Final answer:
During the first quarter phase, the dark half of the Moon sets first as the illuminated side faces eastward and the Moon sets roughly twelve hours after it rises.
Step-by-step explanation:
When a first quarter Moon sets in Chicago, the dark half sets first. This is because during the first quarter phase, the side of the Moon that is illuminated faces eastward, which is also the side that rises first. Since the Moon sets roughly twelve hours after it rises, the dark half would be leading as the Moon moves towards the western horizon during its setting phase.