Square: We known that in a square, angle at the vertices are equal to 90° and all sides are equal. So if we know the measure of one diagonal we can construct a square.
Rhombus : In a rhombus all sides are equal & diagonals bisect at 90°, so if we know the measure of both diagonals we can construct a rhombus.
Parallelogram: In a parallelogram the opposite sides and opposite angles are equal. So to construct a parallelogram we need the measurements of the two adjacent sides of the parallelogram and the angle between them.
To construct any quadrilateral atleast 5 independent measurements are required.