Basic question about angles and measurement in degrees I have a doubt related to angles which I am a bit embarrassaed to ask since I know is something of basic geometry, but nevertheless my question is the following:
As I understand it, an angle between two rays with a common vertex is related to the "openness" that exists between these rays. So, one way to measure this "openness" in a form we all can understand is in what we call degrees, a degree is defined as "the 360th part of a full circle". So here is my big question: 
Suppose I have two circles with different radius, and in each of them I try to determine the openness between two rays as I would do it with a protractor, then, when I divide one circle in 360 segments of equal size, this size is going to be different from the size of the segment obtained by dividing the other circle. Given this difference of sizes, wouldn't the number of degrees of the angle be different when measured in one circle than when measured in the other one? If this is the case, when I talk about the degrees of an angle formed by two rays, what am I talking about? Is is with respect to the division in 360 segments of the unit circle?
Sorry for the lenght of my text but I am extremely messed up with this, I would really appreciate if someone could explain this to me. 
 A: This is a good question because it is asking what really an angle is. An angle is a ratio of the arc intercepting the two rays extended from its vertex angle by it and a circle's circumference which is centered at the vertex of the angle. Obviously, if the rays intersect two circles of different sizes, the angle must still be the same.
If the above explanation doesn't convince you, I can present a more algebraic proof that will explain your confusion. Consider an angle in the plane and a circle of radius $r_1$ centered at the vertex of the angle. The angle by definition is going to be $\theta_1 = 360^{\circ} \cdot \frac{\text{arc length for $r_1$}}{2\pi r_1}$. Now consider another circle of radius $r_2$ centered at the vertex of the angle. The angle formed, call it $\theta_2$, is going to be $\theta_2 = 360^{\circ} \cdot \frac{\text{arc length for $r_2$}}{2\pi r_2}$. Now we may say that $r_1 = r_2k$. Thus, we now have $$\theta_1 = 360^{\circ}\cdot\frac{\text{arc length for $r_1$}}{2\pi r_2k}$$ and $$\theta_2 = 360^{\circ}\cdot\frac{\text{arc length for $r_2$}}{2\pi r_2}.$$ It remains to show that $\text{arc length for $r_1$}$ = $\text{arc length for $r_2$k}$. This is true since $\text{arc length for $r_1$} = 2\pi r_1 \cdot \frac{\theta}{360^{\circ}} = 2 \pi r_2k \cdot \frac{\theta}{360^{\circ}}$ and $\text{arc length for $r_2$} = 2\pi r_1 \cdot \frac{\theta}{360^{\circ}} = 2 \pi r_2 \cdot \frac{\theta}{360^{\circ}}$.
A: An angle is not a measurement of openness; it measures how much you "spin" going from one ray to the other. A full spin - going back to where you started - is 360 degrees. So a quarter spin, more commonly called a right angle, is 90 degrees no matter which protractor you use. Protractor markings are spaced differently based on their size (which you can see by putting one protractor on top of another) so no matter what, the same angle has the same measurement.
360 is a completely arbitrary number - we just chose it because it's easy to divide by a lot of things. You could also think about angles in fractions of a full turn (actually, there are several other systems that don't use 360 degrees as a full turn!) The important thing is what angles measure: they measure how much you turn.
A: You are giving the answer yourself.
"when I divide one circle in 360 segments of equal size" and later "wouldn't the number of degrees of the angle be different".
The size of the segments is such that 360 of them correspond to a full turn. 180 for a half turn, 90 for a quarter... whatever the radius.
