As long as the two quadrilaterals are congruent, yes.
(Congruency ignores position, including rotation and reflection.)
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no they are not rigid.
The answer depends on the quadrilateral. Some have rotational symmetry or reflective symmetry and it is not possible to distinguish between these and translations.
Rigid is immovable, unbending. Semi-rigid can move in a limited way.
Yes, it is non-rigid.
a non rigid is a square and a hexagon