Lines & Angles - Revision history

Khanh: /* Units */

2021-12-13T00:13:25Z

Units

Khanh: /* Measuring angles */

2021-12-13T00:11:29Z

Measuring angles

Khanh: /* Polygon-related angles */

2021-12-13T00:07:24Z

Polygon-related angles

Khanh: /* Combining angle pairs */

2021-12-13T00:06:22Z

Combining angle pairs

Khanh: /* Equivalence angle pairs */

2021-12-13T00:05:02Z

Equivalence angle pairs

Khanh: /* Individual angles */

2021-12-13T00:03:36Z

Individual angles

Khanh: /* Ray */

2021-12-12T20:45:17Z

Ray

Khanh at 20:43, 12 December 2021

2021-12-12T20:43:22Z

Khanh: /* Angles between curves */

2021-12-12T20:36:57Z

Angles between curves

Khanh: /* Astronomical approximations */

2021-12-12T20:35:06Z

Astronomical approximations

@@ Line 391: / Line 391: @@
 |Multiples of <math> \pi </math> ||2||180° || The ''multiples of <math> \pi </math>'' (MUL<math> \pi </math>) unit is implemented in the RPN scientific calculator WP 43S. See also: IEEE 754 recommended operations
 |-
-|Quadrant||4||90°||One ''quadrant'' is a <math> \frac{1}{4}</math> turn and also known as a ''right angle''. The quadrant is the unit used in Euclid's Elements. In German the symbol <sup>∟</sup> has been used to denote a quadrant. It is the unit used in Euclid's Elements. 1 quad = 90° = <math> \tfrac{\pi}{2}</math> rad = <math> \tfrac{1}{4}</math> turn = 100 grad.
+|Quadrant||4||90°||One ''quadrant'' is a <math> \tfrac{1}{4}</math> turn and also known as a ''right angle''. The quadrant is the unit used in Euclid's Elements. In German the symbol <sup>∟</sup> has been used to denote a quadrant. It is the unit used in Euclid's Elements. 1 quad = 90° = <math> \tfrac{\pi}{2}</math> rad = <math> \tfrac{1}{4}</math> turn = 100 grad.
 |-
 |Sextant||6||60°||The ''sextant'' was the unit used by the Babylonians, The degree, minute of arc and second of arc are sexagesimal subunits of the Babylonian unit. It is especially easy to construct with ruler and compasses. It is the ''angle of the equilateral triangle'' or is <math> \tfrac{1}{6}</math> turn. 1 Babylonian unit = 60° = <math> \tfrac{\pi}{3}</math> rad ≈ 1.047197551 rad.

@@ Line 391: / Line 391: @@
 |Multiples of <math> \pi </math> ||2||180° || The ''multiples of <math> \pi </math>'' (MUL<math> \pi </math>) unit is implemented in the RPN scientific calculator WP 43S. See also: IEEE 754 recommended operations
 |-
-|Quadrant||4||90°||One ''quadrant'' is a <math> \frac{1}{4}</math> turn and also known as a ''right angle''. The quadrant is the unit used in Euclid's Elements. In German the symbol <sup>∟</sup> has been used to denote a quadrant. It is the unit used in Euclid's Elements. 1 quad = 90° = <math> \frac{\pi}{2}</math> rad = <math> \frac{1}{4}</math> turn = 100 grad.
+|Quadrant||4||90°||One ''quadrant'' is a <math> \frac{1}{4}</math> turn and also known as a ''right angle''. The quadrant is the unit used in Euclid's Elements. In German the symbol <sup>∟</sup> has been used to denote a quadrant. It is the unit used in Euclid's Elements. 1 quad = 90° = <math> \tfrac{\pi}{2}</math> rad = <math> \tfrac{1}{4}</math> turn = 100 grad.
 |-
-|Sextant||6||60°||The ''sextant'' was the unit used by the Babylonians, The degree, minute of arc and second of arc are sexagesimal subunits of the Babylonian unit. It is especially easy to construct with ruler and compasses. It is the ''angle of the equilateral triangle'' or is <math> \frac{1}{6}</math> turn. 1 Babylonian unit = 60° = <math> \frac{\pi}{3}</math> rad ≈ 1.047197551 rad.
+|Sextant||6||60°||The ''sextant'' was the unit used by the Babylonians, The degree, minute of arc and second of arc are sexagesimal subunits of the Babylonian unit. It is especially easy to construct with ruler and compasses. It is the ''angle of the equilateral triangle'' or is <math> \tfrac{1}{6}</math> turn. 1 Babylonian unit = 60° = <math> \tfrac{\pi}{3}</math> rad ≈ 1.047197551 rad.
 |-
-|Radian||{{math|2''π''}}||57°17′||The ''radian'' is determined by the circumference of a circle that is equal in length to the radius of the circle (''n''&nbsp;=&nbsp;2π&nbsp;=&nbsp;6.283...). It is the angle subtended by an arc of a circle that has the same length as the circle's radius. The symbol for radian is ''rad''. One turn is 2{{math|π}}&nbsp;radians, and one radian is <math> \frac{180^ \circ}{\pi}</math>, or about 57.2958 degrees. In mathematical texts, angles are often treated as being dimensionless with the radian equal to one, resulting in the unit ''rad'' often being omitted. The radian is used in virtually all mathematical work beyond simple practical geometry, due, for example, to the pleasing and "natural" properties that the trigonometric functions display when their arguments are in radians. The radian is the (derived) unit of angular measurement in the SI, which also treats angle as being dimensionless.
+|Radian||{{math|2''π''}}||57°17′||The ''radian'' is determined by the circumference of a circle that is equal in length to the radius of the circle (''n''&nbsp;=&nbsp;2π&nbsp;=&nbsp;6.283...). It is the angle subtended by an arc of a circle that has the same length as the circle's radius. The symbol for radian is ''rad''. One turn is 2{{math|π}}&nbsp;radians, and one radian is <math> \tfrac{180^ \circ}{\pi}</math>, or about 57.2958 degrees. In mathematical texts, angles are often treated as being dimensionless with the radian equal to one, resulting in the unit ''rad'' often being omitted. The radian is used in virtually all mathematical work beyond simple practical geometry, due, for example, to the pleasing and "natural" properties that the trigonometric functions display when their arguments are in radians. The radian is the (derived) unit of angular measurement in the SI, which also treats angle as being dimensionless.
 |-
 | Hexacontade||60 ||6°||The ''hexacontade'' is a unit used by Eratosthenes. It is equal to 6°, so that a whole turn was divided into 60 hexacontades.
 |-
-|Binary degree ||256||1°33'45"  || The ''binary degree'', also known as the ''binary radian'' (or ''brad''). The binary degree is used in computing so that an angle can be efficiently represented in a single byte (albeit to limited precision). Other measures of angle used in computing may be based on dividing one whole turn into 2<sup>''n''</sup> equal parts for other values of ''n''. It is <math> \frac{1}{256}</math> of a turn.
+|Binary degree ||256||1°33'45"  || The ''binary degree'', also known as the ''binary radian'' (or ''brad''). The binary degree is used in computing so that an angle can be efficiently represented in a single byte (albeit to limited precision). Other measures of angle used in computing may be based on dividing one whole turn into 2<sup>''n''</sup> equal parts for other values of ''n''. It is <math> \tfrac{1}{256}</math> of a turn.
 |-
-|Degree ||360 ||1°|| One advantage of this old sexagesimal subunit is that many angles common in simple geometry are measured as a whole number of degrees. Fractions of a degree may be written in normal decimal notation (e.g. 3.5° for three and a half degrees), but the "minute" and "second" sexagesimal subunits of the "degree-minute-second" system are also in use, especially for geographical coordinates and in astronomy and ballistics (''n''&nbsp;=&nbsp;360) The ''degree'', denoted by a small superscript circle (°), is 1/360 of a turn, so one ''turn'' is 360°. The case of degrees for the formula given earlier, a ''degree'' of ''n'' = 360° units is obtained by setting ''k'' = <math> \frac{360^ \circ}{2 \pi}</math>.
+|Degree ||360 ||1°|| One advantage of this old sexagesimal subunit is that many angles common in simple geometry are measured as a whole number of degrees. Fractions of a degree may be written in normal decimal notation (e.g. 3.5° for three and a half degrees), but the "minute" and "second" sexagesimal subunits of the "degree-minute-second" system are also in use, especially for geographical coordinates and in astronomy and ballistics (''n''&nbsp;=&nbsp;360) The ''degree'', denoted by a small superscript circle (°), is 1/360 of a turn, so one ''turn'' is 360°. The case of degrees for the formula given earlier, a ''degree'' of ''n'' = 360° units is obtained by setting ''k'' = <math> \tfrac{360^ \circ}{2 \pi}</math>.
 |-
 | Grad||400 ||0°54′ || The ''grad'', also called ''grade'', ''gradian'', or ''gon''. a right angle is 100 grads. It is a decimal subunit of the quadrant. A kilometre was historically defined as a centi-grad of arc along a meridian of the Earth, so the kilometer is the decimal analog to the sexagesimal nautical mile (''n''&nbsp;=&nbsp;400). The grad is used mostly in triangulation and continental surveying.
 |-
-| Minute of arc||21,600 ||0°1′|| The ''minute of arc'' (or ''MOA'', ''arcminute'', or just ''minute'') is <math> \frac{1}{60} </math> of a degree. A nautical mile was historically defined as a minute of arc along a great circle of the Earth (''n''&nbsp;=&nbsp;21,600).  The ''arcminute'' is <math> \frac{1}{60} </math> of a degree = <math> \frac{1}{21,600} </math> turn. It is denoted by a single prime (&nbsp;′&nbsp;). For example, 3°&nbsp;30′ is equal to 3&nbsp;×&nbsp;60&nbsp;+&nbsp;30&nbsp;=&nbsp;210 minutes or 3&nbsp;+&nbsp;<math> \frac{30}{60} </math> = 3.5 degrees. A mixed format with decimal fractions is also sometimes used, e.g. 3°&nbsp;5.72′ = 3&nbsp;+&nbsp;<math> \frac{5.72}{60} </math> degrees. A nautical mile was historically defined as an arcminute along a great circle of the Earth.
+| Minute of arc||21,600 ||0°1′|| The ''minute of arc'' (or ''MOA'', ''arcminute'', or just ''minute'') is <math> \tfrac{1}{60} </math> of a degree. A nautical mile was historically defined as a minute of arc along a great circle of the Earth (''n''&nbsp;=&nbsp;21,600).  The ''arcminute'' is <math> \tfrac{1}{60} </math> of a degree = <math> \tfrac{1}{21,600} </math> turn. It is denoted by a single prime (&nbsp;′&nbsp;). For example, 3°&nbsp;30′ is equal to 3&nbsp;×&nbsp;60&nbsp;+&nbsp;30&nbsp;=&nbsp;210 minutes or 3&nbsp;+&nbsp;<math> \tfrac{30}{60} </math> = 3.5 degrees. A mixed format with decimal fractions is also sometimes used, e.g. 3°&nbsp;5.72′ = 3&nbsp;+&nbsp;<math> \tfrac{5.72}{60} </math> degrees. A nautical mile was historically defined as an arcminute along a great circle of the Earth.
 |-
-| Second of arc||1,296,000 ||0°0′1″||The ''second of arc'' (or ''arcsecond'', or just ''second'') is <math> \frac{1}{60} </math> of a minute of arc and <math> \frac{1}{3600} </math> of a degree (''n''&nbsp;=&nbsp;1,296,000). The ''arcsecond'' (or ''second of arc'', or just ''second'') is <math> \frac{1}{60} </math> of an arcminute and <math> \frac{1}{3600} </math> of a degree. It is denoted by a double prime (&nbsp;″&nbsp;). For example, 3°&nbsp;7′&nbsp;30″ is equal to 3 + <math> \frac{7}{60} </math> + <math> \frac{30}{3600} </math> degrees, or 3.125&nbsp;degrees.
+| Second of arc||1,296,000 ||0°0′1″||The ''second of arc'' (or ''arcsecond'', or just ''second'') is <math> \tfrac{1}{60} </math> of a minute of arc and <math> \tfrac{1}{3600} </math> of a degree (''n''&nbsp;=&nbsp;1,296,000). The ''arcsecond'' (or ''second of arc'', or just ''second'') is <math> \tfrac{1}{60} </math> of an arcminute and <math> \tfrac{1}{3600} </math> of a degree. It is denoted by a double prime (&nbsp;″&nbsp;). For example, 3°&nbsp;7′&nbsp;30″ is equal to 3 + <math> \tfrac{7}{60} </math> + <math> \tfrac{30}{3600} </math> degrees, or 3.125&nbsp;degrees.
 |}
 ===Other descriptors===
-* Hour angle (''n''&nbsp;=&nbsp;24): The astronomical ''hour angle'' is <math> \frac{1}{24} </math> turn. As this system is amenable to measuring objects that cycle once per day (such as the relative position of stars), the sexagesimal subunits are called ''minute of time'' and ''second of time''.  These are distinct from, and 15 times larger than, minutes and seconds of arc. 1&nbsp;hour = 15° = <math> \frac{\pi}{12} </math> rad = <math> \frac{1}{6} </math> quad = <math> \frac{1}{24} </math>turn = <math> 16 \frac{2}{3} </math> grad.
+* Hour angle (''n''&nbsp;=&nbsp;24): The astronomical ''hour angle'' is <math> \tfrac{1}{24} </math> turn. As this system is amenable to measuring objects that cycle once per day (such as the relative position of stars), the sexagesimal subunits are called ''minute of time'' and ''second of time''.  These are distinct from, and 15 times larger than, minutes and seconds of arc. 1&nbsp;hour = 15° = <math> \tfrac{\pi}{12} </math> rad = <math> \tfrac{1}{6} </math> quad = <math> \tfrac{1}{24} </math>turn = <math> 16 \tfrac{2}{3} </math> grad.
-* (Compass) point or wind (''n''&nbsp;=&nbsp;32): The ''point'', used in navigation, is <math> \frac{1}{32} </math> of a turn. 1&nbsp;point = <math> \frac{1}{8} </math> of a right angle = 11.25° = 12.5&nbsp;grad. Each point is subdivided in four quarter-points so that 1 turn equals 128 quarter-points.
+* (Compass) point or wind (''n''&nbsp;=&nbsp;32): The ''point'', used in navigation, is <math> \tfrac{1}{32} </math> of a turn. 1&nbsp;point = <math> \tfrac{1}{8} </math> of a right angle = 11.25° = 12.5&nbsp;grad. Each point is subdivided in four quarter-points so that 1 turn equals 128 quarter-points.
-* Pechus (''n''&nbsp;=&nbsp;144–180): The ''pechus'' was a Babylonian unit equal to about 2° or <math> {2 \frac{1}{2}} ^ \circ  </math>.
+* Pechus (''n''&nbsp;=&nbsp;144–180): The ''pechus'' was a Babylonian unit equal to about 2° or <math> {2 \tfrac{1}{2}} ^ \circ  </math>.
 * Tau, the number of radians in one turn (1 turn = {{mvar|τ}} rad), ''τ'' = 2''π''.
@@ Line 421: / Line 421: @@
 * Chi, an old Chinese angle measurement.
-* Diameter part (''n''&nbsp;=&nbsp;376.99...): The ''diameter part'' (occasionally used in Islamic mathematics) is <math> \frac{1}{60} </math> radian. One "diameter part" is approximately 0.95493°. There are about 376.991 diameter parts per turn.
+* Diameter part (''n''&nbsp;=&nbsp;376.99...): The ''diameter part'' (occasionally used in Islamic mathematics) is <math> \tfrac{1}{60} </math> radian. One "diameter part" is approximately 0.95493°. There are about 376.991 diameter parts per turn.
-* Milliradian and derived definitions: The true milliradian is defined a thousandth of a radian, which means that a rotation of one turn would equal exactly 2000π mil (or approximately 6283.185 mil), and almost all scope sights for firearms are calibrated to this definition. In addition there are three other derived definitions used for artillery and navigation which are ''approximately'' equal to a milliradian. Under these three other definitions one turn makes up for exactly 6000, 6300 or 6400 mils, which equals spanning the range from 0.05625 to 0.06 degrees (3.375 to 3.6 minutes). In comparison, the true milliradian is approximately 0.05729578 degrees (3.43775 minutes). One "NATO mil" is defined as <math> \frac{1}{6400} </math> of a circle. Just like with the true milliradian, each of the other definitions exploits the mil's useful property of subtensions, i.e. that the value of one milliradian approximately equals the angle subtended by a width of 1 meter as seen from 1&nbsp;km away <math> \left ( \frac{2 \pi}{6400} = 0.0009817 \ldots \approx \frac{1}{1000} \right ) </math>.
+* Milliradian and derived definitions: The true milliradian is defined a thousandth of a radian, which means that a rotation of one turn would equal exactly 2000π mil (or approximately 6283.185 mil), and almost all scope sights for firearms are calibrated to this definition. In addition there are three other derived definitions used for artillery and navigation which are ''approximately'' equal to a milliradian. Under these three other definitions one turn makes up for exactly 6000, 6300 or 6400 mils, which equals spanning the range from 0.05625 to 0.06 degrees (3.375 to 3.6 minutes). In comparison, the true milliradian is approximately 0.05729578 degrees (3.43775 minutes). One "NATO mil" is defined as <math> \tfrac{1}{6400} </math> of a circle. Just like with the true milliradian, each of the other definitions exploits the mil's useful property of subtensions, i.e. that the value of one milliradian approximately equals the angle subtended by a width of 1 meter as seen from 1&nbsp;km away <math> \left ( \tfrac{2 \pi}{6400} = 0.0009817 \ldots \approx \tfrac{1}{1000} \right ) </math>.
 * Akhnam and zam. In old Arabia a turn was subdivided in 32 Akhnam and each akhnam was subdivided in 7 zam, so that a turn is 224 zam.
@@ Line 459: / Line 459: @@
 |-
 !  Hour
-|  h || 15°|| <math> \frac {\pi}{12} </math>  ||<math> \frac{1}{24} </math>||
+|  h || 15°|| <math> \tfrac {\pi}{12} </math>  ||<math> \tfrac{1}{24} </math>||
 |-
 ! Minute
-|  m || 0°15'||<math> \frac {\pi}{720} </math>||<math> \frac {1}{1,440} </math>||<math> \frac {1}{60} </math> hour
+|  m || 0°15'||<math> \tfrac {\pi}{720} </math>||<math> \tfrac {1}{1,440} </math>||<math> \tfrac {1}{60} </math> hour
 |-
 ! Second
-|  s||  0°0'15"|| <math> \frac {\pi}{43,200} </math> || <math> \frac {1}{86,400} </math>  || <math> \frac {1}{60} </math> minute
+|  s||  0°0'15"|| <math> \tfrac {\pi}{43,200} </math> || <math> \tfrac {1}{86,400} </math>  || <math> \tfrac {1}{60} </math> minute
 |}

@@ Line 335: / Line 335: @@
 [[File:ExternalAngles.svg|thumb|300px|right|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 <math> \frac{1}{2}</math> turn; the measures of the interior angles of a simple convex quadrilateral add up to 2{{math|π}} 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''&nbsp;−&nbsp;2){{math|π}}&nbsp;radians, or (''n''&nbsp;−&nbsp;2)180&nbsp;degrees, (''n''&nbsp;−&nbsp;2)2 right angles, or (''n''&nbsp;−&nbsp;2)<math> \frac{1}{2}</math>&nbsp;turn.
+*: In Euclidean geometry, the measures of the interior angles of a triangle add up to π radians, 180°, or <math> \tfrac{1}{2}</math> turn; the measures of the interior angles of a simple convex quadrilateral add up to 2{{math|π}} 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''&nbsp;−&nbsp;2){{math|π}}&nbsp;radians, or (''n''&nbsp;−&nbsp;2)180&nbsp;degrees, (''n''&nbsp;−&nbsp;2)2 right angles, or (''n''&nbsp;−&nbsp;2)<math> \tfrac{1}{2}</math>&nbsp;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.

@@ Line 308: / Line 308: @@
 Three special angle pairs involve the summation of angles:
 [[File:Complement angle.svg|thumb|150px|The ''complementary'' angles <var>a</var> and <var>b</var> (<var>b</var> is the ''complement'' of <var>a</var>, and <var>a</var> is the complement of <var>b</var>).]]
-* ''Complementary angles'' are angle pairs whose measures sum to one right angle (<math> \frac{1}{4}</math> turn, 90°, or <math> \frac{\pi}{2}</math> 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.
+* ''Complementary angles'' are angle pairs whose measures sum to one right angle (<math> \tfrac{1}{4}</math> turn, 90°, or <math> \tfrac{\pi}{2}</math> 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.
@@ Line 322: / Line 322: @@
 [[File:Angle obtuse acute straight.svg|thumb|right|300px|The angles <var>a</var> and <var>b</var> are ''supplementary'' angles.]]
-* Two angles that sum to a straight angle (<math> \frac{1}{2}</math> turn, 180°, or π radians) are called ''supplementary angles''.
+* Two angles that sum to a straight angle (<math> \tfrac{1}{2}</math> 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.
@@ Line 331: / Line 331: @@
 * 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===

@@ Line 286: / Line 286: @@
 * 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 (<math> \frac{1}{2}</math> turn, 180°, or π radians), to the results as necessary, until the magnitude of the result is an acute angle, a value between 0 and <math> \frac{1}{4}</math> turn, 90°, or <math> \frac {\pi}{2}</math> 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).
+* A ''reference angle'' is the acute version of any angle determined by repeatedly subtracting or adding straight angle (<math> \tfrac{1}{2}</math> turn, 180°, or π radians), to the results as necessary, until the magnitude of the result is an acute angle, a value between 0 and <math> \tfrac{1}{4}</math> turn, 90°, or <math> \tfrac {\pi}{2}</math> 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===

@@ Line 218: / Line 218: @@
 * 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 <math> \frac{1}{4}</math> turn (90° or <math> \frac{\pi}{2}</math> radians) is called a ''right angle''. Two lines that form a right angle are said to be ''normal'', ''orthogonal'', or ''perpendicular''.
+* An angle equal to <math> \tfrac{1}{4}</math> turn (90° or <math> \tfrac{\pi}{2}</math> 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 <math> \frac{1}{2}</math> turn (180° or π radians) is called a ''straight angle''.
+* An angle equal to <math> \tfrac{1}{2}</math> turn (180° or π radians) is called a ''straight angle''.
 * An angle larger than a straight angle but less than 1&nbsp;turn (between 180° and 360°) is called a ''reflex angle''.
 * An angle equal to 1 turn (360° or 2π radians) is called a ''full angle'', ''complete angle'', ''round angle'' or a ''perigon''.
@@ Line 247: / Line 247: @@
 |style = "background:#f2f2f2; text-align:center;" | turn&nbsp;&nbsp;
 | 0 turn
-| <math> \left (0, \frac{1}{4} \right ) </math> turn
+| <math> \left (0, \tfrac{1}{4} \right ) </math> turn
-| <math> \frac{1}{4} </math> turn
+| <math> \tfrac{1}{4} </math> turn
-| <math> \left ( \frac{1}{4}, \frac{1}{2} \right ) </math> turn
+| <math> \left ( \tfrac{1}{4}, \tfrac{1}{2} \right ) </math> turn
-| <math> \frac{1}{2} </math> turn
+| <math> \tfrac{1}{2} </math> turn
-| <math> \left ( \frac{1}{2}, 1 \right )</math> turn
+| <math> \left ( \tfrac{1}{2}, 1 \right )</math> turn
 | 1 turn
 |-
 |style = "background:#f2f2f2; text-align:center;" | radian
 | 0 rad
-| <math> \left (0, \frac{1}{2} \pi \right ) </math> rad
+| <math> \left (0, \tfrac{1}{2} \pi \right ) </math> rad
-| <math> \frac{1}{2} \pi </math> rad
+| <math> \tfrac{1}{2} \pi </math> rad
-| <math> \left ( \frac{1}{2} \pi, \pi \right ) </math> rad
+| <math> \left ( \tfrac{1}{2} \pi, \pi \right ) </math> rad
 | <math> \pi </math> rad
 | (<math> \pi </math>, <math> 2 \pi </math>) rad

@@ Line 184: / Line 184: @@
 [[File:Ray (A, B, C).svg|500px|center|Ray]]
-Thus, we would say that two different points, ''A'' and ''B'', define a line and a decomposition of this line into the disjoint union of an open segment {{open-open|''A'', ''B''}} and two rays, ''BC'' and ''AD'' (the point ''D'' is not drawn in the diagram, but is to the left of ''A'' on the line ''AB''). These are not opposite rays since they have different initial points.
+Thus, we would say that two different points, ''A'' and ''B'', define a line and a decomposition of this line into the disjoint union of an open segment (''A'', ''B'') and two rays, ''BC'' and ''AD'' (the point ''D'' is not drawn in the diagram, but is to the left of ''A'' on the line ''AB''). These are not opposite rays since they have different initial points.
 In Euclidean geometry two rays with a common endpoint form an angle.

@@ Line 488: / Line 488: @@
 ==Bisecting and trisecting angles==
-The [[Greek mathematics|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.
+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 vector]]s '''u''' and '''v''' is related to their [[dot product]] and their lengths by the formula
+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
 :<math> \mathbf{u} \cdot \mathbf{v} = \cos(\theta) \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| .</math>
-This formula supplies an easy method to find the angle between two planes (or curved surfaces) from their [[normal vector]]s and between [[skew lines]] from their vector equations.
+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 <math> \langle \cdot , \cdot \rangle </math>, i.e.
+To define angles in an abstract real inner product space, we replace the Euclidean dot product ( '''·''' ) by the inner product <math> \langle \cdot , \cdot \rangle </math>, i.e.
 :<math> \langle \mathbf{u} , \mathbf{v} \rangle = \cos(\theta)\ \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| .</math>
-In a complex [[inner product space]], the expression for the cosine above may give non-real values, so it is replaced with
+In a complex inner product space, the expression for the cosine above may give non-real values, so it is replaced with
 :<math> \operatorname{Re} \left( \langle \mathbf{u} , \mathbf{v} \rangle \right) = \cos(\theta) \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| .</math>
@@ Line 517: / Line 517: @@
 :<math> \left| \langle \mathbf{u} , \mathbf{v} \rangle \right| = \left| \cos(\theta) \right| \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| </math>
-in a [[Hilbert space]] can be extended to subspaces of any finite dimensions. Given two subspaces <math> \mathcal{U} </math>, <math> \mathcal{W} </math> with <math> \dim ( \mathcal{U}) := k \leq \dim ( \mathcal{W}) := l </math>, this leads to a definition of <math>k</math> angles called canonical or [[principal angles]] between subspaces.
+in a Hilbert space can be extended to subspaces of any finite dimensions. Given two subspaces <math> \mathcal{U} </math>, <math> \mathcal{W} </math> with <math> \dim ( \mathcal{U}) := k \leq \dim ( \mathcal{W}) := l </math>, this leads to a definition of <math>k</math> 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 [[tangent]]s. Where ''U'' and ''V'' are tangent vectors and ''g''<sub>''ij''</sub> are the components of the metric tensor ''G'',
+In Riemannian geometry, the metric tensor is used to define the angle between two tangents. Where ''U'' and ''V'' are tangent vectors and ''g''<sub>''ij''</sub> are the components of the metric tensor ''G'',
 :<math>
@@ Line 527: / Line 527: @@
 ===Hyperbolic angle===
-A [[hyperbolic angle]] is an [[argument of a function|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 [[area]]s 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]]''.
+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 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 [[star]]s 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 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 angle|altitude]] /[[elevation angle|elevation]] with respect to the [[horizon]] as well as the [[azimuth]] with respect to [[north]].
+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.
+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.
-== Licensing ==
+= Licensing =
 Content obtained and/or adapted from:
 * [https://en.wikipedia.org/wiki/Line_(geometry) Line (geometry), Wikipedia] under a CC BY-SA license
 * [https://en.wikipedia.org/wiki/Angle Angle, Wikipedia] under a CC BY-SA license

@@ Line 484: / Line 484: @@
 ==Angles between curves==
 [[File:Curve angles.svg|thumb|right|The angle between the two curves at ''P'' is defined as the angle between the tangents <var>A</var> and <var>B</var> at <var>P</var>.]]
-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 [[tangent]]s at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—''amphicyrtic'' (Gr. {{lang|grc|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.
+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.
 ==Bisecting and trisecting angles==