They are both two dimensional. They only have right angles at line intersections. Opposing sides are parallel. Opposing sides are the same size.
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four sides, four right angles, 2 pairs of parallel lines
A square may be classified as a rectangle, a parallelogram, a rhombus, a polygon, and a quadrilateral.
They are both three dementinal
A rectangle is technically a parallelogram and it belongs to the class of polygons known as quadrilaterals
Shapes can be classified in many ways. These ways can be general or specific. Of the terms you mentioned, the most general is the quadrilateral. Parallelograms, squares, rectangles and rhombuses are all quadrilaterals (they have 4 sides) but quadrilaterals are not necessarily any of the others. The next classification is the parallelogram, which adds the qualification that opposite sides be parallel. Squares, rectangles and rhombuses are parallelograms, but a parallelogram does not have to be a square, rectangle or rhombus.