One slice of a circle which has been divided into 360 slices is one degree right? If this is the case, won't bigger circles have bigger slices and therefore bigger degrees? Why is one degree of a small circle equal to one degree of a much larger circle?
Consider two rays with a common endpoint. An angle is a measure of ROTATION of one ray with respect to the other. It doesn't matter then how long you draw the rays on paper. A full rotation (=circle) is agreed to be 360 degrees.