Elementary Geometry problem - Proving the angles $A'B'C'$ and $\beta$ are the same

Question

In the above picture, the line t is tangent to the circle at $C'$. How do I prove the angle $A'B'C'$ is equal to the angle $\beta$?

I tried to do a lot of things, like tracing parallels and perpendiculars through points and using theorems of angles to use similarity of triangles or using the outer angle theorem... nothing worked.

Thanks.

Answer 1

They both intercept the same arc. They are essentially inscribed angles whose measure is one-half the intercepted arc.