Calculate angles of rectangle

Question

How would one go about to solve the exercise on the image? I really have no ansatz to solve it, except that the triangle with vertices $Z_1, Z_2$ and the point south has the same lengths thus has angles of 60 degrees. But what would be the next step to determine $\alpha, \beta,$ and $\gamma$?

Answer 1

I have identified a few more points....

You have that $\triangle Z_1Z_2A$ is equilateral.

I suggest that $\angle Z_1Z_2D = 45^\circ$

An angle on the circumference of a circle has half the measure of the arc length.

Using this logic you can find $\alpha$ and $\angle ADZ_1$

$\triangle Z_1Z_2D$ is isosceles.

Which you can use to find $\angle DZ_1Z_2$ and from there $\beta$

Similar logic will get you the measure of $\gamma$

Answer 2

The angles are determined by employing the following properties: Due to construction arcs, there is a bisection of the right angle. An angle subtended by an arc at the circumference is half the angle subtended by an identical arc at the center. Lastly, the exterior angle of a triangle is equal to the sum of the two interior opposite angles.

Answer 3

Hint:

as a next step I would consider the triangle $Z_{south},Z_2,Z_3$, which is isosceles and you know the angle in $Z_2$, ... you should quickly reach to $\gamma$ ....