1. **State the problem:** We need to determine which triangles are similar to triangle $\triangle ABC$ with sides $AB=0.9$, $BC=1$, and $AC=1.5$. Similar triangles have proportional corresponding sides. 2. **Recall the similarity rule:** Two triangles are similar if their corresponding sides are in the same ratio. 3. **Calculate the side ratios of $\triangle ABC$:** - $\frac{AB}{BC} = \frac{0.9}{1} = 0.9$ - $\frac{BC}{AC} = \frac{1}{1.5} = \frac{2}{3} \approx 0.6667$ - $\frac{AB}{AC} = \frac{0.9}{1.5} = 0.6$ 4. **Check $\triangle DEF$ with sides $DE=4.5$, $EF=5$, $DF=7.5$:** - $\frac{DE}{EF} = \frac{4.5}{5} = 0.9$ - $\frac{EF}{DF} = \frac{5}{7.5} = \frac{2}{3} \approx 0.6667$ - $\frac{DE}{DF} = \frac{4.5}{7.5} = 0.6$ These ratios match exactly the ratios of $\triangle ABC$, so $\triangle DEF$ is similar to $\triangle ABC$. 5. **Check $\triangle GHI$ with sides $HI=9$, $IG=10$, $GH=3$:** - $\frac{HI}{IG} = \frac{9}{10} = 0.9$ - $\frac{IG}{GH} = \frac{10}{3} \approx 3.333$ - $\frac{HI}{GH} = \frac{9}{3} = 3$ These ratios do not match the ratios of $\triangle ABC$, so $\triangle GHI$ is not similar to $\triangle ABC$. **Final answer:** Only $\triangle DEF$ is similar to $\triangle ABC$. $$\boxed{\triangle DEF \text{ only}}$$