# Plane angle | Wikipedia audio article In plane 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 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 (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, although his definitions of right, acute, and obtuse angles are certainly quantitative == Identifying angles == 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, 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 U+2220 ∠ ANGLE) or B A C ^ {\displaystyle {\widehat {\rm {BAC}}}} 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 == Types of angles == === Individual angles === An angle equal to 0° or not turned is called a zero angle Angles smaller than a right angle (less than 90°) are called acute angles (“acute” meaning “sharp”) An angle equal to 1/4 turn (90° or π/2 radians) is called a right angle. Two lines that form a right angle are said to be normal, orthogonal, or perpendicular Angles larger than a right angle and smaller than a straight angle (between 90° and 180°) are called obtuse angles (“obtuse” meaning “blunt”) An angle equal to 1/2 turn (180° or π radians) is called a straight angle Angles larger than a straight angle but less than 1 turn (between 180° and 360°) are called reflex angles An angle equal to 1 turn (360° or 2π radians) is called a full angle, complete angle, round angle or a perigon Angles that are not right angles or a multiple of a right angle are called oblique angles.The names, intervals, and measured units are shown in a 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 which 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 (1/2 turn, 180°, or π radians), to the results as necessary, until the magnitude of result is an acute angle, a value between 0 and 1/4 turn, 90°, or π/2 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, and equals subtracted from equals are equal.In the figure, assume the measure of Angle A = x. When two adjacent angles form a straight line, they are supplementary. Therefore, the measure of Angle C = 180 − x. Similarly, the measure of Angle D = 180 − x. Both Angle C and Angle D have measures equal to 180 − x 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 = 180 − (180 − x) = 180 − 180 + x = x. 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 “Combine angle pairs” 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 === There are three special angle pairs which involve the summation of angles: Complementary angles are angle pairs whose measures sum to one right angle (1/4 turn, 90°, or π/2 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 ninety 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: sin 2 ⁡ A + sin 2 ⁡ B = 1 cos 2 ⁡ A + cos 2 ⁡ B = 1 tan ⁡ A = cot ⁡ B sec ⁡ A = csc ⁡ B {\displaystyle {\begin{aligned}&\sin ^{2}A+\sin ^{2}B=1&&\cos ^{2}A+\cos ^{2}B=1\\[3pt]&\tan A=\cot B&&\sec A=\csc B\end{aligned}}} (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 (1/2 turn, 180°, or π 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 2π 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 π radians, 180°, or 1/2 turn; the measures of the interior angles of a simple convex quadrilateral add up to 2π 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)π radians, or 180(n − 2) degrees, (2n − 4) right angles, or (n/2 − 1) 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 angles 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 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 Geometry 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) 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 simply 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 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 measure of the angle in radians The measure of the angle in another angular unit is then obtained by multiplying its measure in radians by the scaling factor k/2π, where k is the measure of a complete turn in the chosen unit (for example 360 for degrees or 400 for gradians): θ = k s 2 π r {\displaystyle \theta =k{\frac {s}{2\pi r}}.} 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. (Proof. The formula above can be rewritten as k = θr/s. One turn, 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.) === Angle addition postulate === The angle addition postulate states that if B is in the interior of angle AOC, then m ∠ A O C = m ∠ A O B + m ∠ B O C {\displaystyle m\angle AOC=m\angle AOB+m\angle BOC} 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  