Geometry

Question Analysis:
This question involves identifying which option does not correctly apply the Law of Detachment.
The main focus is on understanding the logical structure of the conditional statement and recognizing when the Law of Detachment is misapplied.

Key Concept Explanation:
The Law of Detachment states that if a conditional statement ("If P, then Q") is true, and P is true, then Q must also be true.
However, it does not allow us to reverse the conditional statement ("If Q, then P") without additional information.

Step-by-Step Solution:
Option A:
Statement: Mia worked 40 hours this week. Therefore, she will receive a full paycheck.
Analysis: This correctly applies the Law of Detachment. P is true (Mia worked at least 40 hours), so Q must be true (she will receive a full paycheck).
Conclusion: This is valid.

Option B:
Statement: Ethan received a full paycheck. Therefore, he must have worked at least 40 hours this week.
Analysis: This reverses the conditional statement.
It assumes "If Q, then P," which is not logically valid unless the original statement is biconditional.
Conclusion: This does not apply the Law of Detachment correctly.

Option C:
Statement: Lucas worked 45 hours this week. Therefore, he will receive a full paychec...