A square is a quadrilateral, but not all quadrilaterals are squares. A quadrilateral is just a four-sided figure.
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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.
A square has four sides so it is a quadrilateral. All sides of a square are of equal length so it is equilateral. All angles of a square are of equal measure so it is equiangular. A quadrilateral that is equilateral and equiangular is a regular quadrilateral.