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Yes. All rhombuses are parallelograms, but the reverse is not true. (I.e. there are parallelograms which are rhombuses and there a parallelograms that are not rhombuses. -- Just like Every bus has wheels but not everything with wheels is a bus.)
No. All rhombi (rhombuses) are parallelograms but all parallelograms are not rhombi.
squares, rectangles, rhombuses
parallelograms, rhombuses, & hexagons
yes. rectangles are just the special cases of parallelograms where the angles are all 900