Question

a conditional statement is given.
\if the measures of two angles add to 180°, then the angles are supplementary.\
for each statement below, determine its relationship to the conditional statement.

statement	relationship to the conditional
if two angles are supplementary, then the measures of their angles add to 180°
if two angles are not supplementary, then the measures of their angles do not add to 180°
the measures of two angles add to 180° if and only if the angles are supplementary

Explanation:

1. Let the conditional statement be "If \(p\), then \(q\)" where \(p\) is "the measures of two - angles add to \(180^{\circ}\)" and \(q\) is "the angles are supplementary".

• The first statement "If the measures of two angles do not add to \(180^{\circ}\), then the angles are not supplementary" is the contra - positive of the conditional statement. The contra - positive of "If \(p\), then \(q\)" is "If not \(p\), then not \(q\)" and a conditional statement and its contra - positive are logically equivalent.
• The second statement "If two angles are supplementary, then the measures of their angles add to \(180^{\circ}\)" is the converse of the conditional statement. The converse of "If \(p\), then \(q\)" is "If \(q\), then \(p\)".
• The third statement "If two angles are not supplementary, then the measures of their angles do not add to \(180^{\circ}\)" is the inverse of the conditional statement. The inverse of "If \(p\), then \(q\)" is "If not \(q\), then not \(p\)".
• The fourth statement "The measures of two angles add to \(180^{\circ}\) if and only if the angles are supplementary" is the biconditional of the conditional statement. The biconditional " \(p\) if and only if \(q\)" is equivalent to \((p

ightarrow q)\land(q
ightarrow p)\).

Answer:

1. Contra - positive
2. Converse
3. Inverse
4. Biconditional