Triangle and parallel lines, find angle x

Question

I need to find the minimum angle value of $x$, where $\cal L_1$ and $\cal L_2$ are parallel lines and the angle $\angle ABC$ is acute. I can´t figure out how to relate the angles. So far I only know the adiacent ones as $180°-2\theta-3\beta$ and $180°-3\theta-2\beta$.

Answer 1

Notice first of all that $x=\pi-3(\beta+\theta)$. By angle chasing it is easy to find that $\angle ABC=5(\beta+\theta)-\pi$. But you know that $\angle ABC<\pi/2$, whence $\beta+\theta<3\pi/10$ and $x>\pi/10$.