To "bisect" something is to divide it into two parts. Usually these parts are equal.
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If you are clever you will know that a trapezium does not bisect.
Usually, a parallelogram with not bisect other shapes.
A kite has two pairs of adjacent sides congruent. The diagonals intersect at right angles and bisect one set of angles.
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.
Opposite angles do not bisect any shapes.