Equilateral parallelograms.
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No.
That is true for some parallelograms but not all. For example, the diagonals of a rhombus, kite or square are perpendicular, but those of a rectangle or general parallelogram are not.
No. No. No. No.
A rhombus has 4 equal sides and the diagonals are always perpendicular
No but the diagonals of a square, rhombus and a kite are perpendicular to each other