Angle

two line bent at a point — A green angle formed by two red rays on the Cartesian coordinate system

In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight lines at a point. Formally, an angle is a figure lying in a plane formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.^[1]^[2] More generally angles are also formed wherever two lines, rays or line segments come together, such as at the corners of triangles and other polygons. An angle can be considered as the region of the plane bounded by the sides.^[3]^[4]^[a] Angles can also be formed by the intersection of two planes or by two intersecting curves, in which case the rays lying tangent to each curve at the point of intersection define the angle.

The term angle is also used for the size, magnitude or quantity of these types of geometric figures and in this context an angle consists of a number and unit of measurement. Angular measure or measure of angle are sometimes used to distinguish between the measurement and figure itself. The measurement of angles is intrinsically linked with circles and rotation. For an ordinary angle, this is often visualized or defined using the arc of a circle centered at the vertex and lying between the sides.

^ Hilbert, David. The Foundations of Geometry (PDF). p. 9.
^ Sidorov 2001
^ Evgrafov, M. A. (2019-09-18). Analytic Functions. Courier Dover Publications. ISBN 978-0-486-84366-7.
^ Papadopoulos, Athanase (2012). Strasbourg Master Class on Geometry. European Mathematical Society. ISBN 978-3-03719-105-7.
^ D'Andrea, Francesco (2023-08-19). A Guide to Penrose Tilings. Springer Nature. ISBN 978-3-031-28428-1.
^ Bulboacǎ, Teodor; Joshi, Santosh B.; Goswami, Pranay (2019-07-08). Complex Analysis: Theory and Applications. Walter de Gruyter GmbH & Co KG. ISBN 978-3-11-065803-3.
^ Redei, L. (2014-07-15). Foundation of Euclidean and Non-Euclidean Geometries according to F. Klein. Elsevier. ISBN 978-1-4832-8270-1.

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[:0-1] Hilbert, David. The Foundations of Geometry (PDF). p. 9.

[2] Sidorov 2001

[3] Evgrafov, M. A. (2019-09-18). Analytic Functions. Courier Dover Publications. ISBN 978-0-486-84366-7.

[4] Papadopoulos, Athanase (2012). Strasbourg Master Class on Geometry. European Mathematical Society. ISBN 978-3-03719-105-7.

[5] D'Andrea, Francesco (2023-08-19). A Guide to Penrose Tilings. Springer Nature. ISBN 978-3-031-28428-1.

[6] Bulboacǎ, Teodor; Joshi, Santosh B.; Goswami, Pranay (2019-07-08). Complex Analysis: Theory and Applications. Walter de Gruyter GmbH & Co KG. ISBN 978-3-11-065803-3.

[7] Redei, L. (2014-07-15). Foundation of Euclidean and Non-Euclidean Geometries according to F. Klein. Elsevier. ISBN 978-1-4832-8270-1.

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