Oriented Angles in Euclidean Geometry

Question

I was recently introduced to the idea of oriented angles in a proof of the inscribed angle theorem. While it makes sense to use the oriented angles in that way (as you can look at them through modulo 180), it wasn‘t entirely obvious to me what the general advantages and disadvantages of oriented angles are.

Are there things (lemmas, properties, etc.) in Euclidean Geometry which can only be used when working with unoriented angles? How about oriented angles? Can they just be used interchangeably?

If the two cannot be used interchangeably, how does one notice which ones to use? What is the general advice for that? What are the general advantages/disadvantages of oriented angles?

The proof also mentioned that oriented angles are more useful than unorientef angles in most cases. If that is the case, why are school students introduced to unoriented angles. Something looks off to me...

Answer 1

The main advantage of directed angles modulo $180^\circ$ is that four points $A, B, C, D$ will always lie on a circle if and only if $\measuredangle ABC = \measuredangle ADC$, nevermind any betweenness conditions. Another advantage is that the angle $\measuredangle\left(g;h\right)$ between two lines $g$ and $h$ is well-defined, without having to specify whether you mean "the acute" or "the obtuse" one. All in all, these features allow you to perform angle chasing without having to draw any picture (however, it is still necessary to ensure that the lines in question don't degenerate to a point -- i.e., angles of the form $\measuredangle AAB$ or $\measuredangle ABB$ are still undefined, so it is still necessary to check distinctness of points).

Some examples of the use of directed angles modulo $180^\circ$ can be found in

• Jan van Yzeren, Pairs of Points: Antigonal, Isogonal, and Inverse, Mathematics Magazine 65 (1992), issue 5, pp. 339--347.

• R. A. Johnson, Directed Angles in Elementary Geometry, American Mathematical Monthly 24 (1917) no. 3, pp. 101--105.

• R. A. Johnson, Directed Angles and Inversion, With a Proof of Schoute's Theorem, American Mathematical Monthly 24 (1917) no. 7, pp. 313--317.

• Kiran Kedlaya, Geometry Unbound, 2006-01-17.

• Darij Grinberg, The Neuberg-Mineur circle.

Directed angles modulo $360^\circ$ are mainly useful to specify rotations, and as doubles of directed angles modulo $180^\circ$. For instance, if $A, B, C$ are three points on a circle, and if $O$ is the center of this circle, then $\measuredangle AOB$ (a directed angle modulo $360^\circ$) equals double the angle $\measuredangle ACB$ (a directed angle modulo $180^\circ$).