Question

diana and joby wrote the following proofs to prove that vertical angles are congruent. who is correct?
dianas proof
statement: ( angle 1+angle 2 = 180^circ ), justification: definition of supplementary angles
statement: ( angle 1+angle 4 = 180^circ ), justification: definition of supplementary angles
statement: ( angle 1+angle 2=angle 1+angle 4 ), justification: substitution property of equality
jobys proof
statement: ( angle 2=angle 4 ), justification: vertical angles are congruent.
statement: ( angle 1=angle 3 ), justification: vertical angles are congruent.
statement: vertical angles are congruent., justification: vertical angle theorem
diana is correct, but joby is not.
both diana and joby are correct.
neither diana nor joby is correct.

Explanation:

Step1: Analyze Diana's proof

Diana uses the definition of supplementary angles (\(\angle1+\angle2 = 180^{\circ}\), \(\angle1+\angle4=180^{\circ}\)) and substitution property of equality (\(\angle1+\angle2=\angle1+\angle4\)). Then by subtracting \(\angle1\) from both sides (subtraction property of equality, which is a valid step not shown in her table but can be inferred from the substitution), we can get \(\angle2=\angle4\). Her proof is a valid logical deduction.

Step2: Analyze Joby's proof

Joby directly states \(\angle2 = \angle4\) and \(\angle1=\angle3\) using the vertical - angles - are - congruent property. But in a proof, we should not use the theorem we are trying to prove as a justification in the middle of the proof. His proof is circular (using the conclusion as a premise).

Answer:

Diana is correct, but Joby is not.