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Write an indirect proof to show AB is not equal to AD

To write an indirect proof to show that \( AB \neq AD \) using the provided triangle diagram, we can follow these steps:

### Given:
- A triangle \( \triangle ABC \) is shown where \( \angle ACB = \angle CAD \).
- \( C \) is a point on \( AB \), and \( D \) is a point on \( AC \).

### To Prove:
\( AB \neq AD \)

### Proof by Contradiction (Indirect Proof):

1. **Assume the Opposite**: 
   Suppose \( AB = AD \).

2. **Analyze the Assumption**:
   If \( AB = AD \), then \( \triangle ABC \) and \( \triangle ADC \) would be isosceles triangles with \( AB = AD \).

3. **Use of Congruence Criteria**:
   - Since \( AB = AD \) and \( \angle ACB = \angle CAD \), by the **ASA (Angle-Side-Angle)** congruence postulate, the triangles \( \triangle ABC \) and \( \triangle ADC \) would be congruent.
   - If the triangles are congruent, then \( \angle ABC \) would be equal to \( \angle ADC \), and \( \angle BAC \) would be equal to \( \angle DCA \).

4. **Contradiction**:
   - However, in the diagram, \( \angle BAC \) and \( \angle DCA \) are clearly not equal because \( \angle DCA \) is an exterior angle to \( \triangle ABC \), which should be greater than \( \angle BAC \) (exterior angle theorem).
   - This contradicts our assumption that \( AB = AD \).

5. **Conclusion**:
   Since assuming \( AB = AD \) leads to a contradiction, we must conclude that \( AB \neq AD \).

Thus, the indirect proof shows that \( AB \) is not equal to \( AD \).

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Would you like further details or have any other questions? Here are five related questions:

1. How does the exterior angle theorem apply in indirect proofs?
2. What are other methods to prove \( AB \neq AD \) in triangles?
3. Why is the ASA congruence postulate significant in geometric proofs?
4. Can the assumption \( AB = AD \) lead to other contradictions in different triangles?
5. How can indirect proofs be applied in different branches of mathematics?

**Tip:** When proving geometric theorems indirectly, always look for a logical contradiction that arises from the assumption being false. This helps in confirming the original statement as true.

Write an indirect proof to show AB is not equal to AD.

Explore an indirect proof in geometry that demonstrates why AB is not equal to AD using the concepts of congruent triangles, the ASA postulate, and the exterior angle theorem. Suitable for students in Grades 9-10.

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