  We prove an extension of a ham sandwich theorem for families of lines in the
plane by Dujmovi\'{c} and Langerman. Given two sets $A, B$ of $n$ lines each in
the plane, we prove that it is possible to partition the plane into $r$ convex
regions such that the following holds. For each region $C$ of the partition
there is a subset of $c_r n^{1/r}$ lines of $A$ whose pairwise intersections
are in $C$, and the same holds for $B$. In this statement $c_r$ only depends on
$r$. We also prove that the dependence on $n$ is optimal.
