Notes
two rectangles solution

Solution to the Two Rectangles Puzzle

Solution by Congruent Shapes and Angles in Isosceles Right-Angled Triangles

In the above diagram, point $E$ is so that $A C E G$ is a rectangle. Therefore, $E D$ has length $5 - 3 = 2$ and $E G$ has length $1 + 2 = 3$. Therefore, rectangle $G H D E$ is congruent to rectangle $D I B C$ but rotated by $90^\circ$. Hence angle $G \hat{D} B$ is $90^\circ$, and line segments $G D$ and $B D$ have the same length. Therefore, triangle $G D B$ is an isosceles right-angled triangle. Hence angle $D \hat{B} G$ is $45^\circ$.