Why do you see half of the moon when you can see both the sun and the moon from earth at the same time on some days?
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You see, if you're standing at the south part of the earth in this picture you can only see half the moon, also called a crescent moon, because the part that you can not see is facing the oppsite direction of the sun, therefore not receiving any sunlight.
You don't always see exactly half, sometimes you only see a thin strip of the moon, google for "crescent moon" and you'll see what I mean.