Question

determine if the conditional is true or false. if it is false, find a counterexample. if an angle is acute, then it measures 80 degrees. choose the correct answer below. a. the conditional is true. b. the conditional is false. an angle that measures 75 degrees is also acute. c. the conditional is false. an angle that measures 120 degrees is also acute. d. the conditional is false. an angle can be added to the 80 - degree angle to make it not acute.

Explanation:

Step1: Recall the definition of an acute angle

An acute angle is an angle whose measure is greater than \(0^\circ\) and less than \(90^\circ\).

Step2: Analyze the conditional statement

The conditional statement is "If an angle is acute, then it measures 80 degrees". We need to check if all acute angles must measure 80 degrees.

Step3: Evaluate each option

• Option A: Since there are acute angles with measures other than 80 degrees (e.g., 75 degrees, 45 degrees, etc.), the conditional is not true. So A is incorrect.
• Option B: A 75 - degree angle is greater than \(0^\circ\) and less than \(90^\circ\), so it is acute. This shows that there exists an acute angle that does not measure 80 degrees, so the conditional is false, and this is a valid counterexample.
• Option C: A 120 - degree angle is greater than \(90^\circ\), so it is an obtuse angle, not an acute angle. So this is not a valid counterexample.
• Option D: The statement in D does not provide a counterexample to the original conditional. The original conditional is about the measure of an acute angle being 80 degrees, and D is about adding an angle to an 80 - degree angle, which is irrelevant.

Answer:

B. The conditional is false. An angle that measures 75 degrees is also acute.