Quadrilateral Congruence E017Cf

1. **State the problem:** Determine if quadrilateral $EFGH$ is congruent to quadrilateral $TUVW$ based on the given transformations and properties. 2. **Given points:** - $T(-1,3)$, $W(-1,5)$, $U(3,7)$, $V(3,9)$ - $E(7,8)$, $F(3,4)$, $G(3,2)$, $H(7,6)$ 3. **Check congruence by transformations:** - The first option states a rotation of $180^\circ$ around the origin followed by a translation right $6$ units and up $11$ units maps $EFGH$ onto $TUVW$. - The second option states a translation left $4$ units and down $11$ units followed by a reflection across the x-axis maps $EFGH$ onto $TUVW$. 4. **Verify the first transformation:** - Rotation $180^\circ$ around origin: $(x,y) \to (-x,-y)$. - Apply to $E(7,8)$: $(-7,-8)$, then translate right $6$ units and up $11$ units: $(-7+6,-8+11)=(-1,3)$ which matches $T$. - Similarly for $F(3,4)$: $(-3,-4)$ translate: $(-3+6,-4+11)=(3,7)$ which matches $U$. - For $G(3,2)$: $(-3,-2)$ translate: $(-3+6,-2+11)=(3,9)$ which matches $V$. - For $H(7,6)$: $(-7,-6)$ translate: $(-7+6,-6+11)=(-1,5)$ which matches $W$. 5. **Conclusion:** The first transformation maps $EFGH$ exactly onto $TUVW$, confirming congruence. 6. **Check other options:** - Angles $\angle H$ and $\angle W$ have the same measure because congruence preserves angles. - Lengths $FG$ and $UV$ are equal because congruence preserves side lengths. **Final answer:** Yes, because a rotation $180^\circ$ around the origin followed by a translation right $6$ units and up $11$ units maps $EFGH$ onto $TUVW$.