1. The problem asks to identify the two transformations that show figure ABCD is congruent to figure A'B'C'D'. 2. The original rectangle ABCD has vertices at approximately $(6,6)$, $(8,6)$, $(8,4)$, and $(6,4)$. 3. The transformed rectangle A'B'C'D' has vertices at approximately $(2,-4)$, $(4,-4)$, $(4,-6)$, and $(2,-6)$. 4. To determine the transformations, first observe the change in position and orientation. 5. The rectangle has moved from the first quadrant to the fourth quadrant, indicating a translation combined with a reflection or rotation. 6. Check for rotation: rotating ABCD $180^\circ$ about the origin would map $(x,y)$ to $(-x,-y)$. 7. Applying $180^\circ$ rotation to $(6,6)$ gives $(-6,-6)$, which does not match $(2,-4)$. 8. Check for reflection: reflecting ABCD over the x-axis changes $(x,y)$ to $(x,-y)$. 9. Reflecting $(6,6)$ over the x-axis gives $(6,-6)$, which is close but not equal to $(2,-4)$. 10. Check translation: the figure appears shifted left and up/down after reflection or rotation. 11. Combine reflection over the x-axis and translation by vector $\langle -4, 2 \rangle$. 12. Reflect $(6,6)$ over x-axis: $(6,-6)$. 13. Translate by $\langle -4, 2 \rangle$: $(6-4, -6+2) = (2,-4)$, which matches A'. 14. Therefore, the two transformations are: - Reflection over the x-axis. - Translation by vector $\langle -4, 2 \rangle$. 15. These transformations preserve size and shape, proving ABCD is congruent to A'B'C'D'.