Notes
angle formed by two squares solution

Solution to the Angle Formed by Two Squares Puzzle

Solution by Equilateral Triangles

With the points labelled as above, triangle $C B D$ has the property that it is right-angled at $D$ as that is where the diagonals of the square cross. Also, the length of $C B$ is double that of $B D$. This means that triangle $C B D$ is a half of an equilateral triangle. Therefore, angle $D \hat{C} B$ is $30^\circ$ and so the requested angle is $75^\circ$.