Indeed in problem 23 the angle you want to work with does not go through the diameter, however, that doesn't mean you can't draw some other lines to help you figure it out. Drawing the diameter will let you write the angle you want in terms of two angles you already know how to work with, that is, you can write $\angle APB$ as the difference of two angles that go through the diameter.

On 24 one extra line that should help is $BC$. Try writing the angle you want in terms of this two inscribed angles, and use the fact that the angle measure of these two inscribed angles is half of the angle measure of the arcs they open to.

Try using the fact that $\angle A + \angle C = 180^\circ$, together with the fact that the quadrilateral is a trapezoid. What do you know about the angles of a trapezoid? (Look at the angles formed between parallel lines!)