We have the measure of angle ABC as 6 measures $= (4x - 31)^\circ$ and the measure of angle CDA as 8 measures $= 111^\circ$. Since lines AB and CD are parallel, angle ABC is congruent to angle CDA, so we have the equation $(4x-31)^\circ = 111^\circ$. Solving for $x$, we have $4x-31 = 111$. Adding 31 to both sides gives $4x=142$. Dividing both sides by 4 gives $x=\boxed{35.5}$.
