Question

is the conclusion valid based on the given information? explain your reasoning. given: if two angles are congruent, then they have the same measure. if two angles are vertical angles, then they are congruent. conclusion: if two angles are vertical angles, then they have the same measure. a) valid; law of detachment b) valid; law of syllogism c) invalid; because the conclusion of the first statement is not the hypothesis of the second statement, no valid conclusion can be made. d) invalid; because the hypothesis of the first conditional statement is not met, the conclusion is invalid.

Explanation:

1. First, recall the Law of Syllogism: If we have two conditional statements \( p \to q \) and \( q \to r \), then we can conclude \( p \to r \).

• Let's define the statements:
• Let \( p \): "two angles are vertical angles".
• Let \( q \): "two angles are congruent".
• Let \( r \): "two angles have the same measure".
• The first given statement is \( q \to r \) (If two angles are congruent, then they have the same measure).
• The second given statement is \( p \to q \) (If two angles are vertical angles, then they are congruent).
• By the Law of Syllogism, we can combine \( p \to q \) and \( q \to r \) to get \( p \to r \), which is "If two angles are vertical angles, then they have the same measure" (the conclusion).
• Now let's analyze the options:
• Option A: Law of Detachment is for \( p \to q \) and \( p \) being true to conclude \( q \). Here we are using Law of Syllogism, not Detachment, so A is wrong.
• Option B: The conclusion is valid because we applied the Law of Syllogism correctly. The two conditional statements are \( p \to q \) and \( q \to r \), so we can infer \( p \to r \).
• Option C: The conclusion of the first statement (\( r \)) is not the hypothesis of the second statement (\( p \)), but the hypothesis of the second statement is \( p \) and the conclusion of the first is \( r \), and the middle term is \( q \). The Law of Syllogism uses the middle term \( q \) (conclusion of first, hypothesis of second) to link \( p \) and \( r \), so C is wrong.
• Option D: The hypothesis of the first conditional (\( q \)) is "two angles are congruent" and the second conditional's hypothesis is \( p \) ("two angles are vertical angles"). But we are using the Law of Syllogism, and the conclusion is valid, so D is wrong.

Answer:

B) valid; Law of Syllogism