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Congruence: conditions for congruent triangles

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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

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

SAS

Two sides and the angle between them are the same


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

AS

Two angles and a corresponding side are the same

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

RHS

Both triangles have a right-angle, the same hypotenuse and other common side

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


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 4040^\circ and 8080^\circ and they both have a side of 2.2cm2.2cm.

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 2.2cm2.2cm that is between the two angles of 4040^\circ and 8080^\circ.

The triangle to the right has a side length of 2.2cm2.2cm that is not between the two angles of 4040^\circ and 8080^\circ, so it's not the corresponding sides that are equal.


The triangles are NOT congruent.


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Congruent shapes

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.

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