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A parallelogram is a 4 sided quadrilateral.
The key is the object is quadrilateral with parallel sides. It is a parallelogram. It is also a rhombus.
A rectangle is a type of quadrilateral, and it also is a parallelogram.
It may be. A quadrilateral has four sides, but they need not be all the same length, nor parallel to each other. It's true that a parallelogram *is* a quadrilateral.
A rectangle is a quadrilateral with two pairs of parallel sides. That defines a parallelogram.