# Three Equal Angles in a Rectangle +-- {.image} [[ThreeEqualAnglesinaRectangle.png:pic]] > The three angles are equal. What fraction of the rectangle is shaded? =-- ## Solution by [[rectangle|Angles in a Rectangle]] and [[Angles in a Triangle]] +-- {.image} [[ThreeEqualAnglesinaRectangleLabelled.png:pic]] =-- 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.