Michael and Derrick each completed a separate proof to show that corresponding angles AKG and ELK are congruent. Who completed the proof incorrectly? Explain.
Line AB is parallel to EF, transversal GJ crosses line AB at K and crosses line EF at L.
Michael's Proof
Statement Justification
1. line AB ∥ line EF with transversal segment GJ 1. Given
2. angle AKG is congruent to angle AKL 2. Vertical Angles Theorem
3. angle BKL is congruent to angle ELK 3. Alternate Interior Angles Theorem
4. angle AKG is congruent to angle ELK 4. Transitive Property
Derrick's Proof
Statement Justification
1. line AB ∥ line EF with transversal segment GJ 1. Given
2. angle AKG is congruent to angle BKL 2. Vertical Angles Theorem
3. angle BKL is congruent to angle ELK 3. Alternate Interior Angles Theorem
4. angle AKG is congruent to angle ELK 4. Transitive Property
The proof that is completed incorrectly is Michael's proof of the statement "corresponding angles AKG and ELK are congruent."
Both proofs start with the same given information – lines AB and EF are parallel, and transversal segment GJ crosses line AB at point K and crosses line EF at point L. Both proofs also rely on the same theorems – the vertical angles theorem and the alternate interior angles theorem. However, the difference is in the way the two proofs make the jump from the first three statements to the fourth statement.
In Michael's proof, statement number 3 is incorrect. Statement 3 in Michael's proof states that "angle BKL is congruent to angle ELK" based on the alternate interior angles theorem. However, this statement is not true because the interior angle BKL is not formed by the intersection of two straight lines from a point on the line AB and a point on the line EF, which is required for the alternate interior angles theorem to apply.
In contrast, Derrick's proof uses the vertical angles theorem before applying the transitive property in statement 4. The statement "angle AKG is congruent to angle ELK" in Derrick's proof is a result of applying the transitive property to the statement that "angle AKG is congruent to angle BKL" in statement 3 and the statement that "angle BKL is congruent to angle ELK" in statement 2, which are both results of applying the vertical angles theorem. This is a valid proof.
Therefore, Michael's proof is incorrect because of an incorrect application of the alternate interior angles theorem, while Derrick's proof is correct because it uses the vertical angles theorem and applies the transitive property correctly.
4x + 9y = -14
a. past
b. future perfect
c. present perfect
d. past perfect
Answer:
The coefficient is 2