In a calculus course I took a while ago, we defined an angle between two rays in $\mathbb{R}^2$ sharing a common endpoint as the length of the circle with radius 1, centered at the point of intersection.
Going back to my old notes, I was wondering: Is this definition even well-defined ? Because one has to give additional information at which of the rays forming the angle to start measuring, if I'm not mistaken.
And then there's also the problem of measuring an angle - Why does one use dimension-sounding phraseology, like "degree" or "radian", if at the core of the definition, the measure of an angle, is nothing more than a function, associating every pair of rays sharing a common endpoint a number, by a (strictly mathematically speaking) rather complicated process, of computing the length of a curve ?
Do you know of other definitions of an angle ? It seems to me, like the mathematical object "angle" is getting too little attention - in almost no textbook during the first years of university does it come up (although one needs it for example in complex analysis, for the multiplication of complex numbers), and when it is used, just an intuitive definition of it is assumed.