Infrared spectroscopy of the H alpha emission lines of a subsample of 19 high-redshift (0.8 < z < 2.3) Molonglo quasars, selected at 408 MHz, is presented. These emission lines are fitted with composite models of broad and narrow emission, which include combinations of classical broad-line regions (BLRs) of fast-moving gas clouds lying outside the quasar nucleus, and/or a theoretical model of emission from an optically thick, flattened, rotating accretion disc, with velocity shifts allowed between the components. All bar one of the 19 sources are found to have emission consistent with the presence of an optically emitting accretion disc, with the exception appearing to display complex emission including at least three broad components. 10 of the quasars have strong Bayesian evidence for broad-line emission arising from an accretion disc together with a standard BLR, selected in preference to a model with two simple broad lines. Thus, the best explanation for the complexity required to fit the broad H alpha lines in this sample is optical emission from an accretion disc in addition to a region of fast-moving clouds. We derive estimates of the angle between the rotation axis of the accretion disc and the line of sight. Deprojecting radio sources on the assumption of jets emerging perpendicular to the accretion disc gives rough agreement with expectations of radio source models. The distribution in disc angles is broadly consistent with models in which a Doppler boosted core contributes to the chances of observing a source at low inclination to the line of sight, and in which the radio jets expand at constant speed up to a size of similar to 1 Mpc. A weak correlation is found between the accretion disc angle and the logarithm of the low-frequency radio luminosity. This is direct, albeit tenuous, evidence for the receding torus model in which the opening angle of the torus widens with increasing radio luminosity. The highest accretion disc angle measured is 48 degrees, consistent with the opening angle predicted for radio-luminous sources. Velocity shifts of the broad H alpha components are analysed and the results found to be consistent with a two-component model comprising one single-peaked broad line emitted at the same redshift as the narrow lines, and emission from an accretion disc which appears to be preferentially redshifted with respect to the narrow lines for high-redshift sources and blueshifted relative to the narrow lines for low-redshift sources. An additional analysis is performed in which the disc emission is fixed at the redshift of the narrow-line region; although only two quasars show a robust change in fitted angle, the radio luminosity - disc angle correlation falls sharply in probability, and so is strongly model dependent in this sample.
