In solving two-dimensional one-energy transport problems, it is often necessary to utilize Monte Carlo calculations in situations where this technique converges very slowly. In problems with regionwise constant sources where the required result is the flux at a point or an integral of the flux over a region or surface, the reciprocity theorem can be used to determine an auxiliary problem which yields the same information while in many cases improving the statistics appreciably. The relations required in choosing the auxiliary problem are derived. The required integrals and statistical errors are stated in terms of the results for the auxiliary problem. Examples are given to illustrate the application of these ideas to a flux peaking situation and to the absorption in a small region. The extension of this procedure to energy-dependent cases is discussed briefly.