Angles

An angle is the figure formed by two rays that share a common endpoint, called the vertex. The rays are the sides of the angle, and the amount of rotation from one side to the other determines the angle’s size. The standard unit for measuring angles is the degree, though radians are also used in higher mathematics.

Angles are classified by their measure. An acute angle measures less than ninety degrees, a right angle is exactly ninety degrees, and an obtuse angle measures greater than ninety but less than one hundred eighty degrees. A straight angle measures exactly one hundred eighty degrees, and a reflex angle measures more than one hundred eighty degrees but less than three hundred sixty degrees. A full angle measures three hundred sixty degrees.

Angles are measured with instruments such as a protractor, and can be described by their orientation as well as their magnitude. In many constructions, compass and straightedge techniques are used to create angles of a given size or to bisect them, meaning to divide an angle into two equal angles.

Several basic properties relate angles to each other. Complementary angles have a sum of ninety degrees, and supplementary angles sum to one hundred eighty degrees. Vertical angles, formed by two intersecting lines, are opposite each other and are equal in measure. The sum of adjacent angles on a straight line is a straight angle, and an angle bisector splits an angle into two equal parts.

In polygons, the interior angle sum of an n-gon is (n−2)×180 degrees, while the exterior angles around a point total 360 degrees. Angles play a central role in trigonometry, where their measures relate to ratios of sides, and in the unit circle, where sine and cosine describe coordinates of points on the circle.