No, they do not.
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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.
Parallelogram and rhombus.
It is true only when the parallelogram is in the form of a rhombus, and thus the two diagonals are perpendicular to each other.
The best classification for a parallelogram that has perpendicular diagonals is a rhombus. A rhombus has four sides that are congruent. The also diagonals bisect the vertex angles of this type of parallelogram.
It is a parallelogram with 4 right angles and it has congruent diagonals that bisect.
Rhombus and square are the only quadrilaterals whose diagonals bisect the angles of the quadrilateral. In both these quadrilaterals, the diagonals intersect at right angles, dividing each angle into two equal parts.