The inelastic collisions of He(1s2p ¹P₁) atoms with ground-state noble-gas atoms are studied using the kinetic plasma spectroscopy method. Quenching rate constants are found to be (25+or-5)*10 ^-10, (27+or-5)*10^-10 and (32+or-5)*10 ^-10 cm³s^-1 for Ar, Kr and Xe, respectively (T=600 K). A detailed theoretical interpretation taking the specific features of quasimolecular terms into account is developed. Good agreement between theory and experiment is obtained for He-Ar and He-Kr. Possible reasons for the discrepancy for the He-Xe system are discussed.