No. A trapezoid must have one pair of parallel sides, and a parallelogram must have two pair.
But a quadrilateral in general doesn't necessarily have any parallel sides.
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No trapezoids are parallelograms, and no parallelograms are trapezoids.
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Yes because all parallelograms have four sides. But the opposite is not true. There are some quadrilaterals that aren't parallelograms.
Some are, and some are not. A parallelogram is a special kind of quadrilateral ... one in which every side is parallel to the one across from it. So all parallelograms are quadrilaterals, but there are also a lot of other quadrilaterals that are not parallelograms.
Parallelograms!