Angles in the same segment

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:

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.

To download this Geogebra file click here.