1. **State the problem:** We have a triangle divided into two smaller triangles by a line from one vertex to the opposite side. The left smaller triangle has a top side length of $x + 8$ and a bottom left side length of 20. The right smaller triangle has a top side length of $2x - 5$ and a bottom right side length of 22.5. 2. **Identify the relationship:** Since the two smaller triangles share the same height (the line from the vertex to the opposite side), the ratios of their corresponding sides are equal. This gives us the proportion: $$\frac{x + 8}{20} = \frac{2x - 5}{22.5}$$ 3. **Solve the proportion:** Cross-multiply to solve for $x$: $$22.5(x + 8) = 20(2x - 5)$$ 4. **Expand both sides:** $$22.5x + 180 = 40x - 100$$ 5. **Bring all terms involving $x$ to one side and constants to the other:** $$22.5x - 40x = -100 - 180$$ 6. **Simplify:** $$-17.5x = -280$$ 7. **Divide both sides by $-17.5$ to isolate $x$:** $$x = \frac{-280}{\cancel{-17.5}} \cancel{-1} = 16$$ 8. **Final answer:** $$\boxed{16}$$