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The problem involves an equilateral triangle \( \triangle DEF \), where all three sides are equal. A segment is drawn from point \( F \) to point \( G \), and \( G \) is the midpoint of \( \overline{DE} \).

We are asked to determine which statement about congruence between triangles is **true**:

- **A**: \( \triangle FEG \cong \triangle DEF \)
- **B**: \( \triangle DFG \cong \triangle GFE \)
- **C**: \( \triangle DFG \cong \triangle EFG \)
- **D**: There are no pairs of congruent triangles.

### Let's break it down:

1. \( \triangle DEF \) is equilateral, so each side of the triangle \( \overline{DE} = \overline{EF} = \overline{FD} \).
2. \( G \) is the midpoint of \( \overline{DE} \), so \( \overline{DG} = \overline{GE} \).
3. Drawing the segment \( FG \) creates two smaller triangles \( \triangle DFG \) and \( \triangle EFG \).

Now, examining each statement:

- **Option A** is incorrect because \( \triangle FEG \) is not congruent to \( \triangle DEF \). \( \triangle FEG \) is smaller, with \( \overline{GE} \) being half of \( \overline{DE} \).
- **Option B**: \( \triangle DFG \) and \( \triangle GFE \) are not congruent. Although they share side \( \overline{FG} \), their other sides are not equal.
- **Option C**: \( \triangle DFG \) and \( \triangle EFG \) also cannot be congruent because they have different side lengths.
- **Option D** is correct because none of the triangles formed by drawing the segment \( FG \) are congruent to each other.

Thus, the answer marked as correct is **D: There are no pairs of congruent triangles**.

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Do you want further details on why the other options are incorrect? 

Here are five related questions:
1. What are the properties of an equilateral triangle?
2. How do you prove that two triangles are congruent?
3. What is the significance of a midpoint in triangle geometry?
4. Can two triangles sharing a side be congruent?
5. How does the Side-Side-Side (SSS) congruence rule work?

**Tip**: When proving triangle congruence, always look for shared sides, equal angles, or midpoints that might simplify your analysis.

All three sides of triangle DEF have the same length. You drew a segment from point F to point G, where point G is the midpoint of DE. Which of the following statements is TRUE? A) ∆FEG ≅ ∆DEF B) ∆DFG ≅ ∆GFE C) ∆DFG ≅ ∆EFG D) There are no pairs of congruent triangles.

Solve a geometry problem involving an equilateral triangle where a segment is drawn from one vertex to the midpoint of the opposite side. Explore triangle congruence, the properties of an equilateral triangle, and apply the Midpoint Theorem. Suitable for students in Grades 9-10.

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