Angle measure.svg

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 may also define an angle, which is the angle of the rays lying tangent to the respective curves at their point of intersection.

''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'' 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. In contexts where this is not confusing, an angle may be denoted by the upper case Roman letter denoting its vertex. See the figures in this article for examples.

In geometric figures, angles may also be identified by the three points that define them. For example, the angle with vertex A formed by the rays AB and AC (that is, the lines from point A to points B and C) is denoted or $\widehat$. Where there is no risk of confusion, the angle may sometimes be referred to by its vertex (in this case "angle A").

Potentially, an angle denoted as, say, , 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 '). 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 always refers to the anticlockwise (positive) angle from B to C, and 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 '):
* An angle equal to 0° or not turned is called a zero angle.
* An angle smaller than a right angle (less than 90°) is called an ''acute angle'' ("acute" meaning "sharp").
* An angle equal to  turn (90° or radians) is called a ''right angle''. Two lines that form a right angle are said to be '' normal'', ''orthogonal'', or '' perpendicular''.
* An angle larger than a right angle and smaller than a straight angle (between 90° and 180°) is called an ''obtuse angle'' ("obtuse" meaning "blunt").
* An angle equal to  turn (180° or radians) is called a ''straight angle''.
* An angle larger than a straight angle but less than 1 turn (between 180° and 360°) is called a ''reflex angle''.
* An angle equal to 1 turn (360° or 2 radians) is called a ''full angle'', ''complete angle'', ''round angle'' or a ''perigon''.
* An angle that is not a multiple of a right angle is called an ''oblique angle''.

The names, intervals, and measuring units are shown in the table below:

Equivalence angle pairs

* Angles that have the same measure (i.e. the same magnitude) are said to be ''equal'' or '' congruent''. An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g. all ''right angles'' are equal in measure).
* Two angles that share terminal sides, but differ in size by an integer multiple of a turn, are called ''coterminal angles''.
* A ''reference angle'' is the acute version of any angle determined by repeatedly subtracting or adding straight angle ( turn, 180°, or radians), to the results as necessary, until the magnitude of the result is an acute angle, a value between 0 and turn, 90°, or radians. For example, an angle of 30 degrees has a reference angle of 30 degrees, and an angle of 150 degrees also has a reference angle of 30 degrees (180° − 150°). An angle of 750 degrees has a reference angle of 30 degrees (750° − 720°).

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 pair of angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are called ''vertical angles'' or ''opposite angles'' or ''vertically opposite angles''. They are abbreviated as ''vert. opp. ∠s''.
:The equality of vertically opposite angles is called the ''vertical angle theorem''. Eudemus of Rhodes attributed the proof to Thales of Miletus. The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. According to a historical note, when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal. Thales concluded that one could prove that all vertical angles are equal if one accepted some general notions such as:
:* All straight angles are equal.
:* Equals added to equals are equal.
:* Equals subtracted from equals are equal.

:When two adjacent angles form a straight line, they are supplementary. Therefore, if we assume that the measure of angle ''A'' equals ''x'', then the measure of angle ''C'' would be . Similarly, the measure of angle ''D'' would be . Both angle ''C'' and angle ''D'' have measures equal to and are congruent. Since angle ''B'' is supplementary to both angles ''C'' and ''D'', either of these angle measures may be used to determine the measure of Angle ''B''. Using the measure of either angle ''C'' or angle ''D'', we find the measure of angle ''B'' to be . Therefore, both angle ''A'' and angle ''B'' have measures equal to ''x'' and are equal in measure.

* ''Adjacent angles'', often abbreviated as ''adj. ∠s'', are angles that share a common vertex and edge but do not share any interior points. In other words, they are angles that are side by side, or adjacent, sharing an "arm". Adjacent angles which sum to a right angle, straight angle, or full angle are special and are respectively called ''complementary'', ''supplementary'' and ''explementary'' angles (see ' below).

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:

* ''Complementary angles'' are angle pairs whose measures sum to one right angle ( turn, 90°, or radians). If the two complementary angles are adjacent, their non-shared sides form a right angle. In Euclidean geometry, the two acute angles in a right triangle are complementary, because the sum of internal angles of a triangle is 180 degrees, and the right angle itself accounts for 90 degrees.
:The adjective complementary is from Latin ''complementum'', associated with the verb ''complere'', "to fill up". An acute angle is "filled up" by its complement to form a right angle.
:The difference between an angle and a right angle is termed the ''complement'' of the angle.
:If angles ''A'' and ''B'' are complementary, the following relationships hold:
::
\begin
& \sin^2A + \sin^2B = 1 & & \cos^2A + \cos^2B = 1 \\ pt& \tan A = \cot B & & \sec A = \csc B
\end

:(The tangent of an angle equals the cotangent of its complement and its secant equals the cosecant of its complement.)
:The prefix " co-" in the names of some trigonometric ratios refers to the word "complementary".

* Two angles that sum to a straight angle ( turn, 180°, or radians) are called ''supplementary angles''.
:If the two supplementary angles are adjacent (i.e. have a common vertex and share just one side), their non-shared sides form a straight line. Such angles are called a ''linear pair of angles''. However, supplementary angles do not have to be on the same line, and can be separated in space. For example, adjacent angles of a parallelogram are supplementary, and opposite angles of a cyclic quadrilateral (one whose vertices all fall on a single circle) are supplementary.
:If a point P is exterior to a circle with center O, and if the tangent lines from P touch the circle at points T and Q, then ∠TPQ and ∠TOQ are supplementary.
:The sines of supplementary angles are equal. Their cosines and tangents (unless undefined) are equal in magnitude but have opposite signs.
:In Euclidean geometry, any sum of two angles in a triangle is supplementary to the third, because the sum of internal angles of a triangle is a straight angle.

* Two angles that sum to a complete angle (1 turn, 360°, or 2 radians) are called ''explementary angles'' or ''conjugate angles''.
*: The difference between an angle and a complete angle is termed the ''explement'' of the angle or ''conjugate'' of an angle.

Polygon-related angles

* An angle that is part of a simple polygon is called an '' interior angle'' if it lies on the inside of that simple polygon. A simple concave polygon has at least one interior angle that is a reflex angle.
*: In Euclidean geometry, the measures of the interior angles of a triangle add up to radians, 180°, or turn; the measures of the interior angles of a simple convex quadrilateral add up to 2 radians, 360°, or 1 turn. In general, the measures of the interior angles of a simple convex polygon with ''n'' sides add up to (''n'' − 2) radians, or (''n'' − 2)180 degrees, (''n'' − 2)2 right angles, or (''n'' − 2) turn.
* The supplement of an interior angle is called an '' exterior angle'', that is, an interior angle and an exterior angle form a linear pair of angles. There are two exterior angles at each vertex of the polygon, each determined by extending one of the two sides of the polygon that meet at the vertex; these two angles are vertical and hence are equal. An exterior angle measures the amount of rotation one has to make at a vertex to trace out the polygon. If the corresponding interior angle is a reflex angle, the exterior angle should be considered negative. Even in a non-simple polygon it may be possible to define the exterior angle, but one will have to pick an orientation of the plane (or surface) to decide the sign of the exterior angle measure.
*: In Euclidean geometry, the sum of the exterior angles of a simple convex polygon, if only one of the two exterior angles is assumed at each vertex, will be one full turn (360°). The exterior angle here could be called a ''supplementary exterior angle''. Exterior angles are commonly used in Logo Turtle programs when drawing regular polygons.
* In a triangle, the bisectors of two exterior angles and the bisector of the other interior angle are concurrent (meet at a single point).Johnson, Roger A. ''Advanced Euclidean Geometry'', Dover Publications, 2007.
* In a triangle, three intersection points, each of an external angle bisector with the opposite extended side, are collinear.
* In a triangle, three intersection points, two of them between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended, are collinear.
* Some authors use the name ''exterior angle'' of a simple polygon to mean the ''explement exterior angle'' (''not'' supplement!) of the interior angle. This conflicts with the above usage.

Plane-related angles

* The angle between two planes (such as two adjacent faces of a polyhedron) is called a '' dihedral angle''. It may be defined as the acute angle between two lines normal to the planes.
* The angle between a plane and an intersecting straight line is equal to ninety degrees minus the angle between the intersecting line and the line that goes through the point of intersection and is normal to the plane.

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

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