Directed angles are angles that are directed. There are 2 ways of counting the directed angle
- the angle is positive when the points are in clockwise order, and negative otherwise, or
- the angle is positive when the lines and are in counterclockwise order, and negative otherwise.
You should see that both ways are identical to each other. As an example, see the diagram below:
We take every angle modulo , i.e.
This means that in the diagram above, .
Directed angles might seem really annoying and useless at first, but soon you should find it natural and sometimes better than the normal angles.
Directed angles are useful in combining multiple cases in a statement into one. An example is shown below.
The theorem below has configuration issues; the two cases seem so different and hence we have to differentiate them. By using directed angles, we can significantly reduce the number of words used.
Theorem (Cyclic Quadrilaterals)
Let be any four points, no three collinear.
(i) If and lie on the same side of then the four points are concyclic if and only if
(ii) If and lie on different sides of then the four points are concyclic if and only if
Theorem (Directed Cyclic Quadrilaterals)
Let be four points, no three collinear. Then they are concyclic if and only if
Task: Make sure it works.
Let be any point. Points are collinear if and only if
Task: Prove this yourself. Hint: Show that the assertion is equivalent to
Always remember that this doesn't work for every single case. You should always check your answer after writing it out. An example where directed angles will destroy the solution is shown below:
In the cyclic quadrilateral , let and denote the incenters of and , respectively. Prove that is cyclic.
Do not use directed angles; the problem is false if lie in that order.