Question

michael and derrick each completed a separate proof to show that corresponding angles akg and elk are congruent. who completed the proof incorrectly? explain. michaels proof: statement justification 1. $overline{ab}paralleloverline{ef}$ with transversal $overline{gj}$ 1. given 2. $angle akgcongangle akl$ 2. vertical angles theorem 3. $angle bklcongangle elk$ 3. alternate interior angles theorem 4. $angle akgcongangle elk$ 4. transitive property derricks proof: statement justification 1. $overline{ab}paralleloverline{ef}$ with transversal $overline{gj}$ 1. given 2. $angle akgcongangle bkl$ 2. vertical angles theorem 3. $angle bklcongangle elk$ 3. alternate interior angles theorem 4. $angle akgcongangle elk$ 4. transitive property

Explanation:

Step1: Analyze Michael's proof

In step 2 of Michael's proof, $\angle AKG$ and $\angle AKL$ are not vertical - angles. Vertical angles are formed by two intersecting lines. Here, $\angle AKG$ and $\angle AKL$ are adjacent angles that form a linear pair. So, Michael's proof is incorrect.

Step2: Analyze Derrick's proof

Derrick's proof is correct. $\angle AKG$ and $\angle BKL$ are vertical angles (by the Vertical Angles Theorem as they are formed by the intersection of $\overline{AB}$ and $\overline{GJ}$). $\angle BKL$ and $\angle ELK$ are alternate - interior angles (by the Alternate Interior Angles Theorem since $\overline{AB}\parallel\overline{EF}$ with transversal $\overline{GJ}$). Then, by the Transitive Property, $\angle AKG\cong\angle ELK$.

Answer:

Michael completed the proof incorrectly because in his step 2, $\angle AKG$ and $\angle AKL$ are not vertical angles but adjacent angles forming a linear pair.