Angle

Not to be confused with Angel.

This article is about angles in geometry. For other uses, see Angle (disambiguation).

In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight lines at a point. Formally, an angle is a figure lying in a plane formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.^[1][2] More generally angles are also formed wherever two lines, rays or 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.^[3][4][a] Angles can also be formed by the intersection of two planes or by two intersecting curves, in which case the rays lying tangent to each curve at the point of intersection define the angle.

A green angle formed by two red rays on the Cartesian coordinate system

The term angle is also used for the size, magnitude or 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 of a circle centered at the vertex and lying between the sides. One common measure of the vertex angle of a plane circular sector is the relative arc length, i.e. the length of the circular arc divided by the length of the radius. This is sometimes called the "circular measure" of the angle and is equivalent to the "radian measure".

Fundamentals

An angle is a figure lying in a plane formed by two distinct rays (half-lines emanating indefinitely from an endpoint in one direction), which share a common endpoint. The rays are called the sides or arms of the angle, and the common endpoint is called the vertex. The sides divide the plane into two regions: the interior of the angle and the exterior of the angle.^[1]

Notation

${\displaystyle \angle {\text{BAC}}}$ is formed by rays ${\displaystyle {\vec {\text{AB}}}}$ and ${\displaystyle {\vec {\text{AC}}}}$. ${\displaystyle \theta }$ is the conventional measure of ${\displaystyle \angle {\text{BAC}}}$ and ${\displaystyle \beta }$ is an alternative measure.

An angle symbol (${\displaystyle \angle }$ or ${\displaystyle {\widehat {\quad }}}$, read as "angle") together with one or three defining points is used to identify angles in geometric figures. For example, the angle with vertex A formed by the rays ${\displaystyle {\vec {\text{AB}}}}$ and ${\displaystyle {\vec {\text{AC}}}}$ is denoted as ${\displaystyle \angle {\text{A}}}$ (using the vertex alone) or ${\displaystyle \angle {\text{BAC}}}$ (with the vertex always named in the middle). The size or measure of the angle is denoted ${\displaystyle m\angle {\text{A}}}$ or ${\displaystyle m\angle {\text{BAC}}}$.

In geometric figures and mathematical expressions, it is also common to use Greek letters (α, β, γ, θ, φ, ...) or lower case Roman letters (a, b, c, ...) as variables to represent the size of an angle.^[8]

Conventionally, angle size is measured "between" the sides through the interior of the angle and given as a magnitude or scalar quantity. At other times it might be measured through the exterior of the angle or given as a signed number to indicate a direction of measurement.^{[citation needed]}

Units of measurement

Angles are measured in various units, the most common being the degree (denoted by the symbol °), radian (denoted by the symbol rad) and turn. These units differ in the way they divide up a full angle, an angle where one ray, initially congruent to the other, performs a compete rotation about the vertex to return back to its starting position.^[9]

Degrees and turns are defined directly with reference to a full angle, which measures 1 turn or 360°.^[10] A measure in turns gives an angle's size as a proportion of a full angle and a degree can be considered as a subdivision of a turn. Radians are not defined directly in relation to a full angle (see § Measuring angles), but in such a way that its measure is 2π rad, approximately 6.28 rad.^[11]

Common types of angles

Common angles

• An angle equal to 0° or not turned is called a zero angle.^[12]
• An angle smaller than a right angle (less than 90°) is called an acute angle.^[13]
• An angle equal to ⁠1/4⁠ turn (90° or ⁠π/2⁠ rad) is called a right angle. Two lines that form a right angle are said to be normal, orthogonal, or perpendicular.^[14]
• An angle larger than a right angle and smaller than a straight angle (between 90° and 180°) is called an obtuse angle^[13] ("obtuse" meaning "blunt").
• An angle equal to ⁠1/2⁠ turn (180° or π rad) is called a straight angle.^[12]
• 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 2π rad) is called a full angle, complete angle, round angle or 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:

Name	zero angle	acute angle	right angle	obtuse angle	straight angle	reflex angle	full angle
Unit	Interval
turn	0 turn	(0, ⁠1/4⁠) turn	⁠1/4⁠ turn	(⁠1/4⁠, ⁠1/2⁠) turn	⁠1/2⁠ turn	(⁠1/2⁠, 1) turn	1 turn
degree	0°	(0, 90)°	90°	(90, 180)°	180°	(180, 360)°	360°
radian	0 rad	(0, ⁠1/2⁠π) rad	⁠1/2⁠π rad	(⁠1/2⁠π, π) rad	π rad	(π, 2π) rad	2π rad

Addition and subtraction

The angle addition postulate defines addition and subtraction of angles: θ + α = φ; φ − α = θ.

The angle addition postulate states that if D is a point lying in the interior of ${\displaystyle \angle {\text{BAC}}}$ then:^[15] $${\displaystyle m\angle {\text{BAC}}=m\angle {\text{BAD}}+m\angle {\text{DAC}}.}$$ This relationship defines what it means add any two angles: their vertices are placed together while sharing a side to create a new larger angle. The measure of the new larger angle is the sum of the measures of the two angles. Subtraction follows from rearrangement of the formula.^[15]

Types

"Oblique angle" redirects here. For the cinematographic technique, see Dutch angle.

Adjacent and vertical angles

Angles A and B are adjacent.

"Vertical angle" redirects here; not to be confused with Zenith angle.

Adjacent angles (abbreviated adj. ∠s), are angles that share a common vertex and edge but do not share any interior points. In other words, they are angles 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 § Combining angle pairs below).

Angles A and B are a pair of vertical angles; angles C and D are a pair of vertical angles. Hatch marks are used here to show angle equality.

Vertical angles are formed when two straight lines intersect at a point producing four angles. A pair of angles opposite each other are called vertical angles, opposite angles or vertically opposite angles (abbreviated vert. opp. ∠s),^[16] where "vertical" refers to the sharing of a vertex, rather than an up-down orientation. The vertical angle theorem states that vertical angles are always congruent or equal to each other.^{[citation needed]}

A transversal is a line that intersects a pair of (often parallel) lines and is associated with exterior angles, interior angles, alternate exterior angles, alternate interior angles, corresponding angles, and consecutive interior angles.^[17]

Combining angle pairs

When summing two angles that are either adjacent or separated in space, three cases are of particular importance.^{[citation needed]}

Complementary angles

Angles a and b are complementary angles

Complementary angles are angle pairs whose measures sum to a right angle (⁠1/4⁠ turn, 90°, or ⁠π/2⁠ rad).^[18] If the two complementary angles are adjacent, their non-shared sides form a right angle. In a right-angle triangle the two acute angles are complementary as the sum of the internal angles of a triangle is 180°.

The difference between an angle and a right angle is termed the complement of the angle^[19] which is from the Latin complementum and associated verb complere, meaning "to fill up". An acute angle is "filled up" by its complement to form a right angle.^{[citation needed]}

Supplementary angles

Angles a and b are supplementary angles

Two angles that sum to a straight angle (⁠1/2⁠ turn, 180°, or π rad) are called supplementary angles.^[20] If the two supplementary angles are adjacent, their non-shared sides form a straight angle or straight line and are called a linear pair of angles.^[21] The difference between an angle and a straight angle is termed the supplement of the angle.^[22]

Examples of non-adjacent complementary angles include the consecutive angles of a parallelogram and opposite angles of a cyclic quadrilateral. For a circle with center O, and tangent lines from an exterior point P touching the circle at points T and Q, the resulting angles ∠TPQ and ∠TOQ are supplementary.

Explementary angles

Angles AOB and COD are explementary or conjugate angles

Two angles that sum to a full angle (1 turn, 360°, or 2π radians) are called explementary angles or conjugate angles.^[23] The difference between an angle and a full angle is termed the explement or conjugate of the angle.^{[citation needed]}

Polygon-related angles

Internal and external 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 (n − 2)180 degrees, (n − 2)2 right angles, or (n − 2)⁠1/2⁠ 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 must make at a vertex to trace the polygon.^[24] 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. Still, 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).^[25]: 149
• In a triangle, three intersection points, each of an external angle bisector with the opposite extended side, are collinear.^[25]: 149
• In a triangle, three intersection points, two 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.^[25]: 149
• Some authors use the name exterior angle of a simple polygon to mean the explement exterior angle (not supplement!) of the interior angle.^[26] 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.^[19] 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 complementary to the angle between the intersecting line and the normal to the plane.

Measuring angles

See also: Angle measuring instrument

The angle size ${\displaystyle \theta }$ can be measured as s/r radians or s/C turns

Measurement of angles is intrinsically linked with circles and rotation. An angle is measured by placing it within a circle of any size, with the vertex at the circle's centre and the sides intersecting the perimeter.

An arc s is formed as the shortest distance on the perimeter between the two points of intersection, which is said to be the arc subtended by the angle.

The length of s can be used to measure the angle's size ${\displaystyle \theta }$, however as s is dependent on the size of the circle chosen, it must be adjusted so that any arbitrary circle will give the same measure of angle. This can be done in two ways: by taking the ratio to either the radius r or circumference C of the circle.

The ratio of the length s by the radius r is the number of radians in the angle, while the ratio of length s by the circumference C is the number of turns:^[27] $${\displaystyle \theta _{\mathrm {rad} }={\frac {s}{r}}\,\mathrm {rad} \qquad \qquad \theta _{\mathrm {turn} }={\frac {s}{C}}\ ={\frac {s}{2\pi r}}\,\mathrm {turns} }$$

The measure of angle θ is ⁠s/r⁠ radians.

The value of θ thus defined is independent of the size of the circle: if the length of the radius is changed, then both the circumference and the arc length change in the same proportion, so the ratios ⁠s/r⁠ and ⁠s/C⁠ are unaltered.^{[nb 1]}

Angles of the same size are said to be equal congruent or equal in measure.

Units

In addition to the radian and turn, other angular units exist, typically based on subdivisions of the turn, including the degree (°) and the gradian (grad), though many others have been used throughout history.^[29]

Conversion between units may be obtained by multiplying the angular measure in one unit by a conversion constant of the form ${\displaystyle {\tfrac {k_{a}}{k_{b}}}}$ where ${\displaystyle {k_{a}}}$ and ${\displaystyle {k_{b}}}$ are the measures of a complete turn in units a and b. For example, to convert an angle of ${\displaystyle {\tfrac {\pi }{2}}}$radians to degrees: $${\displaystyle \theta _{\deg }={\frac {k_{\deg }}{k_{\mathrm {rad} }}}\cdot \theta _{\mathrm {rad} }={\frac {360^{\circ }}{2\pi \,\mathrm {rad} }}\cdot {\frac {\pi }{2}}\,\mathrm {rad} =90^{\circ }}$$

The following table lists some units used to represent angles.

Name (symbol)	Number in one turn	1 unit in degrees	Description
turn	1	360°	The turn is the angle subtended by the circumference of a circle at its centre. A turn is equal to 2π or 𝜏 radians.
degree ( ° )	360	1°	The degree is a sexagesimal subunit of the sextant, making one turn equal to 360°.
radian (rad)	2π	57.2957...°	The radian is the angle subtended by an arc of a circle that has the same length as the circle's radius.
grad (gon)	400	0.9°	The grad, also called grade, gradian, or gon, is defined as ⁠1/100⁠ of a right angle. The grad is used mostly in triangulation and continental surveying.
arcminute ( ′ )	21,600	⁠1/60⁠°	The minute of arc (or arcminute, or just minute) is a sexagesimal subunit of a degree.
arcsecond ( ″ )	1,296,000	⁠1/3600⁠°	The second of arc (or arcsecond, or just second) is a sexagesimal subunit of a minute of arc.
milliradian (mrad)	2000π	~0.0573°	The milliradian is a thousandth of a radian. For artillery and navigation a unit is used, often called a 'mil', which are approximately equal to a milliradian. One turn is exactly 6000, 6300, or 6400 mils, depending on which definition is used.
(compass) point	32	11.25°	The point or wind, used in navigation, divides the compass (one turn) into 32 points or compass directions.
binary degree	256	1.40625°	The binary degree, also known as the binary radian or brad or binary angular measurement (BAM).[30]
quadrant	4	90°	One quadrant[dubious – discuss] is a ⁠1/4⁠ turn and also known as a right angle. In German, the symbol ∟ has been used to denote a right angle.
sextant	6	60°	The sextant was the unit used by the Babylonians.[31][32][dubious – discuss]
hexacontade	60	6°	The hexacontade is a unit used by Eratosthenes, with 60 hexacontades in a turn.[citation needed]
diameter part	~376.991	~0.95493°	The diameter part (occasionally used in Islamic mathematics) is ⁠1/60⁠ radian.[citation needed]
zam	224	~1.607°	In old Arabia, a turn was subdivided into 32 akhnam, and each akhnam was subdivided into 7 zam.[citation needed]

Dimensional analysis

Further information: Radian § Dimensional analysis

In mathematics and the International System of Quantities, an angle is defined as a dimensionless quantity, and in particular, the radian is defined as dimensionless in the International System of Units. This convention prevents angles providing information for dimensional analysis.^{[citation needed]}

While mathematically convenient, this has led to significant discussion among scientists and teachers. Some scientists have suggested treating the angle as having its own dimension, similar to length or time. This would mean that angle units like radians would always be explicitly present in calculations, making the dimensional analysis more straightforward. However, this approach would also require changing many well-known mathematical and physics formulas.^{[citation needed]}

Signed angles

Main article: Angle of rotation

See also: Sign (mathematics) § Angles, and Euclidean space § Angle

Measuring from the x-axis, angles on the unit circle count as positive in the counterclockwise direction, and negative in the clockwise direction.

An angle denoted as ∠BAC might refer to any of four angles: the clockwise angle from B to C about A, the anticlockwise angle from B to C about A, the clockwise angle from C to B about A, or the anticlockwise angle from C to B about A, It is therefore frequently helpful to impose a convention that allows positive and negative angular values to represent orientations and/or rotations in opposite directions or "sense" 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 cycles 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 equal to an orientation defined 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 in terms of an orientation, which is typically determined by a normal 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°.

Equivalent angles

• 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.
• The reference angle (sometimes called related angle) for any angle θ in standard position is the positive acute angle between the terminal side of θ and the x-axis (positive or negative).^[33][34] Procedurally, the magnitude of the reference angle for a given angle may determined by taking the angle's magnitude modulo ⁠1/2⁠ turn, 180°, or π radians, then stopping if the angle is acute, otherwise taking the supplementary angle, 180° minus the reduced magnitude. For example, an angle of 30 degrees is already a reference angle, and an angle of 150 degrees also has a reference angle of 30 degrees (180° − 150°). Angles of 210° and 510° correspond to a reference angle of 30 degrees as well (210° mod 180° = 30°, 510° mod 180° = 150° whose supplementary angle is 30°).

Related quantities

For an angular unit, it is definitional that the angle addition postulate holds, however some measurements or quantities related to angles are in use that do not satisfy this postulate:

• The slope or gradient is equal to the tangent of the angle and is often expressed as a percentage ("rise" over "run"). For very small values (less than 5%), the slope of a line is approximately the measure in radians of its angle with the horizontal direction. An elevation grade is a slope used to indicate the steepness of roads, paths and railway lines.
• The spread between two lines is defined in rational geometry 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.
• Although done rarely, one can report the direct results of trigonometric functions, such as the sine of the angle.

Angles between curves

The angle between the two curves at P is defined as the angle between the tangents A and B at P.

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), concavo-convex; amphicoelic (Gr. κοίλη, a hollow) or angulus lunularis, biconcave.^[35]

Bisecting and trisecting angles

Main articles: Bisection § Angle bisector, and Angle trisection

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 this construction could not be performed for most angles.

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 $${\displaystyle \mathbf {u} \cdot \mathbf {v} =\cos(\theta )\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}$$

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 ${\displaystyle \langle \cdot ,\cdot \rangle }$, i.e. $${\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle =\cos(\theta )\ \left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}$$

In a complex inner product space, the expression for the cosine above may give non-real values, so it is replaced with $${\displaystyle \operatorname {Re} \left(\langle \mathbf {u} ,\mathbf {v} \rangle \right)=\cos(\theta )\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}$$ or, more commonly, using the absolute value, with $${\displaystyle \left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=\left|\cos(\theta )\right|\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}$$

The latter definition ignores the direction of the vectors. It thus describes the angle between one-dimensional subspaces ${\displaystyle \operatorname {span} (\mathbf {u} )}$ and ${\displaystyle \operatorname {span} (\mathbf {v} )}$ spanned by the vectors ${\displaystyle \mathbf {u} }$ and ${\displaystyle \mathbf {v} }$ correspondingly.

Angles between subspaces

The definition of the angle between one-dimensional subspaces ${\displaystyle \operatorname {span} (\mathbf {u} )}$ and ${\displaystyle \operatorname {span} (\mathbf {v} )}$ given by $${\displaystyle \left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=\left|\cos(\theta )\right|\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|}$$ in a Hilbert space can be extended to subspaces of finite number of dimensions. Given two subspaces ${\displaystyle {\mathcal {U}}}$, ${\displaystyle {\mathcal {W}}}$ with ${\displaystyle \dim({\mathcal {U}}):=k\leq \dim({\mathcal {W}}):=l}$, this leads to a definition of ${\displaystyle k}$ 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, $${\displaystyle \cos \theta ={\frac {g_{ij}U^{i}V^{j}}{\sqrt {\left|g_{ij}U^{i}U^{j}\right|\left|g_{ij}V^{i}V^{j}\right|}}}.}$$

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.^[36] 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 comparison of the two series corresponding to functions of angles was described by Leonhard Euler in Introduction to the Analysis of the Infinite (1748).

History and etymology

The word angle comes from the Latin word angulus, meaning "corner". Cognate words include 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".^[37]

Euclid defines a plane angle as the inclination to each other, in a plane, of two lines that meet each other and do not lie straight with respect to each other. According to the Neoplatonic metaphysician Proclus, an angle must be either a quality, a quantity, or a relationship. The first concept, angle as quality, was used by Eudemus of Rhodes, who regarded an angle as a deviation from a straight line; the second, angle as quantity, by Carpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third: angle as a relationship.^[38]

Vertical angle theorem

The equality of vertically opposite angles is called the vertical angle theorem. Eudemus of Rhodes attributed the proof to Thales of Miletus.^[39][40] 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,^[40] 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, the measure of angle C would be 180° − x. Similarly, the measure of angle D would be 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 to be 180° − (180° − x) = 180° − 180° + x = x. Therefore, both angle A and angle B have measures equal to x and are equal in measure.

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 and the angular separation between the two stars can be measured.

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 objects' apparent size 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 convert such an angular measurement into a distance/size ratio.

Other astronomical approximations include:

• 0.5° is the approximate diameter of the Sun and of the Moon as viewed from Earth.
• 1° is the approximate width of the little finger at arm's length.
• 10° is the approximate width of a closed fist at arm's length.
• 20° is the approximate width of a handspan at arm's length.

These measurements depend on the individual subject, and the above should be treated as rough rule of thumb approximations only.

In astronomy, right ascension and declination are usually measured in angular units, expressed in terms of time, based on a 24-hour day.

Unit	Symbol	Degrees	Radians	Turns	Other
Hour	h	15°	π⁄12 rad	1⁄24 turn
Minute	m	0°15′	π⁄720 rad	1⁄1440 turn	1⁄60 hour
Second	s	0°0′15″	π⁄43200 rad	1⁄86400 turn	1⁄60 minute

See also

• Angle measuring instrument
• Angles between flats
• Angular statistics (mean, standard deviation)
• Angle bisector
• Angular acceleration
• Angular diameter
• Angular velocity
• Argument (complex analysis)
• Astrological aspect
• Central angle
• Clock angle problem
• Decimal degrees
• Dihedral angle
• Exterior angle theorem
• Golden angle
• Great circle distance
• Horn angle
• Inscribed angle
• Irrational angle
• Phase (waves)
• Protractor
• Solid angle
• Spherical angle
• Subtended angle
• Tangential angle
• Transcendent angle
• Trisection
• Zenith angle

Notes

1. This approach requires, however, an additional proof that the measure of the angle does not change with changing radius r, in addition to the issue of "measurement units chosen". A smoother approach is to measure the angle by the length of the corresponding unit circle arc. Here "unit" can be chosen to be dimensionless in the sense that it is the real number 1 associated with the unit segment on the real line. See Radoslav M. Dimitrić, for instance.^[28]

1. An angular sector can be constructed by the combination of two rotated half-planes, either their intersection or union (in the case of acute or obtuse angles, respectively).^[5][6] It corresponds to a circular sector of infinite radius and a flat pencil of half-lines.^[7]