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 Is the Prime Meridian a full circle or a half circle?

Since the Prime Meridian is a longitude, it is a half circle.

What is the counterpart of the Prime Meridian?

The counterpart of the Prime Meridian is called the International Date Line. Prime Meridian: 0o, International Date Line: 180o 

Is the duration of the day equal to the duration of the night usually?

Usually they are not equal. Only twice a year, they become equal.

What is an equinox?

When the duration of day is equal to the duration of night, it is called an equinox.

What is a solstice?

When the difference between the duration of day and the duration of night is the highest, it is called a solstice.