1. **State the problem:** Given that the ratios of corresponding sides of triangles DEF and GHI are equal, i.e., $$\frac{DE}{GH} = \frac{DF}{GI} = \frac{EF}{HI}$$, prove that $$\angle E \cong \angle H$$. 2. **Recall the Side-Side-Side (SSS) similarity criterion:** If the corresponding sides of two triangles are in proportion, then the triangles are similar, and their corresponding angles are congruent. 3. **Apply the SSS similarity criterion:** Since $$\frac{DE}{GH} = \frac{DF}{GI} = \frac{EF}{HI}$$, triangles DEF and GHI are similar. 4. **Corresponding angles in similar triangles are congruent:** Therefore, $$\angle E$$ in triangle DEF corresponds to $$\angle H$$ in triangle GHI, so $$\angle E \cong \angle H$$. 5. **Conclusion:** We have proved that $$\angle E \cong \angle H$$ using the given side ratios and the SSS similarity criterion.