Chapter 10: Circles

I have learned much from our study of circles in Chpter 10. I have learned that a tangent is a line that intercepts a circle at one point, that inscribed angles are half of their intercepted arc and that central angles are equal to their corresponding arc measurements. Many jobs can apply geometry into their everyday working routine. For example, statisticians must know what percentages and angle measures a pie chart must equal for it to equal 360 degrees and for it to equal 100%. I am still wondering how it is possible for some vertical angles to not equal each other when intercepted in a circle. On many problems, I have noticed that opposite arcs are not equal and I'm wondering how this is happening and how this can be explained. Help!