InEuclidean geometry, anangle can refer to a number of concepts relating to the intersection of two straightlines at apoint. Formally, an angle is a figure lying in aplane formed by tworays, called thesides of the angle, sharing a common endpoint, called thevertex of the angle.[1][2] More generally angles are also formed wherever two lines, rays orline 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 intersectingcurves, in which case the rays lyingtangent to each curve at the point of intersection define the angle.
The termangle is also used for the size,magnitude orquantity of these types of geometric figures and in this context an angle consists of a number and unit of measurement.Angular measure ormeasure 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 thearc of acircle centered at the vertex and lying between the sides.
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: theinterior of the angle and theexterior of the angle.[1]
is formed by rays and. is the conventional measure of and is an alternative measure.
An angle symbol ( or, 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 therays and is denoted as (using the vertex alone) or (with the vertex always named in the middle). The size or measure of the angle is denoted or.
In geometric figures andmathematical expressions, it is also common to useGreek letters (α,β,γ,θ,φ, ...) or lower case Roman letters (a, b, c, ...) asvariables 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 amagnitude orscalar quantity. At other times it might be measured through the exterior of the angle or given as asigned number to indicate a direction of measurement.[citation needed]
Angles are measured in various units, the most common being thedegree (denoted by the symbol°),radian (denoted by the symbolrad) andturn. These units differ in the way they divide up afull 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]
The angle addition postulate defines addition and subtraction of angles:θ +α =φ;φ −α =θ.
Theangle addition postulate states that if D is a point lying in the interior of then:[15] This relationshipdefines 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.[citation needed]
"Vertical angle" redirects here and is not to be confused withZenith angle.
Adjacent angles (abbreviatedadj. ∠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 calledcomplementary,supplementary, andexplementary 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 calledvertical angles,opposite angles orvertically opposite angles (abbreviatedvert. opp. ∠s),[16] where "vertical" refers to the sharing of a vertex, rather than an up-down orientation. Thevertical angle theorem states that vertical angles are always congruent or equal to each other.[citation needed]
Atransversal is a line that intersects a pair of (often parallel) lines and is associated withexterior angles,interior angles,alternate exterior angles,alternate interior angles,corresponding angles, andconsecutive interior angles.[17]
Complementary angles are angle pairs whose measures sum to a right angle (1/4 turn, 90°, orπ/2 rad).[18] If the two complementary angles are adjacent, their non-shared sides form a right angle. In aright-angle triangle the two acute angles are complementary as the sum of the internal angles of atriangle is 180°.
The difference between an angle and a right angle is termed thecomplement of the angle[19] which is from the Latincomplementum and associated verbcomplere, meaning "to fill up". An acute angle is "filled up" by its complement to form a right angle.[citation needed]
Two angles that sum to a straight angle (1/2 turn, 180°, orπ rad) are calledsupplementary angles.[20] If the two supplementary angles areadjacent, their non-shared sides form a straight angle orstraight line and are called alinear pair of angles.[21] The difference between an angle and a straight angle is termed thesupplement of the angle.[22]
Examples of non-adjacent complementary angles include the consecutive angles of aparallelogram and opposite angles of acyclic quadrilateral. For a circle with center O, andtangent lines from an exterior point P touching the circle at points T and Q, the resulting angles ∠TPQ and ∠TOQ are supplementary.
Angles AOB and COD areexplementaryor conjugate angles
Two angles that sum to a full angle (1 turn, 360°, or 2π radians) are calledexplementary angles orconjugate angles.[23] The difference between an angle and a full angle is termed theexplement orconjugate of the angle.[citation needed]
An angle that is part of asimple polygon is called aninterior angle if it lies on the inside of that simple polygon. A simpleconcave polygon has at least one interior angle, that is, a reflex angle. InEuclidean geometry, the measures of the interior angles of atriangle add up toπ radians, 180°, or1/2 turn; the measures of the interior angles of a simpleconvexquadrilateral add up to 2π radians, 360°, or 1 turn. In general, the measures of the interior angles of a simple convexpolygon withn 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 anexterior angle; that is, an interior angle and an exterior angle form alinear 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 considerednegative. Even in a non-simple polygon, it may be possible to define the exterior angle. Still, one will have to pick anorientation of theplane (orsurface) 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 asupplementary exterior angle. Exterior angles are commonly used inLogo Turtle programs when drawing regular polygons.
In atriangle, thebisectors of two exterior angles and the bisector of the other interior angle areconcurrent (meet at a single point).[25]: 149
In a triangle, three intersection points, each of an external angle bisector with the oppositeextended side, arecollinear.[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 nameexterior angle of a simple polygon to mean theexplement exterior angle (not supplement!) of the interior angle.[26] This conflicts with the above usage.
The angle between twoplanes (such as two adjacent faces of apolyhedron) is called adihedral angle.[19] It may be defined as the acute angle between two linesnormal to the planes.
The angle between a plane and an intersecting straight line is complementary to the angle between the intersecting line and thenormal to the plane.
The angle size 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.
Anarcs is formed as the shortest distance on the perimeter between the two points of intersection, which is said to be the arcsubtended by the angle.
The length ofs can be used to measure the angle's size, however ass 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 radiusr or circumferenceC of the circle.
The ratio of the lengths by the radiusr is the number ofradians in the angle, while the ratio of lengths by the circumferenceC is the number ofturns:[27]
The measure of angleθ iss/r radians.
The value ofθ 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 ratioss/r ands/C are unaltered.[nb 1]
Angles of the same size are said to beequalcongruent orequal in measure.
In addition to the radian and turn, other angular units exist, typically based on subdivisions of the turn, including thedegree (°) and thegradian (grad), though many others have been used throughouthistory.[29]
Conversion between units may be obtained by multiplying the angular measure in one unit by a conversion constant of the form where and are the measures of a complete turn in unitsa andb. For example, to convert an angle ofradians to degrees:
The following table lists some units used to represent angles.
The milliradian is a thousandth of a radian. For artillery and navigation a unit is used, often called a 'mil', which areapproximately equal to a milliradian. One turn is exactly 6000, 6300, or 6400 mils, depending on which definition is used.
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]
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 representorientations and/orrotations in opposite directions or "sense" relative to some reference.
In a two-dimensionalCartesian coordinate system, an angle is typically defined by its two sides, with its vertex at the origin. Theinitial side is on the positivex-axis, while the other side orterminal side is defined by the measure from the initial side in radians, degrees, or turns, withpositive angles representing rotations toward the positivey-axis andnegative angles representing rotations toward the negativey-axis. When Cartesian coordinates are represented bystandard position, defined by thex-axis rightward and they-axis upward, positive rotations areanticlockwise, and negative cycles areclockwise.
In many contexts, an angle of −θ is effectively equivalent to an angle of "one full turn minusθ". 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 anorientation, which is typically determined by anormal vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie.
Innavigation,bearings orazimuth 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°.
Angles that have the same measure (i.e., the same magnitude) are said to beequal orcongruent. An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g., allright angles are equal in measure).
Two angles that share terminal sides, but differ in size by an integer multiple of a turn, are calledcoterminal angles.
Thereference angle (sometimes calledrelated angle) for any angleθ in standard position is the positive acute angle between the terminal side ofθ 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 magnitudemodulo1/2 turn, 180°, orπ 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°).
For an angular unit, it is definitional that theangle addition postulate holds, however some measurements or quantities related to angles are in use that do not satisfy this postulate:
Theslope orgradient is equal to thetangent 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. Anelevation grade is a slope used to indicate the steepness of roads, paths and railway lines.
Thespread between two lines is defined inrational 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 oftrigonometric functions, such as thesine of the angle.
The angle between the two curves at P is defined as the angle between the tangentsA andB atP.
The angle between a line and acurve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between thetangents 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) orcissoidal (Gr. κισσός, ivy), biconvex;xystroidal orsistroidal (Gr. ξυστρίς, a tool for scraping), concavo-convex;amphicoelic (Gr. κοίλη, a hollow) orangulus lunularis, biconcave.[35]
Theancient Greek mathematicians knew how to bisect an angle (divide it into two angles of equal measure) using only acompass and straightedge but could only trisect certain angles. In 1837,Pierre Wantzel showed that this construction could not be performed for most angles.
This formula supplies an easy method to find the angle between two planes (or curved surfaces) from theirnormal vectors and betweenskew lines from their vector equations.
To define angles in an abstract realinner product space, we replace the Euclidean dot product (· ) by the inner product, i.e.
In a complexinner product space, the expression for the cosine above may give non-real values, so it is replaced withor, more commonly, using the absolute value, with
The latter definition ignores the direction of the vectors. It thus describes the angle between one-dimensional subspaces and spanned by the vectors and correspondingly.
The definition of the angle between one-dimensional subspaces and given byin aHilbert space can be extended to subspaces of finite number of dimensions. Given two subspaces, with, this leads to a definition of angles called canonical orprincipal angles between subspaces.
InRiemannian geometry, themetric tensor is used to define the angle between twotangents. WhereU andV are tangent vectors andgij are the components of the metric tensorG,
The wordangle comes from theLatin wordangulus, meaning "corner".Cognate words include theGreekἀγκύλος (ankylοs) meaning "crooked, curved" and theEnglish word "ankle". Both are connected with theProto-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 metaphysicianProclus, an angle must be either a quality, a quantity, or a relationship. The first concept, angle as quality, was used byEudemus of Rhodes, who regarded an angle as a deviation from astraight line; the second, angle as quantity, byCarpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third: angle as a relationship.[38]
The equality of vertically opposite angles is called thevertical angle theorem.Eudemus of Rhodes attributed the proof toThales 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 angleA equalsx, the measure of angleC would be180° −x. Similarly, the measure of angleD would be180° −x. Both angleC and angleD have measures equal to180° −x and are congruent. Since angleB is supplementary to both anglesC andD, either of these angle measures may be used to determine the measure of AngleB. Using the measure of either angleC or angleD, we find the measure of angleB to be180° − (180° −x) = 180° − 180° +x =x. Therefore, both angleA and angleB have measures equal tox and are equal in measure.
Inastronomy, a given point on thecelestial sphere (that is, the apparent position of an astronomical object) can be identified using any of severalastronomical coordinate systems, where the references vary according to the particular system. Astronomers measure theangular separation of twostars by imagining two lines through the center of theEarth, each intersecting one of the stars. The angle between those lines and the angular separation between the two stars can be measured.
Astronomers also measure objects'apparent size as anangular diameter. For example, thefull 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." Thesmall-angle formula can convert such an angular measurement into a distance/size ratio.
Other astronomical approximations include:
0.5° is the approximate diameter of theSun and of theMoon as viewed from Earth.
1° is the approximate width of thelittle 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 roughrule of thumb approximations only.
In astronomy,right ascension anddeclination are usually measured in angular units, expressed in terms of time, based on a 24-hour day.
^This approach requires, however, an additional proof that the measure of the angle does not change with changing radiusr, 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]
^An angular sector can be constructed by the combination of two rotatedhalf-planes, either their intersection or union (in the case of acute or obtuse angles, respectively).[5][6] It corresponds to acircular sector of infinite radius and a flatpencil of half-lines.[7]
^Dimitrić, Radoslav M. (2012)."On Angles and Angle Measurements"(PDF).The Teaching of Mathematics.XV (2):133–140.Archived(PDF) from the original on 2019-01-17. Retrieved2019-08-06.
Brinsmade, J. B. (December 1936). "Plane and Solid Angles. Their Pedagogic Value When Introduced Explicitly".American Journal of Physics.4 (4):175–179.Bibcode:1936AmJPh...4..175B.doi:10.1119/1.1999110.
Henderson, David W.; Taimina, Daina (2005),Experiencing Geometry / Euclidean and Non-Euclidean with History (3rd ed.), Pearson Prentice Hall, p. 104,ISBN978-0-13-143748-7
Heiberg, Johan Ludvig (1908),Heath, T. L. (ed.),Euclid, The Thirteen Books of Euclid's Elements, vol. 1,Cambridge: Cambridge University Press.
Wong, Tak-wah; Wong, Ming-sim (2009), "Angles in Intersecting and Parallel Lines",New Century Mathematics, vol. 1B (1 ed.), Hong Kong: Oxford University Press, pp. 161–163,ISBN978-0-19-800177-5
