Any angle divides a plane into two regions

Question

I am studying Euclidian geometry and I noticed that any angle divides a plane into two regions: an inside and an outside. Is there a need for a proof of this (something along the lines of Jordan theorem), or is it just "obvious"?

Browsing the internet, I came across a following simpler version: any line divides a plane into two regions. Maybe someone will find it relevant.

My current understanding is that the PSA plays a key role in this sort of thing. Unless you are doing it the analytic geometry way, in which case the PSA must be somehow already "coded in" ...I think.

Answer 1

Let $\vec v$ (the vertex), $\vec a$, $\vec b$ (the rays) of an angle in $\boldsymbol R^2$. The interior $I$ of the angle may be defined as $I:=\{\vec v+t\cdot\vec a+s\cdot\vec b\mid t,s\in\boldsymbol R_+\cup\{0\}\}$. Can you go from here?

Answer 2

Interesting problem ... Not sure if this is totally correct but I tried using the plane separation axiom (PSA) twice.

Between: undefined (along with point, line, on, and congruent, cf. Hilbert).

Same side: Let $\it{l}$ be a line and let A and B be two points which are not on $\it{l}.$ Points A and B are $\textit{on the same side}$ of $\it{l}$ if either $\it{l}$ and $\leftarrow AB \rightarrow$ do not intersect at all, or if they do intersect but the point of intersection is not between A and B. [cf: Harvey, link below]

PSA: For any line $\it{l}$ and points $A, B, C$ which are not on $\it{l}:$ (i) if A and B are on the same side of $\it{l}$ and A and C are on the same side of $\it{l},$ then B and C are on the same side of $\it{l};$ (ii) If A and B are not on the same side of $\it{l}$ and A and C are not on the same side of $\it{l},$ then B and C are on the same side of $\it{l}.$

The definition of angle interior is

A point lies in the interior or is an interior part of $\angle BAC$ if it is on the same side of AB as C and the same side of AC as B.

(ref: http://www.mcs.uvawise.edu/msh3e/resources/geometryBook/geometryBook.html)

Here's a diagram: