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plane angle[′plān ‚aŋ·gəl]
A plane angle, often called simply an angle, is a geometric figure formed by two rays that extend from a single point. The rays are known as the sides of the angle, and the point is called the vertex of the angle.
An angle with its vertex at the center O of a circle is referred to as a central angle. Such an angle determines on the circle an arc AB that is bounded by the points of intersection of the circle and the sides of the angle. This fact permits the measurement of angles to be reduced to the measurement of the corresponding arcs. Angles are commonly measured in degrees or radians.
Two angles such that each side of one is an extension, through the vertex, of a side of the other are said to be vertical angles. If two angles share a common side and their other sides lie on the same line, the angles are said to be adjacent supplementary angles.
In a number of practical problems it is desirable to treat an angle as a figure obtained by rotating a ray about its origin O to a specified position. In this case the angle is a measure of the rotation of the ray. Such an approach makes it possible to extend the concept of an angle. According to the direction of rotation, for example, a distinction can be made between positive and negative angles. We may speak of angles greater than 180° and angles equal to 0°. In trigonometry this approach permits the study of trigonometric functions for any value of the argument.
If two curves extend from a common point at which each curve has a definite tangent, the angle between the two tangents is taken as the angle between the two curves. The concept of a plane angle can be extended also to various objects considered in solid geometry. For example, the angle between a line and a plane is the smaller angle the line makes with its projection on the plane. The angle between two skew lines is the smaller angle formed by two lines that are parallel to the skew lines and share a common point. (See alsoDIHEDRAL ANGLE, POLYHEDRAL ANGLE, and SOLID ANGLE.)