Two-dimensional Monte-Carlo simulations of lattice polymers were carried out employing Lennard-Jones (LJ) type intramolecular interactions. The shape-dependent properties of these polymers were studied over a large range of interaction parameters and temperatures. The variation of the critical constant was also analysed. Two-dimensional surfaces of sections were used to determine the portions of the parameter space which corresponded to theta-conditions. The behaviour of LJ polymers far from theta-conditions was determined. The investigations were facilitated by a simple scaling argument which resulted in considerable time savings in computations.