Editing Angle (section)

===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:
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[[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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