An association biased Monte Carlo (ABMC) method of simulating associating systems with two bonding sites on each particle is described. The method includes a strategy for frequently forming two simultaneous bonds by a single particle during a Monte Carlo move. This strategy is employed to obtain adequate configuration statistics at each state point and is particularly important when ring formation is possible. A variety of thermodynamic and physicochemical parameters of the associating system were monitored including the compressibility factor, internal potential energy, isothermal compressibility, chain and ring number, and shape were monitored. Our analysis indicated that there is a strong dependence of these monitored quantities upon the angle between vectors representing the bonding sites on each particle. Also presented are results which suggest the existence of a two phase region, which we believe is a gas-liquid coexistence, which is dependent upon density, bonding energy, and the relative angle between the bonding sites.