Short answer: yes
Long answer: quadrilateral is a geometric figure with 4 sides. Both a square and a parallelogram have 4 sides. Therefore, both are a quadrilateral.
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The second statement is false.
Parallelogram quadrilateral trapezoid rectangle square
square, quadrilateral, rectangle, parallelogram
a quadrilateral or a parallelogram.
they can be but dont have to be a square is a parallelogram but trapezoid is not