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To determine whether \( \triangle ABC \) is equilateral, let's analyze the given information and each answer choice.

### Given Information:
- **Angles \( \angle A \) and \( \angle C \) are congruent** in \( \triangle ABC \).

### Analysis:
1. Since \( \angle A \cong \angle C \), \( \triangle ABC \) has two angles that are equal. This suggests that \( \triangle ABC \) is **isosceles** because at least two of its sides are equal.
2. In an equilateral triangle, all three angles must be congruent (each \( 60^\circ \)), which implies all three sides must also be congruent.

### Answer Choices Analysis:
Let's go through each option to see which additional statement would prove that \( \triangle ABC \) is equilateral.

- **Option A: \( AB \cong BC \) only**
  - If \( AB \cong BC \) and \( \angle A \cong \angle C \), it does not necessarily mean that \( AC \) is also congruent to \( AB \) or \( BC \). Therefore, this is **insufficient** to prove the triangle is equilateral.

- **Option B: \( AB \cong AC \) only**
  - If \( AB \cong AC \) and \( \angle A \cong \angle C \), \( \triangle ABC \) is **isosceles** but still might not be equilateral. This does not confirm that \( BC \) is also congruent to \( AB \) or \( AC \). Therefore, this is also **insufficient**.

- **Option C: Either statement is sufficient**
  - Neither of the individual statements (from Options A or B) is sufficient to prove \( \triangle ABC \) is equilateral, so this option is **incorrect**.

- **Option D: We need both statements**
  - If we know both \( AB \cong BC \) and \( AB \cong AC \), we can deduce that all three sides \( AB \), \( BC \), and \( AC \) are congruent, making \( \triangle ABC \) equilateral. Thus, **this is the correct answer**.

- **Option E: Even with both statements, we still could not prove that \( \triangle ABC \) is equilateral**
  - This option is incorrect because with both \( AB \cong BC \) and \( AB \cong AC \), we can indeed conclude that \( \triangle ABC \) is equilateral.

### Conclusion:
The correct answer is:
**D. We need both statements.**

Would you like further clarification or a deeper explanation on any step?

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#### Related Questions:
1. How do we identify an isosceles triangle based on angle congruence?
2. Why is it important to know all three sides are congruent to confirm an equilateral triangle?
3. What other methods can we use to determine if a triangle is equilateral?
4. How does the Triangle Congruence Postulate help in proving triangles are equilateral?
5. What are some properties of equilateral triangles that distinguish them from isosceles triangles?

#### Tip:
In any triangle, if two angles are equal, the sides opposite those angles are also equal, which can be useful when determining whether a triangle is isosceles or equilateral.

Which of the following additional statements would allow us to prove that triangle ABC is equilateral?

Explore how to determine if triangle ABC is equilateral by analyzing angle and side congruence. This problem covers geometry concepts and the properties of equilateral triangles, suitable for Grades 9-10.

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