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Here is a good diagram that pictorially describes what I have been talking about with the "angle between the sun and the moon at the phase of quarter moon."

This is how it looks during the First Quarter moon, which will be coming up in about 2 weeks.
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We just had the Last Quarter moon last week, on Thursday. I measured the angle between the sun and the moon and found it to be greater than 89 degrees and less than 90 degrees.
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There are consequences to these measurements, since as you can see in the diagram, the three corners of the geometrical figure described by the earth, moon and sun is a right triangle.
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Right triangles are very easy to use in determining proportions between the lengths of the sides when one of the not-square interior angles is known. We already know that the angle at the moon is 90 degrees because it is a quarter moon, which see.
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We have measured one of the other two angles, the angle from earth between the sun and moon, and have found it to be consistently close to 90 degrees at the quarter moon phase.
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Therefore, without having to travel to the moon or the sun to check the angles of the other two bodies, we can surmise what the angle between the earth and moon as viewed from the sun would be.
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Since any triangle has 3 interior angles and their sum is always 180 degrees, we subtract the 90 degrees at the moon and the 89-plus degrees at the earth and come up with an angle of less than one degree at the sun.
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ETC............