This work is a generalization of the work of Widom [J. Chem. Phys. 39, 2808 (1963)] and of Henderson [Mol. Phys. 95, 187 (1998)]. Based on geometric analysis and statistical thermodynamics arguments, a set of sum rules for two-component nearest-neighbor interaction models at thermodynamic equilibrium is derived. By choosing the density of one component rho and the unlike-bond density rho (12) as two variables, it is shown that the energy is well-behaved; however, the entropy, (s) over tilde(rho,rho (12)), is independent of rho within two-phase regions, but not outside. Temperature and chemical potentials determine the equilibrium rho and rho (12). The exact entropy function for 1-D systems can be calculated, and an exact free energy density function is formulated. The result shows that s is always dependent on rho except at rho (12)=0, which excludes the possibility of phase transitions at finite temperature.