1. **State the problem:** We need to identify which three transformations among the given options prove that segment $AB$ is congruent to segment $A'B'$. Congruent segments have the same length. 2. **Recall the rule for congruence under transformations:** A transformation that preserves distance (length) is called an isometry. Isometries include translations, rotations, and reflections. Dilations (scaling) change lengths unless the scale factor is 1. 3. **Analyze each transformation:** - A. $(x, y) \to (8x, y)$: This scales the $x$-coordinate by 8, changing lengths. Not an isometry. - B. $(x, y) \to (-y, x)$: This is a rotation by 90 degrees counterclockwise, which preserves lengths. - C. $(x, y) \to (4x, 4y)$: This scales both coordinates by 4, changing lengths. Not an isometry. - D. $(x, y) \to (-x, -y)$: This is a rotation by 180 degrees or a point reflection, preserving lengths. - E. $(x, y) \to (x - 1, y + 3)$: This is a translation, which preserves lengths. 4. **Conclusion:** The three transformations that prove $AB \cong A'B'$ are B, D, and E because they are isometries preserving segment length. **Final answer:** B, D, and E