We report calculations of momentum densities of electron-positron pairs annihilating at various vacancy clusters in Si. The densities integrated along one direction, i.e., those corresponding to the spectra measured by the two-dimensional angular correlation of the annihilation radiation method, are shown to be isotropic if the positron is captured by a small vacancy cluster. The densities integrated along two directions correspond to the one-dimensional angular-correlation distribution or the Doppler-broadened annihilation radiation line shape. The characteristic narrowing of these one-dimensional distributions at vacancy clusters is demonstrated. The line-shape parameters S and W describing mainly the annihilations with valence and core electrons, respectively, are calculated and compared quantitatively with experimental data. We find a systematic increase in the S and a decrease in the W parameter as the size of the vacancy increases.