# Solution to the Five Congruent Rectangles Puzzle +-- {.image} [[FiveCongruentRectangles.png:pic]] > Five congruent rectangles. What’s the angle? =-- ## Solution by [[congruent|Congruent Shapes]] +-- {.image} [[FiveCongruentRectanglesLabelled.png:pic]] =-- 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 $135^\circ$.