[ I figure this to be the last important bit of Trapezoid Week. ]
I have one last bit of proving the area formula for the trapezoid, and figure this should wrap up all the exposition I mean to do here. I’m starting again with a trapezoid, with the two parallel bases AB and DE drawn so they’re horizontal. I’m making no assumptions about the legs AD and BE; they might be parallel, they might not be. This makes a proof that’s improved from the difference-of-triangles proof, since it works for more shapes. It’ll work naturally for parallelograms and rectangles, whether you want them to be trapezoids or not. It’s also improved from the twin-trapezoid proof, because it requires so much less extra work.