How do you construct a 75 degree angle using a compass and straightedge?

Draw a ray $\overrightarrow{PX}$ with endpoint $P$.

With $P$ as the center and any convenient radius, draw an arc that intersects $\overrightarrow{PX}$ at point $A$.
With $A$ as the center and the same radius, draw another arc that intersects the first arc at point $B$.
The angle $\angle BPX$ is $60^\circ$.

With $B$ as the center and the same radius, draw an arc that intersects the first arc at point $C$. (This marks $120^\circ$ from $PX$).
With $B$ and $C$ as centers, and a radius greater than half the distance between $B$ and $C$, draw two arcs that intersect each other at point $D$.
Draw a ray $\overrightarrow{PD}$. The angle $\angle DPX$ is $90^\circ$. Let $Q$ be the point where $\overrightarrow{PD}$ intersects the initial arc.

The angle between the $60^\circ$ line ($\overrightarrow{PB}$) and the $90^\circ$ line ($\overrightarrow{PQ}$) is $90^\circ - 60^\circ = 30^\circ$. Bisecting this angle will add $15^\circ$ to the $60^\circ$ angle.
With $B$ as the center and any convenient radius, draw an arc.
With $Q$ (the intersection of the $90^\circ$ ray and the initial arc) as the center and the same radius, draw another arc that intersects the previous arc at point $R$.

Draw a ray $\overrightarrow{PR}$. The angle $\angle RPX$ is $\boxed{\text{75}^\circ}$.