shouldn't the side of the moon facing the sun be illuminated?

Nope. Go out and observe the progress of the Moon as it moves across the sky. You will find that the terminator is perpendicular to the path it traces.

Jeff's diagram explains phases nicely, but I can't see how it explains why the angle of the terminator is aparently off.

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"The very powerful and the very stupid have one thing in common: They don't alter their views to fit the facts, they alter the facts to fit their views." The Doctor, Doctor Who: The Face of Evil.