shapes

Example 1

Congruent triangles

Example 2

Congruent shapes

In a nutshell

Congruent shapes

Example 1

Congruent triangles

Example 2

Now you’ve read the summary, have a go at the exercises!