We analyze angular momentum effects in heavy-ion-induced fission in terms of the statistical model, including collective effects in the nonadiabatic approach and avoiding all the usual numerical approximations and free parameters but one. We find that the strong increase of the fission probability along with the increasing compound nucleus spin is responsible for several observable effects such as the increase of the fission cross section and the decrease of the multiple-chance fission contribution, with the increase of the projectile mass. It is shown that the fission is particularly sensitive to the fusion cross section and its spin distribution. This feature can be used to determine the latter two from the measured fission cross section.