Yes. "Quadrilateral" means all possible figures that have four sides. Some of them are
parallelograms, some are rhombera, some are rectangles, some are squares, some are
nothing in particular, and some are trapezoids.
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Yes, a parallelogram is always a quadrilateral.
A trapezium is always a quadrilateral.
Yes, because it has 4 sides.
A kite is always a 4 sided quadrilateral.