Let Tn(q) be the ring of lower triangular matrices of order n≥2 with entries from the finite field F(q) of order q≥2 and let Tn2(q) denote its free left module. For n=2,3 it is shown that the projective line over Tn(q) gives rise to a set of (q+1)(n−1)q3(n−1)(n−2)2 affine planes of order q. The points of such an affine plane are non-free cyclic submodules of Tn2(q) not contained in any non-unimodular free cyclic submodule of Tn2(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 Tn2(q) that are generated by non-unimodular pairs. Our approach can be adjusted to address the case of arbitrary n.