Notes
two squares solution

Two Squares

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

As the red square has double the area of the yellow square, the length of the diagonal of the yellow square is the same as the side length of the red square.

This means that a circle drawn with centre $C$ through $A$ also passes through $B$ and $D$. Then as the angle at the circumference is half the angle at the centre, angle $A \hat{D} B$ is half of $A \hat{C} B$, which is $90^\circ$. So angle $A \hat{D} B$ is $45^\circ$.

Solution by Properties of Squares, Isosceles Triangle, and Angles in a Triangle

As above $C D$ has the same length as $C B$ so triangle $C B D$ is isosceles. Since angle $B \hat{C} D$ is $45^\circ$, this means that angle $C \hat{D} B$ is $67.5^\circ$. Triangle $A D C$ is also isosceles and angle $A \hat{C} D$ is $135^\circ$ so angle $C \hat{D} A$ is $22.5^\circ$. This makes angle $A \hat{D} B$ equal to $67.5^\circ - 22.5^\circ = 45^\circ$.