I saw this problem some years ago, don't remember the source:

> A Jordan polygon (i.e. the only points of the plane belonging to two edges are the polygon vertices) can be tiled with parallelograms. 
>
>1) Does it follow that the sum of the areas of all rectangles is the same independent of the tiling?
>2) Does it follow that the sum of the areas of all parallelograms with angles $\alpha$ and $\beta$ is the same independent of the tiling?

I couldn't find a solution, but I believe the answer to both questions is positive.   
Does anyone know how this problem can be solved?