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

In fact, both Amanda's and Stephen's profs are correct; they are just using different supplementary angles. Amanda's is using the supplementary angles ∠1 and ∠4, and ∠3 and ∠4, whereas Stephen is using ∠1 and ∠2, and ∠2 and ∠3.  Please check the picture to visualize this more effectively.

Answer 2

Answer:

Both are correct

Step-by-step explanation:

Both Amanda and Stephan are correct as both are taking two different pairs of supplementary angles which will at the end will prove the same result.

Amanda's proof is as he takes (∠1 and ∠4) and (∠3 and ∠4)as the supplementary angles and writes ∠1 + ∠4 = 180° and ∠3 + ∠4 = 180°, thus  ∠1 + ∠4 = ∠3 + ∠4 by Transitive Property of Equality and then ∠1 ≅ ∠3 by  Subtraction Property of Equality.

Stephan's proof is as he takes (∠1 and ∠2) and (∠3 and ∠4) as the supplementary angle pair, thus ∠1 + ∠2 = 180°by Definition of Supplementary Angles and ∠2 + ∠3 = 180° by Definition of Supplementary therefore,  ∠1 + ∠2 = ∠2 + ∠3 by Transitive Property of Equality and then  ∠1 ≅ ∠3 by Subtraction Property of Equality.