Yes.
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the squares diagonal are all congruent to each other, and the diagonals make 4 isoceles triangles.
A trapezoid with congruent diagonals is an isosceles trapezoid.
Only the diagonals of isosceles trapezoids are congruent.
No, the diagonals of a rhombus are not congruent, but they do intersect each other at 90 degrees.
No not normally only if the parallelogram is in the form of a rectangle will it then have congruent diagonals.