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No but the diagonals of a square, rhombus and a kite are perpendicular to each other
No.
Not always but they are perpendicular in a square, a rhombus and a kite in that the diagonals intersect each other at 90 degrees
If you are talking about the diagonals of a quadrilateral, the only quadrilateral that have diagonals that are perpendicular and bisect each other is a square, because a rectangle has bisecting diagonals, while a rhombus has perpendicular diagonals. And a square fits in both of these categories.
none, no