1. **State the problem:** We need to find the values of $x$ and $y$ in two figures using angle relationships such as supplementary angles, alternate exterior angles, and corresponding angles. 2. **Figure 17:** Given angles are $(9x + 12)^\circ$, $3x^\circ$, and $(4y - 10)^\circ$. We know $x=14$ and $y=37$. 3. **Key angle rules:** - Supplementary angles sum to $180^\circ$. - Alternate exterior angles are equal. - Vertical angles are equal. 4. **Use alternate exterior angles:** $$9x + 12 = 4y - 10$$ Substitute $x=14$ and $y=37$: $$9(14) + 12 = 4(37) - 10$$ $$126 + 12 = 148 - 10$$ $$138 = 138$$ This confirms the equality. 5. **Use supplementary angles:** Angles $3x$ and $(4y - 10)$ are supplementary: $$3x + (4y - 10) = 180$$ Substitute $x=14$, $y=37$: $$3(14) + (4(37) - 10) = 180$$ $$42 + (148 - 10) = 180$$ $$42 + 138 = 180$$ $$180 = 180$$ This confirms the supplementary relationship. 6. **Figure 18:** Given angles are $(5y - 4)^\circ$, $3y^\circ$, and $(2x + 13)^\circ$. We know $x=28$ and $y=23$. 7. **Use corresponding angles:** $3y = 2x + 13$ Substitute $x=28$, $y=23$: $$3(23) = 2(28) + 13$$ $$69 = 56 + 13$$ $$69 = 69$$ This confirms the equality. 8. **Use supplementary angles:** $5y - 4$ and $2x + 13$ are supplementary: $$(5y - 4) + (2x + 13) = 180$$ Substitute $x=28$, $y=23$: $$(5(23) - 4) + (2(28) + 13) = 180$$ $$(115 - 4) + (56 + 13) = 180$$ $$111 + 69 = 180$$ $$180 = 180$$ This confirms the supplementary relationship. **Final answer:** The given values $x=14$, $y=37$ for Figure 17 and $x=28$, $y=23$ for Figure 18 satisfy the angle relationships using supplementary, alternate exterior, and corresponding angles.