Yes.
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Only the diagonals of isosceles trapezoids are congruent.
A trapezoid with congruent diagonals is an isosceles trapezoid.
No. An isosceles trapezium (isosceles triangle with its apex removed) would have congruent diagonals but it is not a parallelogram.
what is the congruent diagonals each of which divides the figure into two congruent isosceles right triangles
If you mean an isosceles triangle then no because all triangles have no diagonals