https://file.aiquickdraw.com/mathbot/file/1728569807185l173xdr8.jpg

Let's walk through the proof based on the image you provided, completing the missing steps and reasons in the two-column proof format.

### **Proof:**
We are given that \(\angle 1\) and \(\angle 3\) are vertical angles. We need to prove that \(\angle 1 \cong \angle 3\).

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**Statements** | **Reasons**  
--- | ---  
1. \(\angle 1\) and \(\angle 3\) are vertical angles. | 1. Given  
2. \(\angle 1\) and \(\angle 2\) form a linear pair. | 2. Definition of linear pair  
3. \(\angle 3\) and \(\angle 2\) form a linear pair. | 3. Definition of linear pair  
4. \(\angle 1\) and \(\angle 2\) are supplementary. | 4. Supplementary angles form a linear pair  
5. \(\angle 3\) and \(\angle 2\) are supplementary. | 5. Supplementary angles form a linear pair  
6. \(m\angle 1 + m\angle 2 = 180^\circ\) | 6. Definition of supplementary angles  
7. \(m\angle 3 + m\angle 2 = 180^\circ\) | 7. Definition of supplementary angles  
8. \(m\angle 1 + m\angle 2 = m\angle 3 + m\angle 2\) | 8. Transitive property (from statements 6 and 7)  
9. \(m\angle 1 = m\angle 3\) | 9. Subtraction property of equality  
10. \(\angle 1 \cong \angle 3\) | 10. Definition of congruent angles (angles with equal measures)

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This completes the proof.

Would you like more details on any step, or do you have any questions?

### 5 Related Questions:
1. What is the definition of vertical angles?
2. Why do vertical angles always have equal measures?
3. Can you provide more examples where linear pairs are used in proofs?
4. What is the transitive property of equality and how does it apply here?
5. How do supplementary angles relate to parallel lines?

### Tip:
When working with geometric proofs, always look for relationships between angles like vertical angles, supplementary angles, and linear pairs. These relationships often provide the necessary reasoning to establish congruence or equality.

Provide reasons for the statements. Given: ∠1 and ∠3 are vertical angles. Prove: ∠1 ≅ ∠3.

Learn how to prove that vertical angles are congruent with this geometry proof. We demonstrate step-by-step how to use theorems such as the Vertical Angles Theorem, Linear Pair Postulate, and properties of equality to show that ∠1 ≅ ∠3.

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