The definition of an angle in Euclidean geometry is given as follows:
Angle. Definition: A shape, formed by two lines or rays diverging from a common point (the vertex).
However, the definition nowhere states how to define the measurement for an angle. With a bit more digging/ thinking its obvious that the definition for a measurement is something like:
Draw a unit circle around the vertex. The angle is then measured as the arc length between the intersection of the two lines and the circle. (in radians)
That is how measuring an angle seems to be always defined, but given that the definition of an angle does't outright state that it is the only way of measurement, I wonder if other measurements could be used that are consistent with Euclidean geometry. For example:
Now, for any non-circle based measurement, it would seem that whenever the "measurement apparatus" is spun the angles change, which means that they have to be always used parallel to some reference line. On the other hand, calculating sin() etc. becomes really easy.
Question: Is there something seriously wrong in altering the methodology of how angle is measured, that might break Euclidean geometry? Is the "circle" definition of an angle the only valid one?