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because only half of the visible moon is lite. we only see half the moon so only a quarter of the moon is lite.

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Another take:

After the New Moon . . .

-- The 'first quarter' is the phase that's seen after 1/4 of the time

until the next New Moon has passed.

-- The 'third quarter' is the phase that's seen after 3/4 of the time

until the next New Moon has passed.

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12y ago

What else can I help you with?