Angle

In Euclidean 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.
Angles formed by two rays lie in the plane that contains the rays. Angles are also formed by the intersection of two planes. These are called dihedral angles. Two intersecting curves define also an angle, which is the angle of the tangents at the intersection point. For example, the spherical angle formed by two great circles on a sphere equals the dihedral angle between the planes containing 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.
History and etymology.
The word Angle came from a sanskrit word Angula, Aṅgula (अङ्गुल, “finger-breadth”) is the Sanskrit name for a unit of measurement. Aṅgula is one of the two basic units of measurement, according to the [Viṣṇudharmottara Purāṇa].
The word "angle" comes from the Latin word "angulus", meaning "corner"; cognate words are the Greek "(ankylοs)", meaning "crooked, curved," and the English word "ankle". Both are connected with the Proto-Indo-European root "*ank-", meaning "to bend" or "bow".
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.
Identifying angles.
In mathematical expressions, it is common to use Greek letters (α, β, γ, θ, φ, . . . ) as variables denoting the size of some angle (to avoid confusion with its other meaning, the symbol is typically not used for this purpose). Lower case Roman letters ("a", "b", "c", . . . ) are also used, as are upper case Roman letters in the context of polygons. 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 (in Unicode ) or formula_1. Where there is no risk of confusion, the angle may sometimes be referred to simply by its vertex (in this case "angle A").
Potentially, an angle denoted as, 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, in which case 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 the anticlockwise (positive) angle from C to B.
Types of angles.
Individual angles.
There is some common terminology for angles, whose measure is always non-negative (see #Positive and negative angles):
The names, intervals, and measuring units are shown in the table below:
Vertical and adjacent angle pairs.
When two straight lines intersect at a point, four angles are formed. Pairwise these angles are named according to their location relative to each other.
A transversal is a line that intersects a pair of (often parallel) lines, and is associated with "alternate interior angles", "corresponding angles", "interior angles", and "exterior angles".
Combining angle pairs.
Three special angle pairs involve the summation of angles:
Measuring angles.
The size of a geometric angle is usually characterized by the magnitude of the smallest rotation that maps one of the rays into the other. Angles that have the same size are said to be "equal" or "congruent" or "equal in measure".
In some contexts, such as identifying a point on a circle or describing the "orientation" of an object in two dimensions relative to a reference orientation, angles that differ by an exact multiple of a full turn are effectively equivalent. In other contexts, such as identifying a point on a spiral curve or describing the "cumulative rotation" of an object in two dimensions relative to a reference orientation, angles that differ by a non-zero multiple of a full turn are not equivalent.
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 ratio of the length s of the arc by the radius r of the circle is the number of radians in the angle. Conventionally, in mathematics and in the SI, the radian is treated as being equal to the dimensionless value 1.
The angle expressed another angular unit may then be obtained by multiplying the angle by a suitable conversion constant of the form , where "k" is the measure of a complete turn expressed in the chosen unit (for example, "k" = 360° for degrees or 400 grad for gradians):
The value of 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. (Proof. The formula above can be rewritten as One turn, for which units, corresponds to an arc equal in length to the circle's circumference, which is 2"r", so . Substituting "n" for "θ" and 2"r" for "s" in the formula, results in ) 
Angle addition postulate.
The angle addition postulate states that if "B" is in the interior of angle "AOC", then
The measure of the angle "AOC" is the sum of the measure of angle AOB and the measure of angle "BOC". In this postulate it does not matter in which unit the angle is measured as long as each angle is measured in the same unit.
Units.
Units used to represent angles are listed below in descending magnitude order. Of these units, the "degree" and the "radian" are by far the most commonly used. Angles expressed in radians are dimensionless for dimensional analysis.
Most units of angular measurement are defined such that one "turn" (i.e. one full circle) is equal to "n" units, for some whole number "n". The two exceptions are the radian and the diameter part.
Positive and negative angles.
Although the definition of the measurement of an angle does not support the concept of a negative angle, it is frequently useful to impose a convention that allows positive and negative angular values to represent orientations and/or rotations in opposite directions relative to some reference.
In a two-dimensional Cartesian coordinate system, an angle is typically defined by its two sides, with its vertex at the origin. The "initial side" is on the positive x-axis, while the other side or "terminal side" is defined by the measure from the initial side in radians, degrees, or turns. With "positive angles" representing rotations toward the positive y-axis and "negative angles" representing rotations toward the negative "y"-axis. When Cartesian coordinates are represented by "standard position", defined by the "x"-axis rightward and the "y"-axis upward, positive rotations are anticlockwise and negative rotations are clockwise.
In many contexts, an angle of −"θ" is effectively equivalent to an angle of "one full turn minus "θ"". For example, an orientation represented as  −45° is effectively equivalent to an orientation represented as 360° − 45° or 315°. Although the final position is the same, a physical rotation (movement) of  −45° is not the same as a rotation of 315° (for example, the rotation of a person holding a broom resting on a dusty floor would leave visually different traces of swept regions on the floor).
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 or azimuth are measured relative to north. By convention, viewed from above, bearing angles are positive clockwise, so a bearing of 45° corresponds to a north-east orientation. Negative bearings are not used in navigation, so a north-west orientation corresponds to a bearing of 315°.
Alternative ways of measuring the size of an angle.
There are several alternatives to measuring the size of an angle by the angle of rotation.
The "slope" or "gradient" is equal to the tangent of the angle, or sometimes (rarely) the sine; a gradient is often expressed as a percentage. For very small values (less than 5%), the grade of a slope is approximately the measure of the angle in radians.
In rational geometry the "spread" between two lines is defined as the square of the sine of the angle between the lines. As the sine of an angle and the sine of its supplementary angle are the same, any angle of rotation that maps one of the lines into the other leads to the same value for the spread between the lines.
Astronomical approximations.
Astronomers measure angular separation of objects in degrees from their point of observation.
These measurements clearly depend on the individual subject, and the above should be treated as rough rule of thumb approximations only.
Angles between curves.
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. , on both sides, κυρτός, convex) or "cissoidal" (Gr. κισσός, ivy), biconvex; "xystroidal" or "sistroidal" (Gr. ξυστρίς, a tool for scraping), concavo-convex; "amphicoelic" (Gr. κοίλη, a hollow) or "angulus lunularis", biconcave.
Bisecting and trisecting angles.
The ancient Greek mathematicians knew how to bisect an angle (divide it into two angles of equal measure) using only a compass and straightedge, but could only trisect certain angles. In 1837 Pierre Wantzel showed that for most angles this construction cannot be performed.
Dot product and generalisations.
In the Euclidean space, the angle "θ" between two Euclidean 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 and between skew lines from their vector equations.
Inner product.
To define angles in an abstract real inner product space, we replace the Euclidean dot product ( · ) by the inner product formula_6, i.e.
In a complex inner product space, the expression for the cosine above may give non-real values, so it is replaced with
or, more commonly, using the absolute value, with
The latter definition ignores the direction of the vectors and thus describes the angle between one-dimensional subspaces formula_10 and formula_11 spanned by the vectors formula_12 and formula_13 correspondingly.
Angles between subspaces.
The definition of the angle between one-dimensional subspaces formula_10 and formula_11 given by
in a Hilbert space can be extended to subspaces of any finite dimensions. Given two subspaces formula_17, formula_18 with formula_19, this leads to a definition of formula_20 angles called canonical or principal angles between subspaces.
Angles in Riemannian geometry.
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",
Hyperbolic angle.
A hyperbolic angle is an argument of a hyperbolic function just as the "circular angle" is the argument of a circular function. The comparison can be visualized as the size of the openings of a hyperbolic sector and a circular sector since the areas of these sectors correspond to the angle magnitudes in each case. Unlike the circular angle, the hyperbolic angle is unbounded. When the circular and hyperbolic functions are viewed as infinite series in their angle argument, the circular ones are just alternating series forms of the hyperbolic functions. This weaving of the two types of angle and function was explained by Leonhard Euler in "Introduction to the Analysis of the Infinite".
Angles in geography and astronomy.
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 center 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 center 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.
In both geography and astronomy, a sighting direction can be specified in terms of a vertical angle such as altitude /elevation with respect to the horizon as well as the azimuth with respect to north.
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's diameter subtends an angle of half a degree." The small-angle formula can be used to convert such an angular measurement into a distance/size ratio.