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techincally 360 degrees

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Not necessary for them to be integer values. A quarter of a right angle (sixteenth of a circle), for example, is 22.5 degrees. Also, later on in mathematics you will learn to measure angles in radians rather than degrees. There are 2*pi radians in a circle - an irrational number. So the measure of an angle need not even be rational.

So the correct answer is infinitely many.

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