Notes
two rectangles in a semi-circle solution

Two Rectangles in a Semi-Circle

Solution by Properties of Chords and Angle at the Circumference is Half the Angle at the Centre

As the rectangles are congruent, the diagonals $O A$ and $O B$ are of equal length and so triangle $A O B$ is isosceles. This means that point $O$ lies on the perpendicular bisector of $A B$. Since $A B$ is a chord, the centre of the circle also lies on that perpendicular bisector. Since $O$ is also on the diameter, then, it is the centre.

Using the fact that the angle at the circumference is half the angle at the centre, angle $A \hat{C} B$ is half of angle $A \hat{O} B$, read anti-clockwise from $A$ to $B$. As the rectangles are congruent, angles $A \hat{O} D$ and $D \hat{O} B$ add up to $90^\circ$, so angle $A \hat{O} B$ is $270^\circ$, and hence angle $A \hat{C} B$ is $135^\circ$.