No, they do not have that property. The quadrilaterals that have that property are the rhombus (and subsequently, the square) and the kite.
The only property I'm aware of diagonals of a trapezoid having is the fact that they cut each other in the same ratio, which happens to be the ratio between the lengths of the parallel sides.
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Not usually.
No.
perpendicular
Not usually.
The diagonals of an isosceles trapezoid are equal in lengths but are not perpendicular to each other at right angles.