Tell me, the properties of a quadrilateral that uniquely determine a rectangle and an isosceles trapezoid.

WIKI says: A trapezoid (in North America) or trapezium (in Britain and elsewhere) is a quadrilateral, which is defined as a shape with four sides, that has one set of parallel sides. In an isosceles trapezoid, the base angles are equal, and so are the pair of non-parallel opposite sides.

WIKI says: In geometry, a rectangle is defined as a quadrilateral where all four of its angles are right angles.

For area, once you do show the above, you can calculate required distances. For example length and breadth are enough for a rectangle.