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Both the 1st Quarter moon and the 3rd (last) Quarter moon can be referred to as the "half moon" since we see half of the sunlit side of the moon.
Full moon,New moon,First quarter,Third quarter or Last quarter.
a 2nd quarter moon is a phase of the moon and the moon goes through 4 phases.
New moon, first quarter, full moon and third quarter.
There is the new moon, first quarter, full moon, and third quarter.