Question

consider the statement an acute angle, a, measures 0° < m∠a < 90°.
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.

d. the conditional statement is false.
c. if the conditional statement is false, which of the following is a counterexample for the conditional?
a. an acute angle with a measure of 180°.
b. an acute angle with a measure of 150°.
c. an acute angle with a measure of 100°.
d. the conditional statement is true.

Explanation:

Step1: Recall definition of conditional statement

A conditional statement is of the form "if p, then q". Here, p is "An angle is acute" and q is "The angle measures \(0^{\circ}

Step2: Recall definition of converse

The converse of "if p, then q" is "if q, then p". So the converse is "If \(0^{\circ}

Step3: Determine truth - value of conditional

By definition, an acute angle is an angle whose measure is between \(0^{\circ}\) and \(90^{\circ}\), so the conditional statement is true.

Step4: Determine truth - value of converse

Since an angle with measure between \(0^{\circ}\) and \(90^{\circ}\) is defined as an acute angle, the converse is true.

Step5: Write biconditional

A biconditional statement is of the form "p if and only if q". So the biconditional is "An angle is acute if and only if \(0^{\circ}

Answer:

a. If an angle is acute, then \(0^{\circ}b. If \(0^{\circ}c. The conditional statement is true.
d. An angle is acute if and only if \(0^{\circ}e. The converse statement is true.