A crossed ion-laser beam apparatus has been used to measure accurate relative total and partial cross sections for photodetachment from Rb-ions with high photon energy resolution (0.1-0.6 cm-1) in the region of the Rb(5p 2P~/2,3/2) thresholds (photon energy range 16,350-16,820cm-1). Satisfactory fits to these data by multichannel quantum defect theory (MQDT) have been obtained, resulting in a reliable set of seven MQDT parameters. The electron angular distribution parameter for the Rb(5s) channel was found to be/~(5s)=2, independent of photon energy. MQDT predicts a sharp, windowtype variation of /~(5s) around the minimum of the 5s-cross section below the Rb(5p 2P1/2) threshold (where an accurate experimental measurement of/~(5s) was not possible) and a similar behaviour of/?(6s) in the case of Cs-photodetachment below the Cs(6p 2P~/2) threshold.