Notes
five congruent rectangles solution

Solution to the Five Congruent Rectangles Puzzle

Solution by Congruent Shapes

By considering the three rectangles on the right-hand side, the basic rectangle in this diagram has sides with lengths in the ratio $2 : 1$. The point $E$ is therefore the midpoint of the side of the vertical rectangle and so the rectangle $A E D F$ is therefore also congruent to the basic rectangle.

The line segments $A D$, $D B$, and $B C$ are then all diagonals of their respective rectangles and since all these rectangles are congruent, they have the same length. The middle of these three rectangles is oriented vertically while the other two are horizontal, so the line segment $D B$ is perpendicular to the other two. So angle $A \hat{D} B$ is a right-angle. Triangle $D B C$ is then an isosceles right-angled triangle so angle $B \hat{D} C$ is $45^\circ$. Putting these together, angle $A \hat{D} C$ is $135^\circ$.