### Using a compass draw an angle of measure $135^\circ.$

Step by Step Explanation:
1. Draw a line $AB.$
2. With $O$ as the centre and any convenient radius, draw an arc, cutting $AB$ at $P$ and $Q.$
3. With $Q$ as the centre and the same radius, draw an arc, cutting the arc drawn in step $2$ at $R.$
4. With $R$ as the centre and the same radius, draw an arc, cutting the arc drawn in step $2$ at $S.$
5. With $R$ as the centre and the same radius, draw an arc.
6. With $S$ as the centre and the same radius, draw an arc, cutting the arc drawn in the previous step at $T.$
7. Draw $OT$ and produce it to $U.$
8. With $P$ as centre and radius more than half of $PX,$ draw an arc in the interior of $\angle AOU.$
9. With $X$ as centre and the same radius, as in the above step, draw another arc intersecting the arc drawn in the previous step at $Y.$
10. Join $OY$ and produce it to $V. \angle BOV$ is the required angle of measure $135^\circ.$