There are many types of angles in geometry. One way of classifying angles is by their magnitude or how large they are. Another method uses the amount of rotation. A third method compares a pair of angles.

### What Is an Angle?

An angle forms when two rays intersect at an endpoint, called a vertex. The angle is the separation between the rays. The most common unit of angle size is in degrees (*°*), but sometimes radians are used. An angle has a lowercase name (like **a** or **b**) or sometimes a Greek letter (like theta **θ** or alpha **α**)

### Parts of an Angle

An angle consists of three parts: the arms, the vertex, and the angle:

**Vertex**: The vertex is the point where two rays (or line segments) meet.**Arms**: The arms are the sides of the angle.**Angle**: The angle is the separation between the arms. If you consider one arm as stationary, the angle is the amount the other arm rotates away from it.

### Types of Angles

There are seven main types of angles, according to their magnitude:

Type of Angle | Description |
---|---|

Zero degree angle | a = 0°; the rays overlap each other in the same direction |

Acute angle | a < 90° |

Right angle | a = 90° |

Obtuse angle | 90 ° < a < 180° |

Straight angle | a = 180°; the rays go in opposite directions |

Reflex angle | a > 180° |

Full rotation angle | a = 360°; looks like a zero degree angle, but one ray rotates exactly 360° to go in the same direction and the other |

#### Zero Degree Angles

The two arms of a zero degree angle point in the same direction from the vertex. In other words, a = 0*°*.

#### Acute Angles

An acute angle measure less than 90*°*. The shape of the letter A forms an acute angle. Other examples of acute angle are 45*°* and 60*°*.

#### Right Angle

A right angle measures exactly 90*°*. The angles that form the interior of a square are right angles. The largest angle in a right triangle is a right angle.

#### Obtuse Angles

An obtuse angle measure greater than 90*°* but less than 180*°*. Examples include 120*°* and 145*°*.

#### Straight Angle

A straight angle measures exactly 180*°*. The rays point in opposite directions.

#### Reflex Angle

A reflex angle is greater than 180*°*, but less than 360*°*. For example, a 270*°* angle is a reflex angle.

#### Full Rotation Angle

A full rotation angle forms when one ray rotates exactly 360*°* (a complete circle) from the other.

### Types of Angles by Rotation

An angle is either a positive angle or a negative angle, depending on this direction the second or terminal arm rotates away from its base.

**Positive angle**: A positive angle moves in a counterclockwise direction from the base. This is the way most angles are drawn in geometry. If you draw a base on a graph, starting from the origin (0,0), a positive angle is in the (+x,+y) plane.**Negative angle**: An negative angle is in a counterclockwise direction from the base. Starting from the origin, a negative angle extends into the (x, -y) plane of a graph.

### Pairs of Angles

Several types of angles form when you compare a pair of angles. In geometry, the key ones to know about are opposite, complementary, adjacent, and supplementary angles.

#### Opposite Angles

When two lines intersect, they form two sets of opposite angles. Opposite angles equal each other.

#### Complementary Angles

Complementary angles add up to 90*°*. While often adjacent angles, complementary angles do not have to be adjacent.

#### Adjacent Angles

Adjacent angles share a common side and vertex, but they do not overlap. In other words, adjacent angles lie next to each other.

#### Supplementary Angles

Supplementary angles add up to 180*°*. As with complementary angles, supplementary angles do not have to be adjacent to one another.

### References

- Henderson, David W.; Taimina, Daina (2005).
*Experiencing Geometry / Euclidean and Non-Euclidean with History*(3rd ed.). Pearson Prentice Hall. ISBN 978-0-13-143748-7. - Jacobs, Harold R. (1974).
*Geometry*. W. H. Freeman. ISBN 978-0-7167-0456-0. - Wong, Tak-wah; Wong, Ming-sim (2009). “Angles in Intersecting and Parallel Lines.”
*New Century Mathematics*(1st ed.). Hong Kong: Oxford University Press. ISBN 978-0-19-800177-5.