Dissections of trapezoids into trapezoids homothetical to trapezoids in a given family

We prove several theorems related to the dissection of trapezoids into trapezoids that are homothetical to given ones.
We prove that, using homotheties of a trapezoid with rational ratio of bases, we can tile any trapezoid with rational ratio of bases and the same angles, and no other trapezoid can be tiled.
We also consider trapezoids whose ratio of bases is a quadratic irrationality. For certain pairs of trapezoids we prove that their homotheties can tile any trapezoid with the same angles and the ratio of bases belonging to the same quadratic field. For some other class of trapezoids with a quadratic-irrational ratio of bases, we present a necessary condition on trapezoids that can be tiled with given ones. This condition is remarkable because it contains a transcendental function. This is the first occurrence of a transcendental function in problems of tiling polygons with similar polygons.
Bibliography: 8 titles.