Tonya and Pearl each completed a separate proof to show that alternate interior angles AKL and FLK are congruent. Who completed the proof incorrectly? Explain.

Question

Line AB is parallel to EF, transversal GJ crosses line AB at K and crosses line EF at L.

Tonya's Proof

Statement Justification
1. line AB ∥ line EF with transversal segment GJ 1. Given
2. angle AKL is congruent to angle GKB 2. Vertical Angles Theorem
3. angle GKB is congruent to angle FLK 3. Corresponding Angles Theorem
4. angle AKL is congruent to angle FLK 4. Transitive Property

Pearl's Proof

Statement Justification
1. line AB ∥ line EF with transversal segment GJ 1. Given
2. angle AKL is congruent to angle GKB 2. Definition of Adjacent Angles
3. angle GKB is congruent to angle FLK 3. Corresponding Angles Theorem
4. angle AKL is congruent to angle FLK 4. Transitive Property

(Please complete in regular font. No titles, heading, etcetera)

Answer 1

Tonya completed the proof incorrectly.

In her proof, statement 2 claims that angle AKL is congruent to angle GKB using the Vertical Angles Theorem, but angle GKB is not the vertical angle corresponding to angle AKL. Vertical angles are formed by two intersecting lines, but in this case, angle AKL and angle GKB are formed by the transversal crossing parallel lines, not intersecting lines. Therefore, this justification is incorrect.

On the other hand, Pearl incorrectly claims that angle AKL is congruent to angle GKB by stating the Definition of Adjacent Angles in statement 2. However, angle AKL and angle GKB are not adjacent. The correct property to use would be the Corresponding Angles Theorem, but her proof maintains logical structure after that mistake.

Ultimately, both proofs contain errors, but Tonya's error is fundamentally more serious because it relies on a misapplication of the Vertical Angles Theorem.