Diameter
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They are arcs of congruent circles.
what is the congruent diagonals each of which divides the figure into two congruent isosceles right triangles
If they're in the same circle or in circles of equal radii (radiuses), then yes.
A bisector.
No. Consider two circles of diameter 10cm and 20cm respectively. For the smaller circle consider the arc formed by half the circle. It has a length of pi*5 (where * means multiply) For the larger circle, consider the arc formed by a quarter of the circle. It has a length of pi*5. Both arc lengths are the same but they are not congruent. They have entirely different shapes.