ANGLE, in plane geometry, a figure formed by two lines which meet. The point of meeting is called the vertex of the angle. The original idea has been extended to include the figure formed by a plane and a line which meets it, by two intersecting planes (a dihedral angle), by three planes having a common point (a triliedral angle), by any number of planes having a common point (a polyhedral angle), or by two intersecting curves (a curvilinear angle) . The angle in the last case is measured by the angle formed by tangents to the curves at their common point; it may also be formed by a curve and a straight line (mixed angle). The concept has also been extended to include the case of two lines in space that are not parallel and yet do not meet, like the upper edge of a box and a lower non-parallel one (skew angle). There are also spherical angles, formed by the intersection of two great circles of a sphere. An angle is measured by the amount of turning neces sary to bring one of the lines (arms, sides, legs) into coincidence with the other, this being measured on a circle with its vertex (from Lat. vertere, to turn) at the centre.
The early idea of angle limited the concept to an angle less than 180°. The demands of science lead modern writers to speak of a straight angle (180° ), a reflex angle, and of angles exceeding 36o°, as in speaking of an angle arising from turning a radius beyond a complete revolution. This is seen in any rotary motion. An angle less than a right angle (90°) is said to be acute; between 9o° and 180°, obtuse; and between 18o° and 36o°, reflex. The angle is the complement of angle a; for example, the com plement of 4o° is or 50°. The angle 18o°—b is the supplement of angle b.
The term angle is itself such a basal term that it is not possible to give a satisfactory definition employing terms more simple than itself. Hilbert's definition represents one of the late efforts : "Let a be any arbitrary plane and h, k any two distinct half-rays lying in a and emanating from the point o so as to form a part of two different straight lines. We shall call the system formed by these two half-rays h, k an angle." BIBLIOGRAPHY.-Sir T. L. Heath, The Thirteen Books of Euclid's Bibliography.-Sir T. L. Heath, The Thirteen Books of Euclid's Elements (1926) ; H. Schotten, Inhalt and Methode des planimet rischen Unterrichts (Leipzig, 189o, 1893) ; J. Tropfke, Geschichte der Elementar-Mathematik (1923).