Congruent Shapes

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Congruent shapes

Here you will learn about congruent shapes, including what they are and how to recognize them.

Students will first learn about congruent shapes as part of geometry in elementary school, but they will expand upon their learning in $8$ th grade.

What are congruent shapes?

Congruent shapes or congruent figures are shapes that are exactly the same.

The corresponding sides are the same and the corresponding angles are the same.

To do this, we need to check all the angles and all the sides of the shapes. If two shapes are congruent, they will fit exactly on top of one another.

For example,

These two polygons are congruent.

The red shape has been translated to give the blue shape.

These two quadrilaterals are congruent.

The red shape has been reflected to give the blue shape.

These two quadrilaterals are congruent.

The red shape has been rotated to give the blue shape.

If two shapes are the same but different sizes, one being an enlargement of the other, these are known as similar shapes or similar figures.

These two polygons are NOT congruent. They are similar.

The red shape has been enlarged by multiplying by a scale factor to give the blue shape.

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Congruent triangles

There are four conditions to be able to prove if a pair of triangles is congruent.

Reasons for congruency

SSS: side-side-side (three sides the same)

RHS: right-hypotenuse-side (right angle, hypotenuse, and a side the same)

ASA or AAS: angle-side-angle or angle-angle-side (two angles and one side the same)

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

For example,

These two triangles are congruent triangles.

They have two angles that are the same.

The side in between the angles is also equal.

The congruence condition would be angle-side-angle (which is abbreviated to ASA).

Step by step guide: Congruent triangles

What are congruent shapes?

Common Core State Standards

How does this relate to $8$ th grade math?

Grade 8 – Geometry (8.G.A.2)
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 recognize congruent shapes

In order to recognize congruent shapes:

Check the type of 2D shape.
Check the corresponding angles and corresponding sides.
State if the shapes are congruent or not.

Congruent shapes examples

Example 1: congruent shapes

Are these 2D shapes congruent?

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 $1.$

The long sides on both rectangles are $3.$

3State if the shapes are congruent or not.

The shapes are the same shape and the same size, so they are congruent shapes.

Example 2: congruent shapes

Are these 2D shapes congruent?

Check the type of 2D shape.

Both shapes are rectangles.

Check the corresponding angles and corresponding sides.

All the angles are $90^{\circ}.$

The short sides on both rectangles are $2.$

The long sides on both rectangles are different.

State if the shapes are congruent or not.

The shapes have differing side lengths, so the shapes are NOT congruent.

Example 3: congruent shapes

Are these 2D shapes congruent?

Check the type of 2D shape.

Both shapes are trapezoids.

Check the corresponding angles and corresponding sides.

The angles are $90^{\circ}, \; 45^{\circ}$ and $135^{\circ}.$

They are in corresponding positions.

The lengths of the corresponding sides are different.

The side lengths of the second shape are double the lengths of the first shape.

State if the shapes are congruent or not.

The shapes are the same shape, but different sizes. They are similar shapes but they are NOT congruent shapes.

Example 4: congruent shapes

Are these 2D shapes congruent?

Check the type of 2D shape.

One shape looks like a capital letter $“C”$ and the other shape looks like a capital letter $“L”.$

Check the corresponding angles and corresponding sides.

There are lots of right angles in both shapes.

There are lots of sides of length $1$ and $3.$

But they are not in corresponding positions as the shapes are different shapes.

State if the shapes are congruent or not.

The shapes are different shapes. They are NOT congruent shapes.

Example 5: congruent shapes

Are these 2D shapes congruent?

Check the type of 2D shape.

It can be tricky to see if these shapes are the same type of 2D shape. They both have $6$ sides so they are irregular hexagons.

Check the corresponding angles and corresponding sides.

The angles are $90^{\circ}, \; 225^{\circ}$ and $135^{\circ}.$

They are in corresponding positions as you go the same direction around the shapes.

Looking at the side lengths, they are in corresponding positions as you go around the shapes in the same direction.

State if the shapes are congruent or not.

The shapes are the same shape and the same size. A rotation is involved. They are congruent shapes.

Example 6: congruent shapes

Are these 2D shapes congruent?

Check the type of 2D shape.

It can be tricky to see if these shapes are the same type of 2D shape. They both have $6$ sides so they are irregular hexagons.

Check the corresponding angles and corresponding sides.

There are $4$ right angles in both shapes.

The two other angles are equal and are in corresponding positions as you go around the shapes. But one is in a clockwise direction, and one is in a counterclockwise direction.

Looking at the side lengths there are lots of sides of length $1$ and $3$ and a diagonal.

They are in corresponding positions as you go around the shapes. But one is in a clockwise direction, and one is in a counterclockwise direction.

State if the shapes are congruent or not.

The shapes are the same shape and the same size. A rotation and a mirror image are involved. They are congruent shapes.

Teaching tips for congruent shapes

Use physical shapes (like geometric tiles or cut-out shapes) and real-life objects that students can manipulate. Let them rotate, reflect, and translate the shapes to discover congruence.

Have students draw a shape on a coordinate plane (graph paper) and trace it onto transparent paper. They can then rotate, reflect, or translate the shape using the transparent paper to explore congruence.

Provide worksheets with a variety of problems, including identifying congruent shapes, and finding missing parts of congruent shapes.

Easy mistakes to make

Thinking shapes must have the same orientation to be congruent
The second shape may be in a different orientation to the first shape. The shapes can still be congruent. Perhaps use tracing paper to help you check.

Not recognizing mirror images as congruence
The second shape may be a mirror image of the first shape. The shapes can still be congruent. Perhaps use tracing paper to help you check.

Not using a grid to check angles and sides
Use the straight lines on the grid to help you identify right angles and work out the side lengths. Be careful with the diagonals.

Thinking diagrams are always drawn to scale
Questions about congruent shapes are often on grids, but sometimes diagrams may have shapes that are NOT drawn to scale. Be sure to use the measurements given, rather than measuring for yourself.

Practice congruent shapes questions

A

B

C

D

The original shape is a rectangle with sides $1$ and $4,$ and so is shape $C.$

A

C

B

D

Shape $A$ is the same as the original shape, but has been rotated.

A

B

C

D

The original shape is a rectangle with sides $3$ and $4,$ and so is shape $D.$

A

B

D

C

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

A

B

C

D

Shape $C$ is the same as the original shape, but is a reflection.

A

B

C

D

Shape $C$ is exactly the same as the original shape.

Congruent shapes FAQs

What are congruent shapes?

Congruent shapes are shapes that are exactly the same. The corresponding sides are the same and the corresponding angles are the same.

Do congruent shapes have congruent angles?

Yes, congruent shapes have congruent angles. When two shapes are congruent, they are identical in size and shape. This means that the corresponding sides of the shapes have equal lengths, and the corresponding angles are also equal.

Can congruent shapes have different names?

Yes, congruent shapes can have different names. For example, a triangle with vertices labeled $ABC$ can be congruent to another triangle with vertices labeled $XYZ,$ as long as the corresponding sides and angles match up.

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