Construct the following angles with your compass and pencil only: 45°

Draw a ray $\vec{OA}$. Place the compass at $O$ and draw a large arc intersecting $\vec{OA}$ at $P$. With the same radius, draw arcs from $P$ and then from the intersection point of the first arc to create points for a $90^\circ$ angle. Specifically, from $P$, draw an arc to intersect the large arc at $Q$. From $Q$, draw another arc to intersect the large arc at $R$. Then, from $Q$ and $R$, draw arcs that intersect above the large arc at $S$. Draw a ray $\vec{OS}$ from $O$ through $S$, such that $\angle SOA = 90^\circ$. Let $T$ be the point where the initial large arc intersects $\vec{OS}$.

Place the compass at $P$ (the intersection of the first arc and $\vec{OA}$) and draw an arc inside $\angle SOA$. Then, place the compass at $T$ (the intersection of the first arc and $\vec{OS}$) and draw another arc that intersects the arc drawn from $P$. Let this intersection point be $U$.

Draw a ray from $O$ through $U$. This ray, $\vec{OU}$, bisects $\angle SOA$.
The angle $\angle UOA$ is the constructed $45^\circ$ angle.