Notes
six rectangles solution

Solution to the Six Rectangles Puzzle

Solution by Properties of Isosceles Right-angled Triangles and Corresponding Angles in Parallel Lines

Consider the diagram labelled as above. By considering rectangle $D E F G$, the sides of the rectangle must be in the ratio $1 : 2$. Therefore, line segments $E H$ and $B D$ have the same length, as to $A B$ and $D E$. Triangle $A B D$ and $D E H$ are therefore related by a rotation of $90^\circ$, so angle $H \hat{D} A$ is $90^\circ$.

This means that triangle $A D H$ is an isosceles right-angled triangle. In particular, angle $A \hat{H} D$ is $45^\circ$. Since line segments $I C$ and $H D$ are parallel, this means that angle $A \hat{J} C$ is also $45^\circ$ since corresponding angles are equal.

Therefore, angle $I \hat{J} A$ is $135^\circ$.