An angle is formed by two rays with a common endpoint.
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Angles are the degree measure between the two rays that make an angle. Vertices are the point in which the two rays meet to create that angle.
There is 1 angle between each pair of rays. So 30 rays equals 30 angles. But, if the angles are likened to the spokes of a bicycle wheel, should the 30 rays not completely fill the circumference of the wheel, the 2 outer rays also form an angle. So the possible total is 31 angles.
Total 180o
There is no relationship between slope and the theorem, however the theorem does deal with the relationship between angles and sides of a triangle.
The sum of the three angles is 180 degrees. There are other relationships that can be derived from this.