Usain is walking for exercise by zigzagging across a $120$-meter by $40$-meter rectangular field, beginning at point $A$ and ending on the segment $\overline{BC}$. He wants to increase the distance walked by zigzagging as shown in the figure below $(APQRS)$. What angle $\theta$ $\angle PAB=\angle QPC=\angle RQB=\cdots$ will produce a length that is $150$ meters? (This figure is not drawn to scale. Do not assume that the zigzag path has exactly four segments as shown. There could be more or fewer.)