Directed Angles
Let’s learn about the concept of directed angles which enables us to write much shorter solutions.
Let’s denote such angles with ∡ . Since this is not a standard notation, if you would like to use it somewhere, do not forget to mention this in your first sentences.
Here is how it works:
First, we consider ∠ABC to be positive if the vertices A, B, C appear in clockwise order, and negative otherwise. In particular, ∠ABC ≠∠CBA; they are negatives.
Then, we are taking the angles modulo 180◦. For example,
−150◦ = 30◦ = 210◦
(From “Euclidian Geometry In Mathematical Olympiads” by Evan Chen)
For example,
By using this concept, we can easily say that a quadrilateral AXBY is cyclic if and only if ∡AXB = ∡AYB
That is easy to prove, without the directed angles we can say ∠AXB = 180 — ∠AYB. If we convert this inequality into directed angles, we will have :
∡AXB = -∡BYA = ∡AYB
Since until this point we have reviewed the fundemental ideas of directed angles, we will leave you some lemmas here and will let you prove them, guaranteeing you that they are not that hard to prove. If you have any questions or you would like to get help, you may contact us via social media or email.
Oblivion. ∡APA = 0.
Anti-Reflexivity. ∡ABC = −∡CBA.
Replacement. ∡PBA = ∡PBC if and only if A, B, C are collinear. (What happens
when P = A?) Equivalently, if C lies on line BA, then the A in ∡PBA may be
replaced by C.
Right Angles. If AP ⊥ BP, then ∡APB = ∡BPA = 90◦.
Directed Angle Addition. ∡APB + ∡BPC = ∡APC.
Triangle Sum. ∡ABC + ∡BCA + ∡CAB = 0.
Isosceles Triangles. AB = AC if and only if ∡ACB = ∡CBA.
Inscribed Angle Theorem. If (ABC) has center P, then ∡APB = 2∡ACB.
Parallel Lines. If AB CD, then ∡ABC + ∡BCD = 0.
(From “Euclidian Geometry In Mathematical Olympiads” by Evan Chen)
This article have introduced you a fundamental topic in your geometry journey. There are pretty many problems where you can use directed angles. If you want to see how, read our article:
