1) A set of right angles corresponds to a set of hypotenuse.

It's HL, there's no need to argue

2) One set of right-angled edges corresponds to another set of right-angled edges.

The right angle in the middle is SAS.

However, this is not the case if right-angled edges correspond to hypotenuse and a set of right-angled edges correspond to hypotenuse.

One side and one angle correspond to the congruence of two right triangles.

Because there is another right angle, which is AAS.

However, if two right triangles with equal right angles and a hypotenuse are not equal.