A regular polygon.
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If the diagonals are cong. then the figure is a rectangle.
what is the congruent diagonals each of which divides the figure into two congruent isosceles right triangles
No. The diagonals of a parallelogram are congruent if and only if the parallelogram is a rectangle.
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If you mean an isosceles triangle then no because all triangles have no diagonals