All regular polygons.
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The diagonals are perpendicular, but not necessarily congruent.
Most quadrilaterals will not have congruent diagonals: rectangles (incl squares) and isosceles trapezium are the exception.
Rhombus.
Square.
Yes the diagonals of a square are always congruent. Their length is a*root 2 where a is the side of the square.