I came upon the following exercise. It seems so obvious, yet I don’t know on which axiom (or more fundamental theorem) to base the proof.
Let $xOz$ be an angle, let $A$ be a random point on the plane. Define the
interior of the angle $xOz$ to be the intersection of the semiplanes
$(Ox, A)$ and $(Oz, A)$. By this definition it follows directly that $A$ is
considered “internal” to the angle $xOz$. Now, let $B$ be a point external
to the angle (i.e. a point not belonging to the intersection of the semiplanes $(Ox, A)$ and $(Oz, A)$. Prove that the line segment $AB$ must cut through one of
the sides of the angle $xOz$.