The question asks whether enough information is provided to prove that triangles $\triangle HGF$ and $\triangle DEF$ are congruent using the SAS (Side-Angle-Side) Congruence Theorem.

Key Analysis:

1. Side Correspondence:

   • $\overline{HG} \cong \overline{DE}$ (denoted by single tick marks).
   • $\overline{GF} \cong \overline{EF}$ (denoted by double tick marks).

2. Angle Correspondence:

   • $\angle HGF \cong \angle DEF$ (shared vertical angles are congruent).

SAS Congruence:

The SAS Congruence Theorem requires two pairs of corresponding sides and the included angle to be congruent. In this case:

• The sides $\overline{HG} \cong \overline{DE}$ and $\overline{GF} \cong \overline{EF}$ are congruent.
• The included angles $\angle HGF \cong \angle DEF$ are also congruent.

Thus, there is enough information to prove $\triangle HGF \cong \triangle DEF$ using SAS. The answer is "Yes".

Would you like further clarification or details on this?

5 Related Questions:

1. What is the SAS Congruence Theorem, and how is it applied?
2. Can triangles be congruent if only one pair of angles and one side are given?
3. How do vertical angles help establish congruence in geometry?
4. What are other triangle congruence theorems apart from SAS?
5. Why is the included angle crucial in the SAS Congruence Theorem?

Tip:

Always check whether the given angle is included between the two sides when applying SAS. This detail is essential for accurate congruence proofs.