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A square or other rhombus (= parallelogram with 4 equal sides, whether or not they're at right angles).

Also, any other regular 4n-gon -- i.e., regular polygon whose number of sides is a multiple of 4 -- will have pairs of perpendicular diagonals. So a regular octagon has two such pairs.

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Q: What polygon has diagonals that are perpendicular?
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