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, and also into angle BEC.

so, "Angles in the same segment are equal"