They bisect each other at an angle of 90 degrees
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The diagonals of any rhombus bisect each other. A square is a special kind of a rhombus.
Yes
Yes. The diagonals of any parallelogram bisect each other. A rectangle is a special case of a parallelogram.
Yes the diagonals of a parallelogram have the same midpoint since they bisect each other.
Not in general. The diagonals of a rectangle are equal length. A rhombus that is also a rectangle would be a square.