# Types of triangles

A triangle is a three-sided polygon that has an interior angle at each of its vertices. Triangles are often classified by their angles or sides.

Triangle △ABC, shown above, has sides AB, BC, and AC, and angles A,B,and C, formed at vertices A, B, and C.

## Triangles classified by their angles

The following table shows the types of triangles classified by their angles.

Type | Angles | Figure |
---|---|---|

Acute | all interior angles < 90° | |

Obtuse | 1 interior angle > 90° | |

Right | 1 angle = 90° | |

Equiangular | each interior angle = 60° |

## Triangles classified by their sides

The following table shows the types of triangles classified by their sides.

Type | Sides | Figure |
---|---|---|

Scalene | no 2 sides are congruent | |

Isosceles | 2 congruent sides | |

Equilateral | all sides are congruent |

## Using the Pythagorean Theorem to classify triangles by their angles

When one of the angles in a triangle is a right angle, the triangle is a right triangle. The lengths of the three sides of a right triangle follow Pythagorean Theorem. The Pythagorean Theorem can also be used to classifies triangles by angles as follows:

For △ABC, given that side c is the longest side:

- If c
^{2}= a^{2}+ b^{2}, then △ABC is a right triangle with right angle C. - If c
^{2}> a^{2}+ b^{2}, then △ABC is an obtuse triangle with obtuse angle C. - If c
^{2}< a^{2}+ b^{2}, then △ABC is an acute triangle with all angles acute.

Example:

Classify a triangle that has side lengths 3, 6, and 8.

Since 8 is the largest side length, let c = 8, a = 3, and b = 6:

c^{2} = 8^{2} = 64 |

a^{2} + b^{2} = 3^{2} +6^{2} = 9 + 36 = 45 |

Since 64 > 45, c^{2} > a^{2} + b^{2}, so the triangle is an obtuse triangle.