Angles measure openings between rays that share a vertex
Two lines meeting at a point form an angle whose sides are those lines and whose vertex is the shared point. The same word names both the figure and its size. Measures link to circles and rotation, using degrees, radians, or turns, though no single formal definition satisfies every context.
Standardly an angle is two rays in a plane with a common endpoint; alternatively one may mean the opening between them, the region between them, or the rotation carrying one ray to the other. Segments meeting at polygon corners, or tangents to intersecting curves, also form angles. Sides typically split the plane into interior and exterior regions, the interior also called an angular sector. Notation uses an angle mark with one or three letters (vertex in the middle), Greek or Roman letters for measures, and sometimes signed measures in physics—positive anticlockwise, negative clockwise.
Common units are the degree, radian, and turn. A full angle—one complete rotation back to the start—measures 1 turn or 360 degrees; historically a straight angle was set at 180 degrees. A full turn is about 2π radians. The angle addition postulate says that if a ray from the vertex lies in the interior, the large angle's measure equals the sum of the two smaller ones. Named sizes include zero, acute (under 90 degrees), right (quarter turn), obtuse (between right and straight), straight (half turn), reflex (between straight and full), and full or perigon (one turn). Oblique angles are those not multiples of a right angle.
Adjacent angles share a vertex and a side without overlapping interiors. Vertical angles at a crossing of two lines are opposite and congruent. A transversal through a pair of lines creates alternate, corresponding, and consecutive interior and exterior angles. Complementary pairs sum to a right angle; supplementary pairs to a straight angle (linear pairs when adjacent); explementary or conjugate pairs to a full turn. In a right triangle the two acute corners complement each other; neighboring corners of a parallelogram, and opposite corners of a cyclic quadrilateral, often form supplementary pairs.
Interior angles of a simple polygon lie inside it; Euclidean triangles sum to a half turn and simple convex quadrilaterals to a full turn. An exterior angle is often the supplement of an interior angle, measuring the turn needed to trace the polygon, though some authors use an explementary convention instead. Dihedral angles between planes may be read as the acute angle of their normals. Measuring angles differs from measuring length: specially named angles such as the right angle structure the unit system itself rather than relying on an arbitrary meter-like choice alone.
Source: Angle
