Yes. Other things about parallelograms:
-opposite sides are equal in length.
-opposite angles are equal in length.
-diagonals bisect each other.
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No.
A quadrilateral is a parallelogram if and only if its diagonals bisect each other (this should be in any geometry book)
Yes, they do. Also, they are congruent to each other. * * * * * They do bisect each other but they are not congruent.
Only if the parallelogram is in the form of a rhombus will its diagonals bisect each other at right angles
Only for a square or rhombus (diamond shape). The diagonals of a rectangle bisect each other, but are not perpendicular and do not bisect the opposite angles they join.