  Let $T_n(q)$ be the ring of lower triangular matrices of order $n \geq 2$
with entries from the finite field $F(q)$ of order $q \geq 2$ and let
${^2T_n(q)}$ denote its free left module. For $n=2,3$ it is shown that the
projective line over $T_n(q)$ gives rise to a set of
$(q+1)^{(n-1)}q^{\frac{3(n-1)(n-2)}{2}}$ affine planes of order $q$. The points
of such an affine plane are non-free cyclic submodules of ${^2T_n(q)}$ not
contained in any non-unimodular free cyclic submodule of ${^2T_n(q)}$ and its
lines are points of the projective line. Furthermore, it is demonstrated that
each affine plane can be extended to the projective plane of order $q$, with
the `line at infinity' being represented by those free cyclic submodules of
${^2T_n(q)}$ that are generated by non-unimodular pairs. Our approach can
straightforwardly be adjusted to address the case of arbitrary $n$.
