A kite is a quadrilateral in which two disjoints pairs of consecutive sides are congruent
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The diagonals are perpendicular to one another. The shorter diagonal is bisected by the longer diagonal. The kite is symmetrical about the longer diagonal. The longer diagonal bisects the angles at each end of the diagonal.
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.
Square, rhombus and a kite have diagonals that bisect each other at 90 degrees
It can be a kite.
A kite is irregular.