A transformation that preserves angle measures and lengths is called a rigid motion or an isometry.
Recall that translations are rigid motions. Hence, they'll preserve the angle measures and lengths whenever they are performed.
It follows that since the rectangle A'B'C'D' is a translation of the rectangle ABCD, angle measures and length will be preserved, and hence, the rectangles must be congruent.
The correct option that justifies congruence is that translations are rigid motions.
The last option is correct.