If a quadrilateral is a parallelogram and a rhombus, does that imply it is a square?

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I have proven that a quadrilateral is a parallelogram, and that it is also a rhombus. Does this imply that it is ultimately a square?

I believe it does because all sides are equal, and opposite sides are parallel, and I believe the only way this is possible is if the angles are all 90, but I'm not sure if that is true. +1 if you can link to a proof that says that (or prove it yourself).

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2 Answers

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Every rhombus is a parallelogram and a rhombus with right angles is a square. Hence a quadrilateral which is a rhombus is a parallelogram, but not necessarily a square.

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A rhombus is always a parallelogram. In order to guarantee that you have a square, you'd have to show that all four angles are congruent.

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