What is the difference between the geometric and trigonometric definition of an angle? I vaguely remember reading that there is a difference between the geometric definition of an angle and the trigonometric definition of an angle. I've tried to search everywhere I can think of but I can't find what I read. 

So what I would like to know is:


Is there a difference between the two? 
and if there is, what are the two definitions?
I have never heard of such a dichotomy, but I can imagine that one might call "two rays with a common vertex" the "geometric definition of an angle" and the "numeric angle measure" the "trigonometric definition of an angle".
The usual terminology is just "angle" for the figure and "angle measure" for the number.
Geometric right angle = $90^\circ$, trigonometric right angle = $\pi/2$?