Final answer:
During the first quarter phase of the Moon, half of the Moon's visible face is illuminated. This phase is called "first quarter" because the Moon is one-quarter of its way around its orbit.
Step-by-step explanation:
During the first quarter phase of the Moon, exactly half of the Moon's visible face that we see from Earth is illuminated by the Sun. This phase occurs after about one week when the Moon is one-quarter of the way around its orbit, hence the term "first quarter".
As the Moon continues its eastward motion, it rises around noon and sets around midnight. The first quarter phase is known for the dramatic shadows it cast on the Moon's topography due to the Sun's oblique angle of illumination, making it an ideal time for lunar observations.