Scattering by enhanced fluctuations from the electromagnetic proton cyclotron instability imposes an upper bound on the proton temperature anisotropy of the form T⊥p/T∥p-1=Sp/&bgr;∥p&agr;p. Here &bgr;∥p is the proton &bgr; based on T∥p and the directional subscripts indicate directions perpendicular and parallel to the background magnetic field. This paper describes the analysis of the proton temperature anisotropy in two-dimensional hybrid simulations of quasi-perpendicular shocks and their associated magnetosheaths. Three simulations are considered: at low, intermediate, and high upstream Mach numbers. Considering results from well downstream of all three shocks as one data set, an upper bound on the proton temperature anisotropy fits the above equation with &agr;p=0.5. This compares favorably to &agr;p values from other simulations and from sheath observations. ¿ American Geophysical Union 1996