## How To Find if Triangles are Congruent

Triangles are congruent when all corresponding sides & interior angles are congruent. The triangles will have the same size & shape, but 1 may be a mirror image of the other.

2 triangles are congruent if they have:

- exactly the same three sides and
- exactly the same three angles.

We don’t have to know all 3 sides and all 3 angles, usually 3 out of the 6 is enough.

There are 4 ways to find if two triangles are congruent. Read the explanation below.

Table of Contents

**How to Tell If Triangles are Congruent**

Any triangle is defined by 6 measures (3 angles and 3 sides). But, to show that 2 triangles are congruent you don’t need to know all of them. Three of them in Various groups will do. Triangles are congruent if:

3 Measures | Detail |

SSS (side side side) | All 3 corresponding sides are equal in length |

SAS (side angle side) | A pair of corresponding sides & the included angle are equal |

ASA (angle side angle) | A pair of corresponding angles and the included side are equal |

AAS (angle angle side) | A pair of corresponding angles and a non-included side are equal |

## SSS (side, side, side)

SSS stands for “*side, side, side*” and means that we have 2 triangles with all three sides equal.

For example:

is congruent to:

If 3 sides of one triangle are equal to 3 sides of another triangle, the triangles are congruent.

## SAS (side, angle, side)

SAS stands for “*side, angle, side*” and means that we have 2 triangles where we know 2 sides and the included angle are equal.

For example:

is congruent to:

If 2 sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, the triangles are congruent.

## ASA (angle, side, angle)

ASA stands for “*angle, side, angle*” and means that we have 2 triangles where we know 2 angles and the included side are equal.

For example:

is congruent to:

If 2 angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

## AAS (angle, angle, side)

AAS stands for “*angle, angle, side*” and means that we have 2 triangles where we know 2 angles and the non-included side are equal.

For example:

is congruent to:

If 2 angles and the non-included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

## Can we use “AAA” ?

AAA “*angle, angle, angle*” means we are given all three angles of a triangle, but no sides.

This is **NOT** enough information to decide if 2 triangles are congruent!

Because the triangles can have the **same angles** but be **different sizes**:

Without knowing at least one side, we can’t be sure if two triangles are congruent.

**C****onstructions**

If you are asked to construct a unique triangle given what you know, for example: If you were given the lengths of 2 angles and the included side (ASA), there is only 1 possible triangle you could draw. If you drew 2 of them, they would be the same size & shape (the definition of congruent).

**Properties of Congruent Triangles**

If 2 triangles are congruent, then each part of the triangle (angle or side) is congruent to the corresponding part in the other triangle. Once you have proved 2 triangles are congruent, you can find the sides or angles of 1 of them from the other.

Use the acronym CPCTC to remember this important idea, which stands for “**C**orresponding **P**arts of **C**ongruent **T**riangles are **C**ongruent”.

In addition to angles & sides, all other properties of the triangle are the same too, such as perimeter, area, location of centers, circles etc.