Question

1. is this a valid argument? given: if an angle is acute, then it cannot be obtuse ∠a is acute. conclusion:∠a cannot be obtuse. image of a right triangle with angles 30°, 60° options: yes. this argument uses the law of detachment. no. this is an improper use of the law of detachment. yes. this argument uses the law of syllogism. no. this is an improper use of the law of syllogism.

Explanation:

Step1: Recall Law of Detachment

The Law of Detachment states that if we have a conditional statement \( p \to q \) (where \( p \) is the hypothesis and \( q \) is the conclusion) and we know that \( p \) is true, then we can conclude that \( q \) is true.

Step2: Identify \( p \) and \( q \)

In the given problem:

• The conditional statement is "If an angle is acute ( \( p \) ), then it cannot be obtuse ( \( q \) )", so \( p \): an angle is acute, \( q \): it cannot be obtuse.
• We are given that \( \angle A \) is acute, which means \( p \) is true for \( \angle A \).

Step3: Apply Law of Detachment

Since \( p \to q \) is true (given conditional) and \( p \) is true ( \( \angle A \) is acute), by the Law of Detachment, \( q \) must be true ( \( \angle A \) cannot be obtuse). Also, the Law of Syllogism involves two conditional statements to form a new conditional, which is not the case here. So the argument uses the Law of Detachment correctly.

Answer:

Yes. This argument uses the Law of Detachment.