  Given a trapezoid dissected into triangles, the area of any triangle
determined by either diagonal of the trapezoid is integral over the ring
generated by the areas of the triangles in the dissection. Given a
parallelogram dissected into triangles, the area of any one of the triangles of
the dissection is integral over the ring generated by the areas of the other
triangles. In both cases, the integrality relations are invariant under
deformation of the dissection.
  The trapezoid theorem implies and provides a new context for Monsky's
Equidissection Theorem that a square cannot be dissected into an odd number of
triangles of equal area.
  A corollary of these results is that the area polynomials for parallelograms
introduced in previous work have all leading coefficients equal to $\pm 1$.
