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To complete the proof that \( \triangle EFH \cong \triangle FGI \), let's analyze the diagram and the given information.

### Steps to complete the proof:

1. **Given:**
   - \( F \) is the midpoint of \( \overline{EG} \), meaning \( EF \cong FG \) (equal segments).
   - \( \overline{EH} \cong \overline{FI} \), as marked in the diagram.
   - \( \overline{FH} \cong \overline{GI} \), also given and marked.

Thus, based on the information, you can now fill in the missing parts of the proof:

1. **Statement:** \( F \) is the midpoint of \( \overline{EG} \) — **Reason:** Given.
2. **Statement:** \( \overline{EH} \cong \overline{FI} \) — **Reason:** Given.
3. **Statement:** \( \overline{FH} \cong \overline{GI} \) — **Reason:** Given.
4. **Statement:** \( \overline{EF} \cong \overline{FG} \) — **Reason:** Definition of midpoint (since \( F \) is the midpoint of \( \overline{EG} \)).
5. **Statement:** \( \triangle EFH \cong \triangle FGI \) — **Reason:** **SSS (Side-Side-Side)** congruence postulate (since all three corresponding sides are congruent).

This completes the proof using the SSS congruence criterion.

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Do you need any further explanation or details on this proof? Here are some related questions to think about:

1. How does the definition of a midpoint help establish congruence between triangles?
2. What other triangle congruence postulates could apply if different information were given?
3. Can you use a different set of triangles from the diagram to apply another congruence criterion?
4. How do we prove corresponding angles are equal in congruent triangles?
5. What if the triangles were not isosceles, how would that change the proof?

**Tip:** Always ensure that corresponding sides or angles in two triangles are clearly identified when proving congruence!

F is the midpoint of EG. Complete the proof that ΔEFH ≅ ΔFGI.

This proof demonstrates how to establish the congruence between ΔEFH and ΔFGI using the SSS (Side-Side-Side) theorem. By identifying the midpoint and congruent sides, students learn how to apply the SSS postulate in geometry. Suitable for high school students in Grades 9-11.

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