1. **Problem Statement:** Find the values of variables $x$ and $y$ in Figure 17, where two intersecting lines form vertical angles labeled $(9x + 12)^\circ$, $3x^\circ$, and $(4y - 10)^\circ$. 2. **Key Concept:** Vertical angles are equal. Also, angles on a straight line sum to $180^\circ$. 3. **Step 1:** Set vertical angles equal: $$9x + 12 = 3x$$ 4. **Step 2:** Solve for $x$: $$9x + 12 = 3x$$ $$9x - \cancel{3x} + 12 = \cancel{3x}$$ $$6x + 12 = 0$$ $$6x = -12$$ $$x = \frac{-12}{6} = -2$$ 5. **Step 3:** Use the straight line angle sum for $x$ and $y$ angles: $$(4y - 10) + 3x = 180$$ Substitute $x = -2$: $$(4y - 10) + 3(-2) = 180$$ $$4y - 10 - 6 = 180$$ $$4y - 16 = 180$$ $$4y = 196$$ $$y = \frac{196}{4} = 49$$ 6. **Final answer:** $$x = -2, \quad y = 49$$ **Explanation:** We used the property that vertical angles are equal to find $x$, then used the fact that angles on a straight line sum to $180^\circ$ to find $y$.