  We show that if $\mathcal{L}$ is a line in the plane containing a badly
approximable vector, then almost every point in $\mathcal{L}$ does not admit an
improvement in Dirichlet's theorem. Our proof relies on a measure
classification result for certain measures invariant under a non-abelian two
dimensional group on the homogeneous space
$\mathrm{SL}_3(\mathbb{R})/\mathrm{SL}_3(\mathbb{Z})$. Using the measure
classification theorem, we reprove a result of Shah about planar nondegenerate
curves (which are not necessarily analytic), and prove analogous results for
the framework of Diophantine approximation with weights. We also show that
there are line segments in $\mathbb{R}^3$, which do contain badly approximable
points, and for which all points do admit an improvement in Dirichlet's
theorem.
