Let the angle be $ \angle AOB$. We assume that $ \angle AOB<180^{\circ}$, otherwise the word "inside" is meaningless. Let $ C$ be on the extension of $ BO$. Now we construct the angle bisector of $ \angle BOC$. Put the center of the compass on $ O$, construct the circle intersecting $ OB$ and $ OC$ at $ X$ and $ Y$ respectively. Construct an arbitrary line through $ X$. Take a point $ M$ on the line, and take a point $ N$ on the extension of $ XM$ such that $ XM=MN$. Join $ NY$, construct a line through $ M$ parallel to $ NY$. This line passing through the midpoint of $ XY$, call this point $ Z$. Therefore $ OZ$ is the angle bisector of $ \angle BOC$. Now it remains to construct a line passing through $ O$ perpendicular to $ OZ$, which is trivial. To avoid marking points inside the angle, we construct the line 'outwardly'.