[en] A consistent many-body theory based on the jellium model is applied for the description of angular resolved photoelectron spectra of metal clusters anions. The results of calculations demonstrate the dominant role of the many-body effects in the formation of angular distributions of photoelectrons emitted from sodium clusters and are in a good agreement with recent experimental data. The concrete comparison of theory and experiment has been performed for the photoionization of Na₇^- and Na₁₉^- anions being characterized by the entirely closed shells of delocalized electrons.