1. **Problem statement:** Iris wants to find the total length of wood needed to construct triangle $\triangle DFG$, which is similar to $\triangle ABC$. Given sides of $\triangle ABC$ are $AB=5$, $BC=6$, and $AC=7$ inches. $\triangle DFG$ has sides $DF=20$, $FG=24$, and side $DG$ unknown. The angle at $B$ in $\triangle ABC$ is congruent to the angle at $F$ in $\triangle DFG$, confirming similarity. 2. **Formula and rules:** For similar triangles, corresponding sides are proportional. The scale factor $k$ between $\triangle ABC$ and $\triangle DFG$ is given by the ratio of any pair of corresponding sides: $$k = \frac{DF}{AB} = \frac{FG}{BC} = \frac{DG}{AC}$$ 3. **Calculate scale factor:** Using sides $DF=20$ and $AB=5$: $$k = \frac{20}{5} = 4$$ 4. **Find missing side $DG$:** Using $DG = k \times AC$: $$DG = 4 \times 7 = 28$$ 5. **Calculate total wood needed:** Sum of sides of $\triangle DFG$: $$DF + FG + DG = 20 + 24 + 28 = 72$$ **Answer:** Iris needs $72$ inches of wood to construct $\triangle DFG$.