Math resources Geometry

Congruence and similarity

Congruent shapes

# 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.

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

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.

### 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

## 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:

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

## Congruent shapes 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 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.

Check the corresponding angles and corresponding sides.

State if the shapes are congruent or not.

### Example 3: congruent shapes

Are these 2D shapes congruent?

Check the type of 2D shape.

Check the corresponding angles and corresponding sides.

State if the shapes are congruent or not.

### Example 4: congruent shapes

Are these 2D shapes congruent?

Check the type of 2D shape.

Check the corresponding angles and corresponding sides.

State if the shapes are congruent or not.

### Example 5: congruent shapes

Are these 2D shapes congruent?

Check the type of 2D shape.

Check the corresponding angles and corresponding sides.

State if the shapes are congruent or not.

### Example 6: congruent shapes

Are these 2D shapes congruent?

Check the type of 2D shape.

Check the corresponding angles and corresponding sides.

State if the shapes are congruent or not.

### 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.

• Congruence and similarity
• Similar shapes
• Scale math
• Scale drawing

### Practice congruent shapes questions

1. Which shape is congruent to shape X?

A

B

C

D

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

2. Which shape is congruent to shape X?

A

C

B

D

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

3. Which shape is congruent to shape X?

A

B

C

D

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

4. Which shape is congruent to shape X?

A

B

D

C

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

5. Which shape is congruent to shape X?

A

B

C

D

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

6. Which shape is congruent to shape X?

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.

## The next lessons are

• Transformations
• Mathematical proof
• Area

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