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'''Directed Angles''' is a method to express [[angles]] that can be very useful in angle chasing problems where there are configuration issues.

== Definition ==
Given any two non-parallel lines <math>l</math> and <math>m</math>, the '''directed angle''' <math>\measuredangle(l,m)</math> is defined as the measure of the angle '''starting''' from <math>l</math> and '''ending''' at <math>m</math>, measured '''counterclockwise''' and '''modulo <math>180^{\circ}</math> (or say it is modulo <math>\pi</math>)'''. With this definition in place, we can define <math>\measuredangle AOB = \measuredangle(AO,BO)</math>, where <math>AO</math> and <math>BO</math> are lines (rather than segments).

An equivalent statement for <math>\measuredangle AOB</math> is that, <math>\measuredangle AOB</math> is '''positive''' if the vertices <math>A</math>, <math>B</math>, <math>C</math> appear in clockwise order, and '''negative''' otherwise, then we take the angles modulo <math>180^{\circ}</math> (or modulo <math>\pi</math>).

[[IMage:50_deg_angle_l_m.png|thumb|Figure 1: The directed angle <math>\measuredangle(l,m)=50^{\circ}</math>, while the directed angle <math>\measuredangle(m,l)=-50^{\circ}=130^{\circ}</math>]]

[[IMage:50_deg_angle_ABC.png|thumb|Figure 2: Here, <math>\measuredangle ABC=50^{\circ}</math> and <math>\measuredangle CBA=-50^{\circ}=130^{\circ}</math>]]

Note that in some other places, regular <math>\angle</math> notation is also used for directed angles. Some writers will also use <math>\equiv</math> sign instead of a regular equal sign to indicate this modulo <math>180^{\circ}</math> nature of a directed angle.

== Warning ==
* The notation introduced in this page for directed angles is still not very well known and standard. It is recommended by many educators that in a solution, it is needed to explicitly state the usage of directed angles.

* Never take a half of a directed angle. Since directed angles are modulo <math>180^{\circ}</math>, taking half of a directed angle may cause unexpected problems.

* Do not use directed angles when the problem only works for a '''certain''' configuration.

== Important Properties == 

* '''Oblivion''': <math>\measuredangle APA = 0</math>.
* '''Anti-Reflexivity''': <math>\measuredangle ABC = -\measuredangle CBA</math>.
* '''Replacement''': <math>\measuredangle PBA = \measuredangle PBC</math> if and only if <math>A</math>, <math>B</math>, <math>C</math> are collinear. 
* '''Right Angles''': If <math>AP \perp BP</math>, then <math>\measuredangle APB = \measuredangle BPA = 90^{\circ}</math>.
* '''Addition''': <math>\measuredangle APB + \measuredangle BPC = \measuredangle APC</math>.
* '''Triangle Sum''': <math>\measuredangle ABC + \measuredangle BCA + \measuredangle CAB = 0</math>.
* '''Isosceles Triangles''': <math>AB = AC</math> if and only if <math>\measuredangle ACB = \measuredangle CBA</math>.
* '''Inscribed Angle Theorem''': If points <math>A</math>, <math>B</math>, <math>C</math> is on a [[circle]] with [[center]] <math>P</math>, then <math>\measuredangle APB = 2\measuredangle ACB</math>.
* '''Parallel Lines''': If <math>AB \parallel CD</math>, then <math>\measuredangle ABC + \measuredangle BCD = 0</math>.
* '''[[Cyclic Quadrilateral]]''': Points <math>A</math>, <math>B</math>, <math>X</math>, <math>Y</math> lie on a circle if and only if <math>\measuredangle AXB = \measuredangle AYB</math>.

== Application ==
The [[slope]] of a line in a [[coordinate system]] can be given as the tangent of the directed angle between <math>x</math>-axis and this line. (Remember the tangent function has a period <math>\pi</math>, so we have our "modulo <math>\pi</math>" part in tangent function)

Other than that, direct angles can be very useful when a geometric (usually angle chasing) problem have a lot of configuration issues. We can avoid solving the same problem twice (sometimes even multiple times) by applying direct angles.

Here are some examples with directed angles:
* Proof of the [[Miquel's Point]]
* Proof of the [[Orthic Triangle]]
* Proof of the [[Pascal's Theorem]]
* [[2002_IMO_Shortlist_Problems/G4|2002 IMO Shortlist Problems G4]]
* [[2010_IMO_Shortlist_Problems/G1|2010 IMO Shortlist Problems G1]]
* [[1998_APMO_Problems/Problem_4|1998 APMO Problem 4]]

== See Also ==
* [https://web.evanchen.cc/handouts/Directed-Angles/Directed-Angles.pdf Handout on Evan Chen's Website]
* [https://www.math.ubbcluj.ro/~didactica/pdfs/2018/didmath2018-03.pdf An article on a Romanian Mathematical Journal, DIDACTICA MATHEMATICA]