Editing Angle (section)

==Types{{anchor|Types of angles}}==
{{Redirect|Oblique angle|the cinematographic technique|Dutch angle}}

===Vertical and {{vanchor|adjacent}} angle pairs===
[[File:Vertical Angles.svg|thumb|150px|right|Angles A and B are a pair of vertical angles; angles C and D are a pair of vertical angles. [[Hatch_mark#Congruency_notation|Hatch marks]] are used here to show angle equality.]]
{{redirect-distinguish|Vertical angle|Zenith angle}}

When two straight lines intersect at a point, four angles are formed. Pairwise, these angles are named according to their location relative to each other.
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| A pair of angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are called ''vertical angles'' or ''opposite angles'' or ''vertically opposite angles''. They are abbreviated as ''vert. opp. ∠s''.<ref name="tb">{{harvnb|Wong|Wong|2009|pp=161–163}}</ref>
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The equality of vertically opposite angles is called the ''vertical angle theorem''. [[Eudemus of Rhodes]] attributed the proof to [[Thales|Thales of Miletus]].<ref>{{cite book|author=Euclid|author-link=Euclid|title=The Elements|title-link=Euclid's Elements}} Proposition I:13.</ref>{{sfn|Shute| Shirk|Porter|1960|pp=25–27}} 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,{{sfn|Shute| Shirk|Porter|1960|pp=25–27}} 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 {{nowrap|180° − ''x''}}. Similarly, the measure of angle ''D'' would be {{nowrap|180° − ''x''}}. Both angle ''C'' and angle ''D'' have measures equal to {{nowrap|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 {{nowrap|1=180° − (180° − ''x'') = 180° − 180° + ''x'' = ''x''}}. Therefore, both angle ''A'' and angle ''B'' have measures equal to ''x'' and are equal in measure.

[[File:Adjacentangles.svg|right|thumb|225px|Angles ''A'' and ''B'' are adjacent.]]
| ''Adjacent angles'', often abbreviated as ''adj. ∠s'', are angles that share a common vertex and edge but do not share any interior points. In other words, they are angles 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 ''{{section link|#Combining angle pairs}}'' below).
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A [[Transversal (geometry)|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''.{{sfn|Jacobs|1974|p=255}}

===Combining angle pairs===
{{anchor|Angle addition postulate}}The '''angle addition postulate''' states that if B is in the interior of angle AOC, then

<math display="block"> m\angle \mathrm{AOC} = m\angle \mathrm{AOB} + m\angle \mathrm{BOC} </math>

I.e., the measure of the angle AOC is the sum of the measure of angle AOB and the measure of angle BOC.

Three special angle pairs involve the summation of angles:
{{anchor|complementary angle}}
[[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>.)]]
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| ''Complementary angles'' are angle pairs whose measures sum to one right angle ({{sfrac|4}} turn, 90°, or {{sfrac|{{math|π}}|2}} radians).<ref>{{Cite web|title=Complementary Angles|url=https://www.mathsisfun.com/geometry/complementary-angles.html|access-date=2020-08-17 | website=www.mathsisfun.com}}</ref> 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 accounts for 90 degrees.
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The adjective complementary is from the Latin ''complementum'', associated with the verb ''complere'', "to fill up". An acute angle is "filled up" by its complement to form a right angle.
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The difference between an angle and a right angle is termed the ''complement'' of the angle.<ref name="Chisholm 1911">{{harvnb|Chisholm|1911}}</ref>
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If angles ''A'' and ''B'' are complementary, the following relationships hold: <math display="block">
\begin{align}
& \sin^2A + \sin^2B = 1 & & \cos^2A + \cos^2B = 1 \\[3pt]
& \tan A = \cot B  & & \sec A = \csc B
\end{align}</math>
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(The [[tangent]] of an angle equals the [[cotangent]] of its complement, and its secant equals the [[cosecant]] of its complement.)
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The [[prefix]] "[[co (function prefix)|co-]]" in the names of some trigonometric ratios refers to the word "complementary".
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[[File:Angle obtuse acute straight.svg|thumb|right|300px|The angles <var>a</var> and <var>b</var> are ''supplementary'' angles.]]
| {{anchor|Linear pair of angles|Supplementary angle}}Two angles that sum to a straight angle ({{sfrac|2}} turn, 180°, or {{math|π}} radians) are called ''supplementary angles''.<ref>{{Cite web|title=Supplementary Angles|url=https://www.mathsisfun.com/geometry/supplementary-angles.html|access-date=2020-08-17 | website=www.mathsisfun.com}}</ref>
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If the two supplementary angles are [[#adjacent|adjacent]] (i.e., have a common [[vertex (geometry)|vertex]] and share just one side), their non-shared sides form a [[line (geometry)|straight line]]. Such angles are called a ''linear pair of angles''.{{sfn|Jacobs|1974|p=97}} 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.
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If a point P is exterior to a circle with center O, and if the [[tangent lines to circles|tangent lines]] from P touch the circle at points T and Q, then ∠TPQ and ∠TOQ are supplementary.
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The sines of supplementary angles are equal. Their cosines and tangents (unless undefined) are equal in magnitude but have opposite signs.
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In Euclidean geometry, any sum of two angles in a triangle is supplementary to the third because the sum of the internal angles of a triangle is a straight angle.
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{{anchor|explementary angle}}
[[File:Conjugate Angles.svg|thumb|Angles AOB and COD are conjugate as they form a complete angle. Considering magnitudes, 45° + 315° = 360°.]]
| Two angles that sum to a complete angle (1 turn, 360°, or 2{{math|π}} radians) are called ''explementary angles'' or ''conjugate angles''.<ref>{{cite book |last=Willis |first=Clarence Addison |year=1922 |publisher=Blakiston's Son |title=Plane Geometry |page=8 |url=https://archive.org/details/planegeometryexp00willrich/page/8/ }}</ref>
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The difference between an angle and a complete angle is termed the ''explement'' of the angle or ''conjugate'' of an angle.
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===Polygon-related angles===
[[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. {{pb}} <!-- --> In [[Euclidean geometry]], the measures of the interior angles of a [[triangle]] add up to {{math|π}} radians, 180°, or {{sfrac|2}} turn; the measures of the interior angles of a simple [[convex polygon|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){{sfrac|1|2}}&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|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.{{sfn|Henderson|Taimina|2005|p=104}} If the corresponding interior angle is a reflex angle, the exterior angle should be considered [[Negative number|negative]]. Even in a non-simple polygon, it may be possible to define the exterior angle. Still, one will have to pick an [[orientation (space)|orientation]] of the [[plane (mathematics)|plane]] (or [[surface (mathematics)|surface]]) to decide the sign of the exterior angle measure. {{pb}} <!--
--> 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 (programming language)|Logo Turtle programs]] when drawing regular polygons.
* In a [[triangle]], the [[bisection|bisectors]] of two exterior angles and the bisector of the other interior angle are [[concurrent lines|concurrent]] (meet at a single point).<ref name=Johnson>Johnson, Roger A. ''Advanced Euclidean Geometry'', Dover Publications, 2007.</ref>{{rp|p=149}}
* In a triangle, three intersection points, each of an external angle bisector with the opposite [[extended side]], are [[collinearity|collinear]].<ref name=Johnson/>{{rp|p=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.<ref name=Johnson/>{{rp|p=149}}
* Some authors use the name ''exterior angle'' of a simple polygon to mean the ''explement exterior angle'' (''not'' supplement!) of the interior angle.<ref>{{citation|editor=D. Zwillinger|title=CRC Standard Mathematical Tables and Formulae|place=Boca Raton, FL|publisher=CRC Press | year=1995 | page= 270}} as cited in {{MathWorld |urlname=ExteriorAngle |title=Exterior Angle}}</ref> This conflicts with the above usage.

===Plane-related angles===
* The angle between two [[Plane (mathematics)|planes]] (such as two adjacent faces of a [[polyhedron]]) is called a ''[[dihedral angle]]''.<ref name="Chisholm 1911"/> It may be defined as the acute angle between two lines [[Normal (geometry)|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 (geometry)|normal]] to the plane.