The answer depends on the shape of the quadrilateral and the form in which that information is given: for example, lengths of sides and angles, coordinates of vertices.
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A square is a quadrilateral, but not all quadrilaterals are squares. A quadrilateral is just a four-sided figure.
Yes. A square is a rectangle and a rectangle is a quadrilateral.
A rhombus is a quadrilateral.
All squares are quadrilaterals. All quadrilaterals are not squares. There is no square that is not a quadrilateral.
The second statement is false.