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.