Usually, a parallelogram with not bisect other shapes.
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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.
Yes every parallelogram has bisecting diagonals
yes
They do in some parallelograms, not in others.
That's a rhombus...