1. **State the problem:** Given that $\frac{DE}{GH} = \frac{DF}{GI} = \frac{EF}{HI}$, prove that $\triangle DEF \cong \triangle GHI$. 2. **Understand the given:** We have three equal ratios of corresponding sides of the two triangles: $$\frac{DE}{GH} = \frac{DF}{GI} = \frac{EF}{HI}$$ This means the sides of $\triangle DEF$ are proportional to the sides of $\triangle GHI$. 3. **Recall the criteria for triangle congruence:** To prove two triangles are congruent, we can use criteria such as SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA, AAS, or RHS. 4. **Analyze the problem:** Since the ratios of all three pairs of sides are equal, the triangles are similar by SSS similarity criterion. But the problem asks to prove congruence, not similarity. 5. **Check if the triangles are equilateral or isosceles:** From the graph description, $DE \cong EF$ and $GH \cong HI$, meaning: $$DE = EF \quad \text{and} \quad GH = HI$$ Since the sides are equal in pairs, and the ratios of corresponding sides are equal, the triangles are not just similar but also have equal corresponding sides. 6. **Use the given equal ratios and equal sides to prove congruence:** Since $DE = EF$ and $GH = HI$, and $\frac{DE}{GH} = \frac{EF}{HI}$, it follows that: $$\frac{DE}{GH} = \frac{EF}{HI} = 1$$ Therefore: $$DE = GH \quad \text{and} \quad EF = HI$$ 7. **From the given $\frac{DE}{GH} = \frac{DF}{GI} = \frac{EF}{HI}$, and now knowing $DE = GH$ and $EF = HI$, it follows that:** $$DF = GI$$ 8. **Conclusion:** All three pairs of corresponding sides are equal: $$DE = GH, \quad DF = GI, \quad EF = HI$$ Therefore, by the SSS congruence criterion, $\triangle DEF \cong \triangle GHI$. **Final answer:** $$\triangle DEF \cong \triangle GHI$$