Notes
angle in two overlapping squares solution

Solution to the Angle in Two Overlapping Squares Puzzle

Solution by Angle at the Circumference is Half that at the Centre

In the diagram above, the circle has centre $O$ and passes through point $A$. As the $O B$ is another side of the square, $O B$ has the same length as $O A$ and so the circle also passes through $B$. The squares are the same size, so $O C$ is also the same length and so the circle passes through $C$.

Angle $A \hat{C} B$ is formed by joining a point on the circumference of the circle, $C$, to two other points, $A$ and $B$, and so since the angle at the circumference is half that at the centre, it is half the angle formed by joining $A$ and $B$ to the centre, $O$. This angle is the corner of the square and so is a right-angle.

Hence angle $A \hat{C} B = 45^\circ$.