Final answer:
John Wayne Gacy's murders were primarily under the jurisdiction of local authorities in Des Plaines, Illinois, with cooperation from other jurisdictions and involvement of state and federal agencies for the trial in Cook County court system.
Step-by-step explanation:
The jurisdiction in the John Wayne Gacy murders primarily fell to the local authorities in Des Plaines, Illinois, where Gacy resided and where many of his crimes were committed. However, because of the nature and extent of the crimes, involving multiple victims across different places, the investigations did involve cooperation with other local jurisdictions, as well as state and possibly federal agencies. Gacy's trial was held in the Cook County court system since Cook County was where the majority of his crimes were uncovered. The extensive investigation and legal proceedings required coordinating various law enforcement and justice departments to ensure a comprehensive trial.