third quarter
half moon
When the moon looks like a half moon, it is in its first or third quarter phase. This happens when the moon is at a 90-degree angle to the sun as seen from Earth, with half of the side facing the sun illuminated and the other half in shadow.
It is exactly like when the moon looks like when it is half full.
A week after the Full Moon the Moon looks like a semi circle again, this phase is called ''Third Quarter''
first or last quarter
It actually looks more like a triangle.
first or last quarter
Mercury
The last quarter looks like half a moon, whereas a crescent moon (aka waning crescent or waxing crescent ) looks like a quarter of the moon. So no they are not the same. type in on Google "moon phases" and it'll show pictures of all the moon phases.
About a week after the full moon, the moon is called the last quarter moon and it's "face" appears to be half lit. More than half lit is a gibbous and less than half lit is a crescent.
A crescent moon looks most like
That's the "gibbous" phase.