Given a point $P,$ not on the line $l,$ it's possible to use a compass and straightedge to create a line perpendicular to $l$ that passes through $P.$ That is, the unique line described by the Perpendicular Postulate.

First, place the sharp end of the compass at point $P,$ and draw an arc that intersects $l$ at two distinct points. These points will be named $A$ and $B,$ for later referencing.

Draw two arcs that intersect on the side of $l$ opposite to $P$ by placing the sharp end in $A$ and $B$ respectively, using the same distance on the compass for each arc.

Using a straightedge to draw a line through $P$ and the intersection of the arcs gives the desired perpendicular line.

The image can now be cleaned up to get the final result.