Question

write the following statement as a biconditional statement and determine if it is valid or not. a pair of angles where one is acute and one is obtuse are supplementary. answer attempt 1 out of 3 ~ , if and only if, ~ .

Explanation:

Step1: Recall Biconditional Structure

A biconditional statement has the form "P if and only if Q", where P and Q are statements. First, identify P (hypothesis) and Q (conclusion) from the original statement. The original statement: "A pair of angles where one is acute and one is obtuse are supplementary." So, P: "A pair of angles has one acute and one obtuse angle", Q: "The pair of angles is supplementary".

Step2: Form Biconditional

Write the biconditional as: "A pair of angles is supplementary if and only if one angle is acute and one angle is obtuse."

Step3: Test Validity

To check validity, we need to see if both "If P, then Q" and "If Q, then P" are true.

• Test "If P, then Q": Take an acute angle (e.g., \(30^\circ\)) and an obtuse angle (e.g., \(100^\circ\)). Their sum is \(30 + 100 = 130^\circ

eq 180^\circ\), so they are not supplementary. Thus, "If P, then Q" is false.

• Test "If Q, then P": Supplementary angles can be two right angles (\(90^\circ + 90^\circ = 180^\circ\)), which are neither acute nor obtuse (right angles are \(90^\circ\)). So, "If Q, then P" is false. Since both implications are not true, the biconditional is invalid.

Answer:

A pair of angles is supplementary if and only if one angle is acute and one angle is obtuse. This biconditional is invalid.