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Not for every parallelogram. Only for a rhombus (diamond) or square will the diagonals bisect the opposite angles they connect, and diagonals are perpendicular. In rectangles, the diagonals do not bisect the angles and are notperpendicular, but they do bisect each other.
An isosceles trapezoid, or any trapezoid, does not have diagonals that bisect each other.
they are called parallelogram in which diagonals are equal
A rhombus satisfies this. Also a rectangle. And a square (which is both a rhombus and a rectangle). There may be others.
Not necessarily - the diagonals of a rhombus bisect each other (they are perpendicular bisectors of each other), but are not equal.