Here is an activity to try before you look at the proof;

1) print off a pdf of this page (download here).

2) cut out the circle carefully.

3) now cut out the grey sector ABC, and fold it in half with grey side showing, so that edges AB and AC line up.

The angle in this new, half size sector should fit exactly into angle BDC,

which means α = 2 β

so, "Angle at the centre is twice the angle at the circumference"

(A note: I first tried this method of illustrating the theorem when I was tutoring a student & neither of us had a protractor handy! I think it is far more convincing than measuring the angle, where any small error hides the beautiful truth ;-)