1. **State the problem:** We need to determine which triangles are similar to triangle $\triangle ABC$ with sides $AB=36$, $BC=32$, and $CA=24$. The candidates are $\triangle DEF$ with sides $DE=27$, $EF=24$, $FD=18$ and $\triangle GHI$ with sides $GH=9$, $HI=8$, and $IG=6$. 2. **Recall the similarity rule for triangles:** Two triangles are similar if their corresponding sides are in proportion, i.e., the ratios of their corresponding sides are equal. 3. **Calculate the side ratios for $\triangle ABC$:** $$\frac{AB}{BC} = \frac{36}{32} = \frac{9}{8}, \quad \frac{BC}{CA} = \frac{32}{24} = \frac{4}{3}, \quad \frac{CA}{AB} = \frac{24}{36} = \frac{2}{3}$$ 4. **Check similarity with $\triangle DEF$:** Sides of $\triangle DEF$ are $DE=27$, $EF=24$, $FD=18$. Calculate ratios: $$\frac{DE}{EF} = \frac{27}{24} = \frac{9}{8}, \quad \frac{EF}{FD} = \frac{24}{18} = \frac{4}{3}, \quad \frac{FD}{DE} = \frac{18}{27} = \frac{2}{3}$$ These ratios exactly match those of $\triangle ABC$. 5. **Check similarity with $\triangle GHI$:** Sides of $\triangle GHI$ are $GH=9$, $HI=8$, $IG=6$. Calculate ratios: $$\frac{GH}{HI} = \frac{9}{8}, \quad \frac{HI}{IG} = \frac{8}{6} = \frac{4}{3}, \quad \frac{IG}{GH} = \frac{6}{9} = \frac{2}{3}$$ These ratios also match those of $\triangle ABC$. 6. **Conclusion:** Both $\triangle DEF$ and $\triangle GHI$ have side lengths proportional to $\triangle ABC$, so both are similar to $\triangle ABC$. **Final answer:** C Both