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

Let $P$ be a Jordan polygon (i.e. the only points of the plane belonging to two edges are the polygon vertices) that can be tiled with parallelograms.

1. Does it follow that the sum of the areas of all the rectangles among those parallelograms 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?