Geometry

Question Analysis:
This question involves identifying which option represents an incorrect use of the Law of Detachment.
The main focus is on understanding the logical structure of conditional statements 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: If a quadrilateral is a square, then it has four right angles. Quadrilateral ABCD is a square, so it has four right angles.
Analysis: This correctly applies the Law of Detachment. P is true (Quadrilateral ABCD is a square), so Q must be true (it has four right angles).
Conclusion: This is valid.

Option B:
Statement: If a student studies every day, then they will pass the exam. Sophia studied every day, so she passed the exam.
Analysis: This correctly applies the Law of Detachment. P is true (Sophia studied every day), so Q must be true (she passed the exam).
Conclusion: This is valid.

Option C:
Statement: If a number is divisible by 10, then it is also divisible by 5. The number 30 is divisible by 10, so it is also divi...