## ::Angle

### ::concepts

**Angle**::angles Sfrac::right Measure::**angle** Style::between ''n''::width Equal::geometry

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In planar geometry, an **angle** is the figure formed by two rays, called the *sides* of the angle, sharing a common endpoint, called the *vertex* of the angle.<ref>{{#invoke:citation/CS1|citation
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Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in Euclidean and other spaces. These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles.

*Angle* is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation.

The word *angle* comes from the Latin word *angulus*, meaning "corner"; cognate words are the Greek ἀγκύλος{{#invoke:Category handler|main}} *(ankylοs)*, meaning "crooked, curved," and the English word "ankle". Both are connected with the Proto-Indo-European root **ank-*, meaning "to bend" or "bow".<ref>{{#invoke:citation/CS1|citation
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Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. According to Proclus an angle must be either a quality or a quantity, or a relationship. The first concept was used by Eudemus, who regarded an angle as a deviation from a straight line; the second by Carpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept, although his definitions of right, acute, and obtuse angles are certainly quantitative.<ref>{{#invoke:Footnotes|harvard_citation_no_bracket}}; {{#invoke:Footnotes|harvard_citation_no_bracket}}</ref>

**Angle sections**

Intro Identifying angles Types of angles Angles between curves Bisecting and trisecting angles Dot product and generalisation Inner product Angles between subspaces Angles in Riemannian geometry Angles in geography and astronomy See also Notes References External links

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