Question: Given that $DE \cong HG$ and $\angle D \cong \angle H$, which of the following rules could be used to prove $\triangle DEF \cong \triangle HGF$? A. Side-Side-Side B. Angle-Angle-Side C. Side-Angle-Side D. Angle-Side-Angle Graph: Two triangles are shown, triangle $DEF$ on the left and triangle $HGF$ on the right, connected by the diagonal line $D-F-H$ crossing the horizontal line $E-F-G$; position_hint = top-center

1. **Problem Statement:** We are given two triangles $\triangle DEF$ and $\triangle HGF$ with $DE \cong HG$ and $\angle D \cong \angle H$. We need to determine which congruence rule can be used to prove $\triangle DEF \cong \triangle HGF$. 2. **Recall Congruence Rules:** - Side-Side-Side (SSS): All three sides of one triangle are congruent to all three sides of another triangle. - Angle-Angle-Side (AAS): Two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle. - Side-Angle-Side (SAS): Two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle. - Angle-Side-Angle (ASA): Two angles and the included side of one triangle are congruent to the corresponding parts of another triangle. 3. **Analyze Given Information:** - $DE \cong HG$ (side) - $\angle D \cong \angle H$ (angle) 4. **Look for Additional Congruent Parts:** - The triangles share the segment $EF$ and $GF$ respectively, which are corresponding sides. - Since $D-F-H$ is a diagonal line crossing $E-F-G$, $F$ is a common vertex. 5. **Check if the angle is included between the two sides:** - The angle $\angle D$ is between sides $DE$ and $DF$. - The angle $\angle H$ is between sides $HG$ and $HF$. 6. **Conclusion:** - We have one side $DE \cong HG$, one angle $\angle D \cong \angle H$, and the side $DF \cong HF$ (common side or given by the diagram). - This matches the Side-Angle-Side (SAS) congruence rule. **Final answer:** C. Side-Angle-Side