Geometry

Question Analysis:
This question tests the ability to identify a correct application of the Law of Detachment.
It challenges students to recognize logical errors and correct reasoning in conditional statements.

Key Concept Explanation:
The Law of Detachment states that if a conditional statement is true (“If p, then q”) and the hypothesis (p) is known to be true, then the conclusion (q) must also be true.
It is invalid to assume the converse (i.e., “If q, then p”).

Step-by-Step Solution:
Analyze each choice and check if it follows the correct pattern:

A: "If a student studies, then they will pass the test. James passed the test, so he must have studied."
Incorrect. This assumes the converse is true, which is not necessarily correct.
A student can pass without studying.

B: "If a polygon is a square, then it has four right angles. A polygon has four right angles, so it must be a square."
Incorrect. Again, this assumes the converse is true.
A rectangle also has four right angles but is not necessarily a square.

C: "If an animal is a dog, then it has four legs. Rex is a dog, so Rex has four legs."
Correct. This follows the Law of Detachment:
Conditional statement: If an animal is a dog, then it has four legs. ✅ (True)
Given: Rex is a dog. ✅ (Matches the hypothesis)
Conclusion: Rex has four legs. ✅ (Follows from the Law of Detachment)

D: "If two angles are vertical angles, then they are congruent. Two angles are congruent, so th...