Questions: Which law can be used to determine that the conclusion is valid based on the given statements? Given: If an angle is acute, then it cannot be obtuse. angle A is acute. Conclusion: angle A cannot be obtuse. (A) Law of Detachment (B) Law of Syllogism (C) Law of Converse (D) The conclusion is not valid.

Solution Steps

To determine the validity of the conclusion, we need to identify the logical structure of the given statements. The first statement is a conditional statement: "If an angle is acute, then it cannot be obtuse." The second statement affirms the antecedent: "$\angle A$ is acute." The conclusion is that "$\angle A$ cannot be obtuse." This follows the Law of Detachment, which states that if a conditional statement is true and its antecedent is true, then the consequent must also be true.

We are given the conditional statement: "If an angle is acute, then it cannot be obtuse," which can be expressed as: \[ P \implies \neg Q \] where \( P \) is "an angle is acute" and \( Q \) is "an angle is obtuse." We also know that \( \angle A \) is acute, which means \( P \) is true.

According to the Law of Detachment, if the conditional statement \( P \implies \neg Q \) is true and \( P \) is true, then we can conclude that \( \neg Q \) is also true. Since \( P \) (i.e., \( \angle A \) is acute) is true, we can conclude: \[ \neg Q \text{ (i.e., } \angle A \text{ cannot be obtuse)} \]

The conclusion states that \( \angle A \) cannot be obtuse, which is equivalent to \( \neg Q \). Since we have established that \( \neg Q \) is true based on the given statements, the conclusion is valid.

Final Answer