A self-consistent evolution of quasi-perpendicular collisionless laminar magnetosonic shock waves with subcritical Mach number is investigated by using a 2-1/2 dimension, relativistic, electromagnetic particle code with full ion and electron dynamics. It is shown that significant ion acceleration occurs within the shock ramp by resonant interactions. The ions are heated primarily in the direction perpendicular to a magnetic field. Our results are in good agreement with the recent observations of interplanetary shocks by ISEE spacecraft.