Notes
angle formed by three rectangles solution

Solution to the Angle Formed by Three Rectangles Puzzle

Solution by Triangles

With the points labelled as in the diagram above, the line segments $A D$, $D F$, and $F A$ all have the same length since they are diagonals of congruent rectangles. This means that triangle $A D F$ is an equilateral triangle and so angle $A \hat{F} D$ is $60^\circ$. In triangle $D F I$, angle $D \hat{I} F$ is $90^\circ$ and sides $I D$ and $I F$ have the same length. Therefore, triangle $D F I$ is an isosceles right-angled triangle so angle $I \hat{F} D$ is $45^\circ$. Putting these together, angle $A \hat{F} B = 60^\circ - 45^\circ = 15^\circ$. This is the same as angle $K \hat{F} D$ and so angle $I \hat{F} K$ is $45^\circ - 15^\circ = 30^\circ$. Therefore, angle $E \hat{F} G = 360^\circ - 90^\circ - 90^\circ - 30^\circ = 150^\circ$.