I'm not quite sure why you (ckfan) WANT to include the general trapezoid in your classification!
I don't know of any interesting theorems on the trapezoid that don't work for more general quadrilaterals, and it's always rather puzzled me why anyone ever thought it worth while to give this particular kind of quadrilateral its own special name.
I now think I know the reason, and am not too impressed by it. Proclus (who's the guy responsible) says that the area is the mean of the lengths of the two parallel sides times the distance between them, and I think this was the point - the trapezoid is the most general quadrilateral with a simple area formula. So it has some practical point - you can compute the area of a polygon by dividing it up into trapezoids.
But there's not much theoretical point - this area formula is trivially deducible from the one for a triangle by just drawing a diagonal. Does anyone know of any more interesting theorem about trapezoids?