I have recently been introduced to Hilbert's axioms of geometry. Right now I am studying angles. A text I have been using as a guide defines an angle and its opening in the following way:
An angle is the union of two half-lines which share a common endpoint and do not lie in the same line. The angle formed by the half-lines $OA$ and $OB$ is denoted by $\angle AOB$.
The opening of an angle $\angle AOB$ is the set of points which lie on the same side of the line $OA$ as $B$ and at the same time lie on the same side of the line $OB$ as $A$.
I however have some problems with this definition. First, the definition seems to discard the concept of a straight or a full angle as the half-lines defining the angle must be on separate lines. Secondly the definition also seems to treat an angle and its explementary angle as the same angle. Because of this I have been thinking of an alternative definition:
- An angle is a pair of half-lines which share a common endpoint. The angle $(OA,OB)$ is denoted by $\angle AOB$ and we call call the half-line $OA$ the left side of the angle $\angle AOB$ and call the half-line $OB$ the right side of the same angle.
As the half-lines may be on the same line, the concept of an opening of an angle doesn't seem to have a obvious replacement.
I am now particularly interested in how we could order these kinds of angles. In the definition of the text that I am using, in the way I have interpreted it, they define an ordering of an angle in terms of its opening: We say that $\angle ABC <\angle PQR$ if and only if there exists a point $S$ in the opening of the angle $\angle PQR$ such that the angles $\angle ABC$ and $\angle PQS$ are congruent.
Attempting to define a similar kind of concept of order has led me to difficulty in the other definition as the concept of an opening of an angle isn't there. Instead of a linear order, I have also been trying to define a cyclic order these kinds of angles (Wikipedia has an article on cyclic orders if they are unfamiliar: https://en.wikipedia.org/wiki/Cyclic_order). However my attempts have not yet succeeded. I would appreciate any ideas for trying to formulate a concept of order for these kinds of angles or other ways to think about angles and their ordering