No because cubes do not have opposite sides that are different lengths. Cubes have all equal sides and all equal coners. A cube is not a rectangle.
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Cubes are not rectangles. The reason for this is because cubes are 3-dimensional and rectangles and 2-dimensional. However, you could say that a cube is a rectangular prism.
Yes, all rectangles are parallelograms. However, not all parallelograms are rectangles.
In fact, some are rectangles, but not all are.
All squares are rectangles also, but not all rectangles are squares, only equilateral rectangles are considered square.
All rectangles are quadrilaterals. A quadrilateral has 4 sides; all rectangles have 4 sides; thus all rectangles are quadrilaterals.