NVT Monte Carlo simulations are reported for chemically associating two-dimensional fluids which can polymerize for certain interaction energies, due to the presence of two attractive sites per monomeric hard disc. The sites are fixed inside a hard core at a given valence angle and mutual penetration of discs is permitted. The type of products of polymerization depends on the parameters of the model; we observe the formation of small associates, as well as of extended chains, bent chains, and rings with different number of monomers. The values of valence angles and of association energy are of primary importance. The dependence of the structural properties of the model on fluid density and association energy is investigated. We performed detailed analysis of the clusters formed due to association in terms of fractions of singly and doubly bonded particles, of average numbers of chains and rings, and of their size. We also obtain the average end to end distance, the radius of gyration, and the persistent length of the products of polymerization. The pressure is calculated from the density profiles of particles of the polymerizing fluid near a hard "wall." The data can be used to develop the equation of state for chemically associating two-dimensional fluids.