Question

consider the statement \an obtuse angle, b, measures ( 90^circ < mangle b < 180^circ ).\ a. write the statement as a conditional statement. b. if the conditional statement is true, write the converse of the conditional. c. if the conditional statement is false, give a counterexample for the conditional. d. if the converse statement is true, write a true biconditional statement. e. if the converse statement is false, give a counterexample for the converse. ... c. if the conditional statement is false, which of the following is a counterexample for the conditional? a. an obtuse angle with a measure of ( 70^circ ). b. an obtuse angle with a measure of ( 50^circ ). c. an obtuse angle with a measure of ( 90^circ ) d. the conditional statement is true.

Explanation:

First, recall the definition of an obtuse angle: an angle whose measure is greater than \(90^\circ\) and less than \(180^\circ\) (\(90^\circ < m\angle B < 180^\circ\)). So the conditional statement "If an angle \(B\) is obtuse, then \(90^\circ < m\angle B < 180^\circ\)" is true because it matches the definition.

Now, analyze the options for the counterexample (though the conditional is true, let's check each option):

• Option A: An obtuse angle with \(70^\circ\) – but \(70^\circ\) is acute (less than \(90^\circ\)), so it can't be an obtuse angle. So this is invalid as a counterexample (and the angle isn't obtuse).
• Option B: An obtuse angle with \(50^\circ\) – \(50^\circ\) is acute, so it's not an obtuse angle. Invalid.
• Option C: An obtuse angle with \(90^\circ\) – \(90^\circ\) is a right angle, not obtuse (obtuse is greater than \(90^\circ\), not equal). So it's not an obtuse angle. Invalid.
• Option D: The conditional statement is true – since the definition of an obtuse angle is exactly \(90^\circ < m\angle B < 180^\circ\), the conditional "If angle \(B\) is obtuse, then \(90^\circ < m\angle B < 180^\circ\)" is true. So there is no counterexample needed because the conditional is true, which matches option D.

Answer:

D. The conditional statement is true.