No, not necessarily. It would have to also be a square or a kite in order to have perpendicular diagonals.
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If the parallelogram happens to also be a rhombus (i.e. has all sides equal in length) then yes, otherwise no.
The diagonals of a square are always perpendicular.
Rhombus and Square (since a square is just a "special" rhombus, with right angles)
Yes, to each other.
We would need to have the parallelogram to answer this question.