90 degrees
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One answer is a kite.
They are perpendicular and one diagonal is bisected.
It can be but a square and a rhombus diagonals are also perpendicular and therefore intersect at 90 degrees and they too are both quadrilaterals.
Perpendicular and the one for which each endpoint touches only sides of equal length bisects the other.
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.