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.