Yes if they both have the same dimensions and angles
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what is the congruent diagonals each of which divides the figure into two congruent isosceles right triangles
Absolutely. Any two congruent right triangles will form a rectangle, and if the right triangles are isosceles right triangles, they will form a square.
The hypotenuse leg theorem states that any two right triangles that have a congruent hypotenuse and a corresponding, congruent leg arecongruent triangles.
No isosceles triangle in the world is congruent to any equilateral triangle. No equilateral triangle in the world is congruent to any right triangle.
Similar, YES. Congruent, NO.