The line AB is a chord, dividing the circle into two segments.

While the points C and D lie in the same segment the angles ACB and ADB will be equal.

Try the following:

- Move C or D. Neither of the measured angles should change.
- Move A or B. The angles should change - but remain equal.

Obviously this assumes you keep C and D in the same segment ie on the same side of AB; otherwise you have a cyclic quadrilateral and then the two angles will add to 180°.

The reason behind this is basically the angle at the centre is double angle at circumference (see the hint). If A and B do not move then the angle at the centre does not change so neither does the one at the circumference.