Absolutely; a parallelogram is a form of quadrilateral, in that it has four sides. If the four sides come in two sets of parallel lines, you have a parallelogram.
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A shape that is a parallelogram but not a quadrilateral doesn't exist.
No. A parallelogram is a special kind of quadrilateral.
It's not a matter of 'finding' it. Either the quadrilateral is a parallelogram,or else it is not one.If the opposite sides of the quadrilateral are parallel, then it's a parallelogram.
A quadrilateral is a parallelogram if it has two pairs of (opposite) sides that are parallel.
The second statement is false.