It depends on what you mean by bisect. All rectangles have diagonals that bisect the other one. Only certain rectangles (Squares) have diagonals that bisect its vertex, the ninety degree angle.
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Yes
Parallelograms (including rectangles and squares)
The diagonals are equal in length and bisect each other forming vertical opposte equal angles
A square has two diagonals that bisect each other at 90 degrees
A rhombus or a square. In rectangles (unequal side length) this does not occur.