Amanda and Stephen wrote the following proofs to prove that vertical angles are congruent. Who is correct?

Line segment NT intersects line segment MR, forming four angles. Angles 1 and 3 are vertical angles. Angles 2 and 4 are vertical angles.

Amanda's Proof

Statement	Justification
∠1 + ∠4 = 180°	Definition of Supplementary Angles
∠3 + ∠4 = 180°	Definition of Supplementary Angles
∠1 + ∠4 = ∠3 + ∠4	Substitution Property of Equality
∠1 ≅ ∠3	Subtraction Property of Equality

Stephen's Proof

Statement	Justification
∠1 + ∠2 = 180°	Definition of Supplementary Angles
∠2 + ∠3 = 180°	Definition of Supplementary Angles
∠1 + ∠2 = ∠2 + ∠3	Substitution Property of Equality
∠1 ≅ ∠3	Subtraction Property of Equality
Stephen is correct, but Amanda is not.
Amanda is correct, but Stephen is not.
Both Amanda and Stephen are correct.
Neither Amanda nor Stephen is correct.

To determine who is correct, let's analyze the proofs provided by Amanda and Stephen.

### Amanda's Proof:
1. **Statement**: \( \angle 1 + \angle 4 = 180^\circ \)  
   **Justification**: Definition of Supplementary Angles (angles on a straight line).
2. **Statement**: \( \angle 3 + \angle 4 = 180^\circ \)  
   **Justification**: Definition of Supplementary Angles (angles on a straight line).
3. **Statement**: \( \angle 1 + \angle 4 = \angle 3 + \angle 4 \)  
   **Justification**: Substitution Property of Equality (since both sums are equal to 180°).
4. **Statement**: \( \angle 1 \cong \angle 3 \)  
   **Justification**: Subtraction Property of Equality (subtracting \( \angle 4 \) from both sides).

### Stephen's Proof:
1. **Statement**: \( \angle 1 + \angle 2 = 180^\circ \)  
   **Justification**: Definition of Supplementary Angles (angles on a straight line).
2. **Statement**: \( \angle 2 + \angle 3 = 180^\circ \)  
   **Justification**: Definition of Supplementary Angles (angles on a straight line).
3. **Statement**: \( \angle 1 + \angle 2 = \angle 2 + \angle 3 \)  
   **Justification**: Substitution Property of Equality (since both sums are equal to 180°).
4. **Statement**: \( \angle 1 \cong \angle 3 \)  
   **Justification**: Subtraction Property of Equality (subtracting \( \angle 2 \) from both sides).

### Conclusion:
Both Amanda and Stephen have correctly proven that vertical angles are congruent. They have used different pairs of supplementary angles and employed the same logical steps involving the substitution and subtraction properties of equality.

**Answer:** Both Amanda and Stephen are correct.

Would you like further details or have any questions?

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Here are 5 related questions for further exploration:
1. How would you prove that angles 2 and 4 are congruent?
2. Can the proof of vertical angles being congruent be generalized to any intersecting lines?
3. What are some real-life applications of congruent vertical angles?
4. How does the concept of linear pairs relate to the proofs given by Amanda and Stephen?
5. Could a different pair of supplementary angles be used to prove the same result?

**Tip:** When proving geometric theorems, it's crucial to clearly define and justify each step to ensure the logical flow of the argument.

Explore the proofs by Amanda and Stephen on the congruence of vertical angles. Understand the use of supplementary angles and properties of equality in geometry proofs. Suitable for students in Grades 9-10 studying basic geometry concepts.

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