#QuarterMoon
When you look at a quarter moon (AKA half moon), one that looks like this: D , you are looking at the sunrise or sunset line, AKA terminator. Yes, you can see the terminator if you look at a crescent moon, or a gibbous moon, but it’s only when you are directly over the sunrise/sunset line that you see that kind of a half moon. When you look at the moon at your local sunset you are standing on the sunset line of the Earth, also called the terminator. If you think about it, the only way this can happen is if the moon is directly behind the Earth in its orbit around the Sun. Go do a drawing. You have the moon and the Earth in line, like this: o O –> with the arrow indicating the movement of the Earth in its orbit. To complete the picture, the Sun would be off the top of the page, and the bottom half of those circles would be dark.
Now, the Earth travels about 29.6km/sec in its orbit around the sun. And the moon, on average, is 382,500 from the Earth. (Bear with me here, the math is almost over). So, the Earth covers the Earth-Moon distance in about 3.5hours.
That means that when you look at a First Quarter Moon, at sunset, you are looking at the very spot in space that the Earth was at, 3.5 hours ago. In the morning, should you be up early enough to see a Last Quarter Moon at local sunrise, you will be looking at the very spot in space where the Earth will be about the time you sit down for your pre-luncheon snack.