The diagonals of a square are always perpendicular.
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No, not necessarily. It would have to also be a square or a kite in order to have perpendicular diagonals.
Yes, to each other.
If the parallelogram happens to also be a rhombus (i.e. has all sides equal in length) then yes, otherwise no.
Rhombus and Square (since a square is just a "special" rhombus, with right angles)
You could draw in the two diagonals (from corner to opposite corner). You could draw two perpendicular lines to develop four squares inside the existing square. You could draw three parallel lines to develop four equally-sized rectangles within the square.