1. **State the problem:** We need to solve for $x$ given the equation involving angles $m\angle DBA = (4x + 2)^\circ$ and $m\angle ABC = (5x - 9)^\circ$ that sum to $m\angle ABD = 78^\circ$. 2. **Use the angle addition rule:** The sum of the two smaller angles equals the larger angle: $$m\angle DBA + m\angle ABC = m\angle ABD$$ 3. **Set up the equation:** $$(4x + 2) + (5x - 9) = 78$$ 4. **Simplify the equation:** $$4x + 2 + 5x - 9 = 78$$ $$9x - 7 = 78$$ 5. **Solve for $x$:** $$9x - 7 = 78$$ $$9x = 78 + 7$$ $$9x = 85$$ $$x = \frac{85}{9}$$ 6. **Calculate each angle:** $$m\angle DBA = 4x + 2 = 4\times \frac{85}{9} + 2 = \frac{340}{9} + \frac{18}{9} = \frac{358}{9} \approx 39.78^\circ$$ $$m\angle ABC = 5x - 9 = 5\times \frac{85}{9} - 9 = \frac{425}{9} - \frac{81}{9} = \frac{344}{9} \approx 38.22^\circ$$ 7. **Check the sum:** $$39.78^\circ + 38.22^\circ = 78^\circ$$ **Final answers:** $$m\angle ABC = \frac{344}{9} \approx 38.22^\circ$$ $$m\angle DBC = m\angle DBA = \frac{358}{9} \approx 39.78^\circ$$