Final answer:
The lay investiture controversy was settled through a ban on simony, the excommunication of Emperor Henry IV, and a compromise in England known as the Settlement.
Step-by-step explanation:
The lay investiture controversy was settled through various means. One of the key resolutions was the ban on simony by Pope Gregory VII in 1075. This practice involved kings appointing church officials in exchange for bribes. The pope excommunicated Holy Roman Emperor Henry IV for continuing simony, leading to a rebellion against him by German princes. This resulted in a loss of power for the kings and vassals, and a significant gain in power for the popes.
Another settlement occurred in England in 1559, known as the Settlement. It consisted of two acts of Parliament, one conferring the title of Supreme Head of the Church upon Queen Elizabeth and the other creating the Anglican prayer book and defining the Church of England. This settlement reflected a compromise between Catholicism and Calvinist theology, and it established Anglicanism as a distinct form of Christianity.
Overall, the lay investiture controversy was resolved through a combination of papal authority, political pressure, and compromises between different religious factions.