1. **State the problem:** We need to find values of $t$ and $u$ such that the right triangles $\triangle GHI$ and $\triangle RQS$ are congruent by the Hypotenuse-Leg (HL) Theorem. 2. **Recall the HL Theorem:** Two right triangles are congruent if their hypotenuses and one corresponding leg are equal. 3. **Identify sides:** - For $\triangle GHI$, the sides are $t + 9u - 32$ and $6u - 39$ with a right angle at $H$. - For $\triangle RQS$, the sides are $13t + u$ and $3u$ with a right angle at $Q$. 4. **Set hypotenuses equal:** The hypotenuse is the longest side. We assume $t + 9u - 32$ corresponds to $13t + u$ (hypotenuses equal): $$t + 9u - 32 = 13t + u$$ 5. **Set legs equal:** The other sides correspond, so: $$6u - 39 = 3u$$ 6. **Solve the leg equation:** $$6u - 39 = 3u$$ $$6u - 3u = 39$$ $$3u = 39$$ $$u = 13$$ 7. **Substitute $u=13$ into the hypotenuse equation:** $$t + 9(13) - 32 = 13t + 13$$ $$t + 117 - 32 = 13t + 13$$ $$t + 85 = 13t + 13$$ $$85 - 13 = 13t - t$$ $$72 = 12t$$ $$t = 6$$ 8. **Final answer:** $$t = 6, \quad u = 13$$