Olympiad Geometry · Challenge problem

Two parallel lines are $6$ units apart. A transversal crosses them at points $P$ and $Q$, making a $30^\circ$ angle with each of the lines. The four interior angles (two at $P$, two at $Q$) are bisected, and the four bisectors form a quadrilateral with $P$ and $Q$ as opposite vertices. Find the area of this quadrilateral.