Question

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.
tonyas proof

1. \\( \overline { a b } \parallel \overline { e f } \\) with transversal \\( \overline { g j } \\) 1. given
2. \\( \angle a k l \cong \angle g k b \\) 2. vertical angles theorem
3. \\( \angle g k b \cong \angle f l k \\) 3. corresponding angles theorem
4. \\( \angle a k l \cong \angle f l k \\) 4. transitive property

pearls proof

1. \\( \overline { a b } \parallel \overline { e f } \\) with transversal \\( \overline { g j } \\) 1. given
2. \\( \angle a k l \cong \angle g k b \\) 2. definition of adjacent angles
3. \\( \angle g k b \cong \angle f l k \\) 3. corresponding angles theorem

Explanation:

• In Tonya's proof, she correctly used the Vertical Angles Theorem (which states that vertical angles are congruent) in step 2.
• In Pearl's proof, she made an error. The definition of adjacent angles (angles that share a common side and a common vertex and do not overlap) is not relevant for showing that \( \angle AKL\cong\angle GKB\). \( \angle AKL\) and \( \angle GKB\) are vertical angles, not adjacent angles.

Answer:

Pearl completed the proof incorrectly. She used the wrong justification (Definition of Adjacent Angles) in step 2 of her proof. The correct justification for \( \angle AKL\cong\angle GKB\) is the Vertical Angles Theorem.