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In rhombuses and squares the diagonals bisect opposite angles.
No, the diagonals of a parallelogram do not necessarily bisect the angles. The diagonals of a parallelogram divide it into four congruent triangles, but they do not necessarily bisect the angles of those triangles.
The diagonals will not always bisect opposite angles in the rectangle.
Opposite angles do not bisect any shapes.
The diagonals of a rectangular shape will only bisect opposite angles if, in fact, the shape is a square. Otherwise they will not bisect them.