In a circle, the diameter $\overline{EB}$ is parallel to the chord $\overline{DC}$, and the chord $\overline{AB}$ is parallel to the chord $\overline{ED}$. The angles $AEB$ and $ABE$ are in the ratio $2:7$. What is the degree measure of angle $BCD$?

1. Since $EB$ is a diameter, $\angle BAE=90^\circ$, so $\angle AEB+\angle ABE=90^\circ$. With the ratio $2:7$, $\angle AEB=20^\circ$ and $\angle ABE=70^\circ$.
2. Since $AB\parallel ED$, $\angle BED=\angle ABE=70^\circ$ (alternate interior angles).
3. Since $EB\parallel DC$ and $B,E,D,C$ lie on a circle, $BEDC$ is an isosceles trapezoid, so $\angle CDE=\angle BCD$ and $\angle BCD=180^\circ-\angle BED$ (consecutive angles between parallel sides).
4. $\angle BCD=180^\circ-70^\circ=\boxed{110^\circ}$.