Can unequal right-angled triangles have two pairs of equal legs?

If two right-angled triangles have correspondingly equal legs, we also automatically obtain an angle between them equal to 90 °, since there is always a right angle between the legs, which means that the triangles are equal by the first sign of equality of the triangles (two sides and the angle between them).
The conclusion suggests itself: no, this cannot be, – right-angled triangles that have correspondingly identical (equal to each other) legs cannot be different.