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Directed angles

(Redirected from Directed Angle)

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 $l$ and $m$, the directed angle $\measuredangle(l,m)$ is defined as the measure of the angle starting from $l$ and ending at $m$, measured counterclockwise and modulo $180^{\circ}$ (or say it is modulo $\pi$). With this definition in place, we can define $\measuredangle AOB = \measuredangle(AO,BO)$, where $AO$ and $BO$ are lines (rather than segments).

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

Figure 1: The directed angle $\measuredangle(l,m)=50^{\circ}$, while the directed angle $\measuredangle(m,l)=-50^{\circ}=130^{\circ}$
Figure 2: Here, $\measuredangle ABC=50^{\circ}$ and $\measuredangle CBA=-50^{\circ}=130^{\circ}$

Note that in some other places, regular $\angle$ notation is also used for directed angles. Some writers will also use $\equiv$ sign instead of a regular equal sign to indicate this modulo $180^{\circ}$ 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 $180^{\circ}$, 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: $\measuredangle APA = 0$.
  • Anti-Reflexivity: $\measuredangle ABC = -\measuredangle CBA$.
  • Replacement: $\measuredangle PBA = \measuredangle PBC$ if and only if $A$, $B$, $C$ are collinear.
  • Right Angles: If $AP \perp BP$, then $\measuredangle APB = \measuredangle BPA = 90^{\circ}$.
  • Addition: $\measuredangle APB + \measuredangle BPC = \measuredangle APC$.
  • Triangle Sum: $\measuredangle ABC + \measuredangle BCA + \measuredangle CAB = 0$.
  • Isosceles Triangles: $AB = AC$ if and only if $\measuredangle ACB = \measuredangle CBA$.
  • Inscribed Angle Theorem: If points $A$, $B$, $C$ is on a circle with center $P$, then $\measuredangle APB = 2\measuredangle ACB$.
  • Parallel Lines: If $AB \parallel CD$, then $\measuredangle ABC + \measuredangle BCD = 0$.
  • Cyclic Quadrilateral: Points $A$, $B$, $X$, $Y$ lie on a circle if and only if $\measuredangle AXB = \measuredangle AYB$.

Application

The slope of a line in a coordinate system can be given as the tangent of the directed angle between $x$-axis and this line. (Remember the tangent function has a period $\pi$, so we have our "modulo $\pi$" 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:

See Also

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