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.

What does it mean for two shapes to be congruent?

Two shapes are congruent if they are exactly the same shape, just rotated and/or mirrored.

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Chapter overview

Learning goals

Maths

Congruence: conditions for congruent triangles

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Summary

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Congruence: conditions for congruent triangles

In a nutshell

Congruent shapes are shapes that are exactly the same apart from rotational/mirror differences. There are four main conditions for two triangles to be congruent.

Congruency

Two shapes are congruent if they have the same side lengths and the same angles. It doesn't matter if the shapes are rotated slightly differently or if they're mirrored.

Congruency conditions

There are four ways to prove that 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
AS	Two angles and a corresponding side are the same
RHS	Both triangles have a right-angle, the same hypotenuse and other common side

Example 1

Are the two triangles shown congruent to one another?

Maths; Shapes and area; KS4 Year 10; Congruence: conditions for congruent triangles

At first glance, the two triangles look to be congruent due to the ASA rule - both triangles have angles of $40^\circ$ and $80^\circ$ and they both have a side of $2.2cm$ .

However, upon closer inspection, this is not a case of ASA. This is because the two sides aren't corresponding.

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

The triangle to the right has a side length of $2.2cm$ that is not between the two angles of $40^\circ$ and $80^\circ$ , so it's not the corresponding sides that are equal.

The triangles are NOT congruent.

Congruence: conditions for congruent triangles

In a nutshell

Congruent shapes are shapes that are exactly the same apart from rotational/mirror differences. There are four main conditions for two triangles to be congruent.

Congruency

Two shapes are congruent if they have the same side lengths and the same angles. It doesn't matter if the shapes are rotated slightly differently or if they're mirrored.

Congruency conditions

There are four ways to prove that 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
AS	Two angles and a corresponding side are the same
RHS	Both triangles have a right-angle, the same hypotenuse and other common side

Example 1

Are the two triangles shown congruent to one another?

At first glance, the two triangles look to be congruent due to the ASA rule - both triangles have angles of $40^\circ$ and $80^\circ$ and they both have a side of $2.2cm$ .

However, upon closer inspection, this is not a case of ASA. This is because the two sides aren't corresponding.

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

The triangle to the right has a side length of $2.2cm$ that is not between the two angles of $40^\circ$ and $80^\circ$ , so it's not the corresponding sides that are equal.

The triangles are NOT congruent.

Want to find out more? Check out these other lessons!

Congruent shapes

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Frequently Asked Questions (FAQ)

FAQs

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.
Question: What does it mean for two shapes to be congruent?

Answer: Two shapes are congruent if they are exactly the same shape, just rotated and/or mirrored.

Theory

Exercises

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Congruence: conditions for congruent triangles

Congruence: conditions for congruent triangles

​​In a nutshell

Congruency

Congruency conditions

name

description

example

Example 1

Congruence: conditions for congruent triangles

​​In a nutshell

Congruency

Congruency conditions

name

description

example

Example 1

Want to find out more? Check out these other lessons!

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

In a nutshell

In a nutshell