1. **State the problem:** Given triangles DEF and HGI with DE = EF, GH = HI, and $\angle E \cong \angle H$, prove that EF = DF and HI = GI. 2. **Analyze given information:** - DE = EF (Given) - GH = HI (Given) - $\angle E \cong \angle H$ (Given) 3. **Identify what to prove:** - EF = DF - HI = GI 4. **Use triangle properties:** Since DE = EF, triangle DEF is isosceles with DE = EF, so angles opposite these sides are equal. Similarly, GH = HI implies triangle HGI is isosceles. 5. **Apply Isosceles Triangle Theorem:** In triangle DEF, since DE = EF, $\angle D = \angle F$. In triangle HGI, since GH = HI, $\angle G = \angle I$. 6. **Use given angle congruence:** $\angle E \cong \angle H$ is given. 7. **Conclude side equalities:** Since $\angle D = \angle F$ and DE = EF, by the Isosceles Triangle Theorem, EF = DF. Similarly, since GH = HI and $\angle G = \angle I$, HI = GI. **Final answer:** $$EF = DF \quad \text{and} \quad HI = GI$$