In geometry, an angle (in full, plane angle) is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.^{[1]} The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide with the other (see "Measuring angles", below). Where there is no possibility of confusion, the term "angle" is used interchangeably for both the geometric configuration itself and for its angular magnitude (which is simply a numerical quantity).
The word angle comes from the Latin word angulus, meaning "a corner". The word angulus is a diminutive, of which the primitive form, angus, does not occur in Latin. Cognate words are the Greek ἀγκύλος (ankylοs), meaning "crooked, curved," and the English word "ankle." Both are connected with the ProtoIndoEuropean root *ank, meaning "to bend" or "bow".^{[2]}
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.^{[3]}
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In order to measure an angle θ, a circular arc centered at the vertex of the angle is drawn, e.g. with a pair of compasses. The length of the arc s is then divided by the radius of the circle r, and possibly multiplied by a scaling constant k (which depends on the units of measurement that are chosen):
The value of θ thus defined is independent of the size of the circle: if the length of the radius is changed then the arc length changes in the same proportion, so the ratio s/r is unaltered.
In many geometrical situations, angles that differ by an exact multiple of a full circle are effectively equivalent (it makes no difference how many times a line is rotated through a full circle because it always ends up in the same place). However, this is not always the case. For example, when tracing a curve such as a spiral using polar coordinates, an extra full turn gives rise to a quite different point on the curve.
Angles are considered dimensionless, since they are defined as the ratio of lengths. There are, however, several units used to measure angles, depending on the choice of the constant k in the formula above. Of these units, treated in more detail below, the degree and the radian are by far the most common.
With the notable exception of the radian, most units of angular measurement are defined such that one full circle (i.e. one revolution) is equal to n units, for some whole number n. For example, in the case of degrees, n = 360. A full circle of n units is obtained by setting k = n/(2π) in the formula above. (Proof. The formula above can be rewritten as k = θr/s. One full circle, for which θ = n units, corresponds to an arc equal in length to the circle's circumference, which is 2πr, so s = 2πr. Substituting n for θ and 2πr for s in the formula, results in k = nr/(2πr) = n/(2π).)
A convention universally adopted in mathematical writing is that angles given a sign are positive angles if measured anticlockwise, and negative angles if measured clockwise, from a given line. If no line is specified, it can be assumed to be the xaxis in the Cartesian plane. In many geometrical situations a negative angle of −θ is effectively equivalent to a positive angle of "one full rotation less θ". For example, a clockwise rotation of 45° (that is, an angle of −45°) is often effectively equivalent to an anticlockwise rotation of 360° − 45° (that is, an angle of 315°).
In three dimensional geometry, "clockwise" and "anticlockwise" have no absolute meaning, so the direction of positive and negative angles must be defined relative to some reference, which is typically a vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie.
In navigation, bearings are measured from north, increasing clockwise, so a bearing of 45 degrees is northeast. Negative bearings are not used in navigation, so northwest is 315 degrees.
These measurements clearly depend on the individual subject, and the above should be treated as rough approximations only.
In mathematical expressions, it is common to use Greek letters (α, β, γ, θ, φ, ...) to serve as variables standing for the size of some angle. (To avoid confusion with its other meaning, the symbol π is typically not used for this purpose.) Lower case roman letters (a, b, c, ...) are also used. See the figures in this article for examples.
In geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB and AC (i.e. the lines from point A to point B and point A to point C) is denoted ∠BAC or BÂC. Sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex ("angle A").
Potentially, an angle denoted, say, ∠BAC might refer to any of four angles: the clockwise angle from B to C, the anticlockwise angle from B to C, the clockwise angle from C to B, or the anticlockwise angle from C to B, where the direction in which the angle is measured determines its sign (see Positive and negative angles). However, in many geometrical situations it is obvious from context that the positive angle less than or equal to 180° degrees is meant, and no ambiguity arises. Otherwise, a convention may be adopted so that ∠BAC always refers to the anticlockwise (positive) angle from B to C, and ∠CAB to the anticlockwise (positive) angle from C to B.




A Euclidean angle is completely determined by the corresponding right triangle. In particular, if θ is a Euclidean angle, it is true that
and
for two numbers x and y. So an angle in the Euclidean plane can be legitimately given by two numbers x and y.
To the ratio y/x there correspond two angles in the geometric range 0 < θ < 2π, since
Suppose we have two unit vectors and in the euclidean plane . Then there exists one positive isometry (a rotation), and one only, from to that maps u onto v. Let r be such a rotation. Then the relation defined by is an equivalence relation and we call angle of the rotation r the equivalence class , where denotes the unit circle of . The angle between two vectors will simply be the angle of the rotation that maps one onto the other. We have no numerical way of determining an angle yet. To do this, we choose the vector (1,0), then for any point M on at distance θ from (1,0) (on the circle), let . If we call r_{θ} the rotation that transforms (1,0) into , then is a bijection, which means we can identify any angle with a number between 0 and .
The angle between a line and a curve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangents at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—amphicyrtic (Gr. ἀμφί, on both sides, κυρτός, convex) or cissoidal (Gr. κισσός, ivy), biconvex; xystroidal or sistroidal (Gr. ξυστρίς, a tool for scraping), concavoconvex; amphicoelic (Gr. κοίλη, a hollow) or angulus lunularis, biconcave.^{[8]}
In the Euclidean plane, the angle θ between two vectors u and v is related to their dot product and their lengths by the formula
This formula supplies an easy method to find the angle between two planes (or curved surfaces) from their normal vectors. It also allows a calculation of the angle between skew lines from their vector equations, and it allows one to define angles in any real inner product space, replacing the Euclidean dot product ( · ) by the Hilbert space inner product .
In Riemannian geometry, the metric tensor is used to define the angle between two tangents. Where U and V are tangent vectors and g_{ij} are the components of the metric tensor G,
In geography, the location of any point on the Earth can be identified using a geographic coordinate system. This system specifies the latitude and longitude of any location in terms of angles subtended at the centre of the Earth, using the equator and (usually) the Greenwich meridian as references.
In astronomy, a given point on the celestial sphere (that is, the apparent position of an astronomical object) can be identified using any of several astronomical coordinate systems, where the references vary according to the particular system. Astronomers measure the angular separation of two stars by imagining two lines through the centre of the Earth, each intersecting one of the stars. The angle between those lines can be measured, and is the angular separation between the two stars.
Astronomers also measure the apparent size of objects as an angular diameter. For example, the full moon has an angular diameter of approximately 0.5°, when viewed from Earth. One could say, "The Moon subtends an angle of half a degree." The smallangle formula can be used to convert such an angular measurement into a distance/size ratio.
This article incorporates text from the Encyclopædia Britannica, Eleventh Edition, a publication now in the public domain.
ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. evyKOs, a bend; both connected with the Aryan root ank, to bend: see Angling), in geometry, the inclination of one line or plane to another. Euclid (Elements, book 1) 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 (see Geometry, Euclidean). According to Proclus an angle must be either a quality or a quantity, or a relationship. The first concept was utilized 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. A discussion of these concepts and the various definitions of angles in Euclidean geometry is to be found in W. B. Frankland, The First Book of Euclid's Elements (1905). Following Euclid, a right angle is formed by a straight line standing upon another straight line so as to make the adjacent angles equal; any angle less than a right angle is termed an acute angle, and any angle greater than a right angle an obtuse angle. The difference between an acute angle and a right angle is termed the complement of the angle, and between an angle and two right angles the supplement of the angle. The generalized view of angles and their measurement is treated in the article Trigonometry. A solid angle is definable as the space contained by three or more planes intersecting in a common point; it is familiarly represented by a corner. The angle between two planes is termed dihedral, between three trihedral, between any number more than three polyhedral. A spherical angle is a particular dihedral angle; it is the angle between two intersecting arcs on a sphere, and is measured by the angle between the planes containing the arcs and the centre of the sphere.
The angle between a line and a curve (mixed angle) or between two curves (curvilinear angle) is measured by the angle between the line and the tangent at the point of intersection, or between the tangents to both curves at their common point. Various names (now rarely, if ever, used) have been given to particular cases: amphicyrtic (Gr. d 4 L, on both sides, KuprO , convex) or cissoidal (Gr. Klqvos, ivy), biconvex; xystroidal or sistroidal (Gr. uoTpis, a tool for scraping), concavoconvex; amphicoelic (Gr. KoLA77, a hollow) or angulus lunularis, biconcave.
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Categories: ANEANT  Mathematics
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Singular 
Plural 
Angle (plural Angles)
An angle is the space between two lines that meet each other. When two lines meet at a point, an angle is formed. The two lines are called the arms of the angle. The point they meet is called the vertex.
To measure the size of an angle, we use degree. Degree is a standard unit. Degree is denoted by the symbol '°'. A degree can be divided into 60 minutes (1° = 60'), and a minute can be also be divided into 60 seconds (1' = 60"). In mathematics, angles are usually measured in radians.
An acute angle is an angle less than 90°. A right angle is an angle equal to 90°. An obtuse angle is an angle greater than 90°. A straight angle (or straight line) is an angle equal to 180°. A reflex angle is an angle greater than 180° but less than 360°.
Supplementary angles are two angles with the sum equal to 180°.
Two angles that sum to one right angle (90°) are called complementary angles.
Two angles that sum to one full circle (360°) are called explementary angles or conjugate angles.
People usually use a protractor to measure and draw angles. Sometimes, people use an angle ruler to measure angles.
