Construct the angles 45° and 90° solution

Draw a straight line segment, let's call it $\vec{OA}$, with endpoint $O$. This will be one arm of your angle.

• Place the compass at point $O$ and draw an arc that intersects $\vec{OA}$ at a point, say $P$.
• With the same compass width, place the compass at $P$ and draw an arc that intersects the first arc at a point, say $Q$.
• With the same compass width, place the compass at $Q$ and draw another arc that intersects the first arc at a point, say $R$.
• Now, place the compass at $Q$ and draw an arc above $O$.
• With the same compass width, place the compass at $R$ and draw another arc that intersects the arc drawn from $Q$. Let this intersection point be $S$.
• Draw a ray from $O$ through $S$. This ray, $\vec{OS}$, forms a $90^\circ$ angle with $\vec{OA}$ (i.e., $\angle SOA = 90^\circ$).

• Place the compass at $P$ (the intersection of the first arc and $\vec{OA}$) and draw an arc inside the $90^\circ$ angle.
• Place the compass at the point where the first arc intersects $\vec{OS}$ (let's call this point $T$) 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.