# 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.*