No, a trapezoid is not a parallelogram. Irregular polygons are generally not parallel.
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It is true that every trapezoid is a quadrilateral
Yes every trapezoid is a quadrilateral because they have 4 sides
Every parallelogram must, by definition, be a quadrilateral.
Yes. Every rhombus has four sides (of equal length). This makes it a quadrilateral.
Yes any polygon that has 4 sides is a quadrilateral.