Question 4(Multiple Choice Worth 1 points)
(01.07 MC)

Diana and Joby 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.

Diana's Proof


Statement	Justification
∠1 + ∠2 = 180°	Definition of Supplementary Angles
∠1 + ∠4 = 180°	Definition of Supplementary Angles
∠1 + ∠2 = ∠1 + ∠4	Substitution Property of Equality


Joby's Proof

Statement	Justification
∠2 = ∠4	Vertical angles are congruent.
∠1 = ∠3	Vertical angles are congruent.
Vertical angles are congruent.	Vertical Angle Theorem
 Diana is correct, but Joby is not.
 Both Diana and Joby are correct.
 Neither Diana nor Joby is correct.
 Joby is correct, but Diana is not.

The correct answer is: **Joby is correct, but Diana is not.**

Here's the reasoning:

- **Joby's proof** is correct because the "Vertical Angle Theorem" states that vertical angles are congruent. So, ∠2 = ∠4 and ∠1 = ∠3 based on this theorem. This is a straightforward application of the rule.

- **Diana's proof**, however, has an issue. While her statements about supplementary angles are true (∠1 + ∠2 = 180° and ∠1 + ∠4 = 180° by definition of supplementary angles), her conclusion that ∠1 + ∠2 = ∠1 + ∠4 via the "Substitution Property of Equality" does not directly lead to the conclusion that vertical angles are congruent. The Substitution Property of Equality doesn't help to prove vertical angles congruent here. 

Thus, **Joby's proof** is the correct one.

Would you like further details on this? Here are 5 related questions to explore:

1. What is the Substitution Property of Equality in geometry?
2. How does the Vertical Angle Theorem work in other geometric proofs?
3. Can you prove the congruence of vertical angles using a different method?
4. What are supplementary angles, and how do they relate to adjacent angles?
5. How are angles formed when two lines intersect, and what are their properties?

**Tip:** When proving geometric theorems, always ensure that each step logically follows from the previous one based on established properties and definitions.

Diana and Joby 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. Diana's Proof: Statement: ∠1 + ∠2 = 180° (Definition of Supplementary Angles), ∠1 + ∠4 = 180° (Definition of Supplementary Angles), ∠1 + ∠2 = ∠1 + ∠4 (Substitution Property of Equality). Joby's Proof: Statement: ∠2 = ∠4 (Vertical angles are congruent), ∠1 = ∠3 (Vertical angles are congruent), Vertical angles are congruent (Vertical Angle Theorem). Who is correct?

Explore the proofs by Diana and Joby regarding the congruence of vertical angles using the Vertical Angle Theorem and the Substitution Property of Equality. This geometry problem highlights the differences in their approaches and clarifies who is correct. Suitable for students in Grades 9-10.

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