No, and all parrallelograms are not quadralaterals.
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No but they are all 4 sided quadrilaterals
Not necessarily. A quadrilateral is a plane shape with four straight sides. There is no requirement for a side to be parallel to another.
A circle and an ellipse. Neither of them are polygons and so cannot be quadrilaterals, still less parallelograms.
All quadrilaterals are polygons, but not all polygons are quadrilaterals.
Not necessarily. All parallelograms are quadrilaterals but not all quadrilaterals are parallelograms. Parallelograms are more specific instances of quadrilaterals. Its like saying all squares are rectangles but not all rectangles are squares.