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

Question

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.

Answer 1

To determine who is correct, let's analyze both Amanda's and Stephen's proofs regarding the vertical angles.

Vertical Angles: When two lines intersect, they form pairs of opposite angles, known as vertical angles. Vertical angles are always equal in measure.

Amanda's Proof:

1. Statement: ∠1 + ∠4 = 180°
   Justification: Definition of Supplementary Angles - Angles ∠1 and ∠4 are supplementary because they form a linear pair.

2. Statement: ∠3 + ∠4 = 180°
   Justification: Definition of Supplementary Angles - Angles ∠3 and ∠4 also form a linear pair and are supplementary.

3. Statement: ∠1 + ∠4 = ∠3 + ∠4
   Justification: Substitution Property of Equality - Since both sums are equal to 180°, you can equate them.

4. Statement: ∠1 ≅ ∠3
   Justification: Subtraction Property of Equality - Subtract ∠4 from both sides, leading to ∠1 = ∠3.

Conclusion: Amanda's proof correctly shows that vertical angles ∠1 and ∠3 are congruent.

Stephen's Proof:

1. Statement: ∠1 + ∠2 = 180°
   Justification: Definition of Supplementary Angles - Angles ∠1 and ∠2 are supplementary as they form a linear pair.

2. Statement: ∠2 + ∠3 = 180°
   Justification: Definition of Supplementary Angles - Angles ∠2 and ∠3 are supplementary because they form a linear pair.

3. Statement: ∠1 + ∠2 = ∠2 + ∠3
   Justification: Substitution Property of Equality - Since both sums equal 180°, they can be set equal to each other.

4. Statement: ∠1 ≅ ∠3
   Justification: Subtraction Property of Equality - Subtract ∠2 from both sides, leading to ∠1 = ∠3.

Conclusion: Stephen's proof correctly leads to the conclusion that ∠1 and ∠3 are congruent.

Final Assessment:

Both Amanda and Stephen have valid proofs that show vertical angles are congruent:

• Amanda's proof demonstrates the congruence of ∠1 and ∠3 by using the properties of supplementary angles involving angles ∠4.
• Stephen's proof ensures the congruence of ∠1 and ∠3 using a different approach, involving angles ∠2.

Answer:

Both Amanda and Stephen are correct.