Notes
three equal angles in a rectangle solution

Three Equal Angles in a Rectangle

Solution by Angles in a Rectangle and Angles in a Triangle

With the points labelled as in the diagram, point $F$ is such that angle $B \hat{F} D$ is a right-angle.

As the outer shape is a rectangle, angle $E \hat{D} C$ is $90^\circ$. Since the three marked angles are equal, they must therefore each by $30^\circ$. Angle $C \hat{A} D$ is also $30^\circ$, either by alternate angles from angle $E \hat{D} A$ or angles in a triangle. The triangles $A F B$, $D F B$, and $D C B$ are therefore all similar, but then by considering the edges in common they are in fact congruent. The shaded area is therefore $\frac{2}{3}$ of triangle $D C A$ and so $\frac{1}{3}$ of the full rectangle.