Sun, Moon: Relative Distance from Crescent?
This is exactly how the ancient Greeks attempted to estimate the relative distances to the sun and moon.

The difficulty is the angles involved are rather small

Aristarchus measured the Moon-Earth-Sun angle at half moon to be 87 degrees. (or 3 degrees less than a right angle) Which implies the Sun is about 19 times further than the moon. And from this (and an estimate of the sizes of the Earth and moon made at a lunar eclipe) he estimated the distance of the sun.

The actual angle is more like 89.85 degrees (which is very hard to distinguish from 90 degrees) (0.15 degrees less than a right angle). 

So this method can be done, but it requires very exact measurement to succeed.