Question

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 Transitive 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 Transitive Property of Equality
∠1 ≅ ∠3 Subtraction Property of Equality

Answer 1

Amanda's proof is correct because she used the definition of supplementary angles to show that angles 1 and 4 and angles 3 and 4 are supplementary, and then used the transitive property of equality to prove that angles 1 and 3 are congruent. Stephen's proof is incorrect because he used angles 1 and 2 instead of angles 1 and 4 in his first equation.

Step-by-step explanation:

Vertical angles are congruent. This means that when two lines intersect, the angles that are opposite each other (or across from each other) are equal in measure. Amanda's proof is correct because she used the definition of supplementary angles to show that angles 1 and 4 and angles 3 and 4 are supplementary. Then, she used the transitive property of equality to prove that angles 1 and 3 are congruent. Stephen's proof is incorrect because he incorrectly used angles 1 and 2 instead of angles 1 and 4 in his first equation. Therefore, Amanda's proof is correct.