Final answer:
Both Diana and Joby correctly prove that vertical angles are congruent using different methods. Diana's proof involves the definition of supplementary angles and substitution property, while Joby's proof invokes the vertical angle theorem.
Step-by-step explanation:
The question asks to determine which proof correctly proves that vertical angles are congruent when two lines intersect. Diana's proof uses the definition of supplementary angles and the substitution property of equality to show congruence of angles, whereas Joby's proof directly states that vertical angles are congruent, invoking the vertical angle theorem.
Diana's proof is shown in steps:
- Angle 1 and Angle 2 are supplementary, hence their sum is 180 degrees (Definition of supplementary angles).
- Angle 1 and Angle 4 are also supplementary, so their sum is also 180 degrees (Definition of supplementary angles).
- Since Angle 1 + Angle 2 equals Angle 1 + Angle 4, by the Substitution Property of Equality, Angle 2 must be equal to Angle 4 (Substitution Property of Equality).
Joby's proof claims the congruence of vertical angles directly:
- Angle 2 is congruent to Angle 4 (Vertical Angles are Congruent).
- Angle 1 is congruent to Angle 3 (Vertical Angles are Congruent).
- By stating that the vertical angles theorem shows the congruence of vertical angles, Joby's proof is also correct.
Therefore, both proofs are valid, and the correct answer is that both Diana and Joby are correct in their claim that vertical angles are congruent.