Question

write the biconditional as a conditional and its converse. then determine whether the biconditional is true or false. an angle is acute if and only if it has a measure less than 90°. conditional: if select choice, then select choice. converse: if an angle measures less than 90°, then it is select choice. the conditional and the converse are select choice, so the biconditional is select choice.

Explanation:

Step1: Define Conditional Statement

A conditional statement is in the form "If \( p \), then \( q \)". For the biconditional "An angle is acute if and only if it has a measure less than \( 90^\circ \)", the conditional part is: If an angle is acute, then it has a measure less than \( 90^\circ \). So the first "Select Choice" for the conditional's \( p \) (hypothesis) is "an angle is acute", and \( q \) (conclusion) is "it has a measure less than \( 90^\circ \)".

Step2: Define Converse Statement

The converse of a conditional "If \( p \), then \( q \)" is "If \( q \), then \( p \)". So the converse here is: If an angle has a measure less than \( 90^\circ \), then it is acute. So the "Select Choice" for the converse's \( p \) is "an angle has a measure less than \( 90^\circ \)" and \( q \) is "it is acute".

Step3: Check Validity of Conditional and Converse

• Conditional: If an angle is acute, then its measure is less than \( 90^\circ \). By definition of an acute angle, this is true.
• Converse: If an angle's measure is less than \( 90^\circ \), then it is acute. Angles with measure between \( 0^\circ \) and \( 90^\circ \) are acute (negative angles or zero angles are not typically considered, but in standard angle definitions for this context, angles less than \( 90^\circ \) and greater than \( 0^\circ \) are acute; assuming we are talking about positive angles in the context of basic geometry, this is true. Wait, actually, in standard terms, an acute angle is defined as an angle whose measure is between \( 0^\circ \) and \( 90^\circ \). So if an angle has measure less than \( 90^\circ \) (and greater than \( 0^\circ \)), it is acute. So the converse is also true.
• Since both the conditional and the converse are true, the biconditional is true.

Step4: Fill in the Blanks

• Conditional: If \(\boldsymbol{\text{an angle is acute}}\), then \(\boldsymbol{\text{it has a measure less than } 90^\circ}\).
• Converse: If \(\boldsymbol{\text{an angle has a measure less than } 90^\circ}\), then \(\boldsymbol{\text{it is acute}}\).
• The conditional and the converse are both \(\boldsymbol{\text{true}}\), so the biconditional is \(\boldsymbol{\text{true}}\).

Answer:

• Conditional: If \(\boldsymbol{\text{an angle is acute}}\), then \(\boldsymbol{\text{it has a measure less than } 90^\circ}\).
• Converse: If \(\boldsymbol{\text{an angle has a measure less than } 90^\circ}\), then \(\boldsymbol{\text{it is acute}}\).
• The conditional and the converse are both \(\boldsymbol{\text{true}}\), so the biconditional is \(\boldsymbol{\text{true}}\).