When we talk about measurements, we use much. I would guess that this is because we're applying a countable measurement to an uncountable quantity. After all, angle is uncountable, as is distance, area and so on. Thus, you're not really talking about a number of degrees, but a quantity of angle that is represented by the number of degrees.
Telling when this is 'correct' and when it's a common 'error' is difficult, because errors are very common. Where talking about genuinely discrete objects, even abstract ones, it 'ought' to be many rather than much. However, people will use the 'wrong' one a lot, which really calls into question the idea that it's 'wrong'. However, if you're trying to avoid people who "know the rules" from correcting you, it's good to understand them as best as you can.