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To solve the given problem, let's analyze the triangle and apply relevant theorems.

### Problem Summary:
You are given a triangle \( \triangle ABC \) with the following conditions:
- \( AB > AC \)
- \( AC > BC \)

You are asked to prove that \( m \angle C > m \angle A \).

### Step-by-Step Solution:

1. **Understanding the Inequalities:**
   - \( AB > AC \) implies that the side opposite angle \( C \) (which is \( AB \)) is longer than the side opposite angle \( B \) (which is \( AC \)).
   - \( AC > BC \) implies that the side opposite angle \( B \) (which is \( AC \)) is longer than the side opposite angle \( A \) (which is \( BC \)).

2. **Applying the Triangle Inequality Theorem:**
   - The triangle inequality states that in any triangle, the side opposite the larger angle is the longer side.
   - From \( AB > AC \), since \( AB \) is opposite angle \( C \) and \( AC \) is opposite angle \( B \), we can infer that \( m \angle C > m \angle B \).
   - Similarly, \( AC > BC \) implies \( m \angle B > m \angle A \).

3. **Combining the Results:**
   - Since \( m \angle C > m \angle B \) and \( m \angle B > m \angle A \), it follows from transitivity that:
   \[
   m \angle C > m \angle A
   \]

Thus, you have proved that \( m \angle C > m \angle A \) based on the given conditions.

### Conclusion:
The proof is complete, showing that in triangle \( \triangle ABC \), angle \( C \) is larger than angle \( A \).

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. What is the relationship between the angles in a triangle and the lengths of the sides?
2. Can this approach be applied to all triangles, including obtuse and right triangles?
3. How does the Triangle Inequality Theorem generalize to other polygons?
4. How would the proof change if the inequalities were reversed (e.g., \( AB < AC \))?
5. What are some real-life applications of the Triangle Inequality Theorem?

**Tip:** Always remember that in any triangle, the larger angle is always opposite the longer side, and this can be a useful tool in many geometric proofs.

Learn how to prove that angle C is greater than angle A in triangle ABC using the Triangle Inequality Theorem and detailed step-by-step solution. Understand the relationship between side lengths and angles in triangles. Suitable for high school students studying geometry.

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