1. **Stating the problem:** We have a triangle EFG with a segment CD inside it parallel to FG. Given lengths are $EC=32$, $ED=40$, and $FC=24$. We need to find the length of segment $DG$. 2. **Understanding the problem:** Since $CD$ is parallel to $FG$, triangles $ECD$ and $EFG$ are similar by the Basic Proportionality Theorem (Thales' theorem). 3. **Using similarity ratios:** The ratio of corresponding sides in similar triangles is equal. So, $$\frac{EC}{EF} = \frac{ED}{EG} = \frac{CD}{FG}$$ 4. **Finding $EF$ and $EG$:** Since $F$ and $G$ are points on the base, and $FC=24$, then $EF = EC + FC = 32 + 24 = 56$. 5. **Setting up the ratio for $ED$ and $EG$:** We know $ED=40$, and $EG = ED + DG$. We want to find $DG$. 6. **Using the ratio:** $$\frac{EC}{EF} = \frac{ED}{EG} \Rightarrow \frac{32}{56} = \frac{40}{40 + DG}$$ 7. **Cross-multiplying:** $$32(40 + DG) = 56 \times 40$$ 8. **Expanding:** $$1280 + 32 \times DG = 2240$$ 9. **Isolating $DG$:** $$32 \times DG = 2240 - 1280 = 960$$ 10. **Solving for $DG$:** $$DG = \frac{960}{32} = 30$$ **Final answer:** The length of segment $DG$ is $30$.