1. **State the problem:** Prove that triangles $\triangle DEG$ and $\triangle FEG$ are congruent given that they are right triangles, $DG \cong FG$, and they share side $EG$. 2. **Given:** - $\triangle DEG$ and $\triangle FEG$ are right triangles. - $DG = FG$ (given). - $EG = EG$ (common side, reflexive property). 3. **Goal:** Prove $\triangle DEG \cong \triangle FEG$. 4. **Use the Right Angle-Hypotenuse-Side (RHS) congruence theorem:** - For right triangles, if the hypotenuse and one side are equal, the triangles are congruent. 5. **Identify the right angles:** Both triangles have a right angle at vertex $E$. 6. **Identify the hypotenuses:** In $\triangle DEG$, the hypotenuse is $DG$; in $\triangle FEG$, the hypotenuse is $FG$. 7. **Check the given equal sides:** $DG = FG$ (given). 8. **Check the shared side:** $EG = EG$ (reflexive property). 9. **Apply RHS theorem:** Since both triangles have a right angle at $E$, hypotenuses $DG$ and $FG$ are equal, and side $EG$ is common, by RHS congruence theorem, $$\triangle DEG \cong \triangle FEG$$ **Final answer:** $\triangle DEG \cong \triangle FEG$ by RHS congruence theorem.