an angle

Open a pair of scissors, swing a door on its hinge, or watch the hands of a clock spread apart: in each case two straight arms share a fixed corner and the gap between them widens or narrows. That amount of turning between two arms is an angle.

Formally, an angle is the figure formed by two rays that share a common endpoint. The shared endpoint is the vertex, and the two rays are the sides (or arms) of the angle. We name an angle by three points — angle ABC, with the vertex letter B always in the middle — or by the vertex alone (angle B) when there is no ambiguity, or by a number or single Greek letter marked inside it. The size of an angle, written m(angle ABC), measures how much one side is rotated from the other and does not depend on how long the rays are drawn.

Angles are everywhere in geometry — every polygon corner, every intersection of lines, every triangle carries them — and they come with their own arithmetic. The angle addition postulate lets neighbouring angles combine, an angle bisector splits an angle into two equal parts, and angles are sorted by size into acute, right, obtuse, straight, and reflex. A subtle point: the same two rays bound two angles, a smaller one and a reflex one summing to 360 degrees; by convention 'the angle' means the smaller unless the reflex is meant explicitly.