Editing Angle (section)

{{Short description|Figure formed by two rays meeting at a common point}}
{{Distinguish|Angel}}
{{About|angles in geometry}}
[[File:Angle.svg|alt=two line bent at a point|thumb|upright=1.25|A green angle formed by two red [[Ray (geometry)|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 [[Line (geometry)|lines]] at a point. Formally, an angle is a figure lying in a [[Euclidean plane|plane]] formed by two [[Ray (geometry)|rays]], called the ''[[Side (plane geometry)|sides]]'' of the angle, sharing a common endpoint, called the ''[[vertex (geometry)|vertex]]'' of the angle.<ref>{{Cite book |last=Hilbert |first=David |url=https://math.berkeley.edu/~wodzicki/160/Hilbert.pdf |title=The Foundations of Geometry |pages=9}}</ref><ref>{{harvnb|Sidorov|2001|ignore-err=yes}}</ref> More generally angles are also formed wherever two lines, rays or  [[Line segment|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.<ref>{{Cite book |last=Evgrafov |first=M. A. |url=https://books.google.com/books?id=N8-wDwAAQBAJ&dq=angle+and+%2522angular+sector%2522+domain&pg=PA126 |title=Analytic Functions |date=2019-09-18 |publisher=Courier Dover Publications |isbn=978-0-486-84366-7 |language=en}}</ref><ref>{{Cite book |last=Papadopoulos |first=Athanase |url=https://books.google.com/books?id=f6yZeVMqhNEC&dq=angle+and+%2522angular+sector%2522+region&pg=PA12 |title=Strasbourg Master Class on Geometry |date=2012 |publisher=European Mathematical Society |isbn=978-3-03719-105-7 |language=en}}</ref>{{efn|An angular sector can be constructed by the combination of two rotated [[half-plane]]s, either their intersection or union (in the case of acute or obtuse angles, respectively).<ref>{{Cite book |last=D'Andrea |first=Francesco |url=https://books.google.com/books?id=BszREAAAQBAJ&dq=angle+and+%2522angular+sector%2522&pg=PA68 |title=A Guide to Penrose Tilings |date=2023-08-19 |publisher=Springer Nature |isbn=978-3-031-28428-1 |language=en}}</ref><ref>{{Cite book |last1=Bulboacǎ |first1=Teodor |url=https://books.google.com/books?id=r0miDwAAQBAJ&dq=angle+and+%2522angular+sector%2522+half-planes&pg=PT22 |title=Complex Analysis: Theory and Applications |last2=Joshi |first2=Santosh B. |last3=Goswami |first3=Pranay |date=2019-07-08 |publisher=Walter de Gruyter GmbH & Co KG |isbn=978-3-11-065803-3 |language=en}}</ref> It corresponds to a [[circular sector]] of infinite radius and a flat [[pencil of half-lines]].<ref>{{Cite book |last=Redei |first=L. |url=https://books.google.com/books?id=XMTSBQAAQBAJ&dq=half-pencil+of+lines&pg=PA45 |title=Foundation of Euclidean and Non-Euclidean Geometries according to F. Klein |date=2014-07-15 |publisher=Elsevier |isbn=978-1-4832-8270-1 |language=en}}</ref>}} Angles can also be formed by the intersection of two planes or by two intersecting [[curve]]s, 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 (mathematics)|magnitude]] or [[Physical quantity|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 (geometry)|arc]] of a [[circle]] centered at the vertex and lying between the sides.