We study the feeding of massive galaxies at high redshift through streams from the cosmic web using the Mare Nostrum hydrocosmological simulation. Our statistical sample consists of 350 dark matter haloes of ≃1012 M⊙ at z= 2.5. We find that ∼70 per cent of the influx into the virial radius Rv is in narrow streams covering 10 per cent of the virial shell. On average 64 per cent of the stream influx is in one stream, and 95 per cent is in three dominant streams. The streams that feed a massive halo tend to lie in a plane that extends from half to a few Rv, hereafter ‘the stream plane’ (SP). The streams are typically embedded in a thin sheet of low-entropy gas, a Zel’dovich pancake, which carries ∼20 per cent of the influx into Rv. The filaments-in-a-plane configuration about the massive haloes at the nodes of the cosmic web differs from the large-scale structure of the web where the filaments mark the intersections of slanted sheets. The SP is only weakly aligned with the angular momentum (AM) near Rv, consistent with the fact that typically 80 per cent of the AM is carried by one dominant stream. The galactic disc plane shows a weak tendency to be perpendicular to the large-scale SP, consistent with tidal-torque theory. Most interesting, the direction of the disc AM is only weakly correlated with the AM direction at Rv. This indicates a significant AM exchange at the interphase between streams and disc in the greater environment of the disc inside an ‘AM sphere’ of radius ∼0.3Rv. The required large torques are expected based on the perturbed morphology and kinematics within this interaction sphere. This AM exchange may or may not require a major modification of the standard disc modelling based on AM conservation, depending on the extent to which the amplitude of the disc AM is affected, which is yet to be studied.