If two angles are supplementary, then the measures of the angles sum to 180^(∘).
True
Converse statement
If the measures of two angles sum to 180^(∘), then the two angles are supplementary.
True
Inverse statement
If two angles are not supplementary, then the measures of the angles do not sum to 180^(∘).
True
Contrapositive statement
If the measures of two angles do not sum to 180^(∘), then the two angles are not supplementary.
True
Practice makes perfect
Let's consider each of the statements one at a time using the given p and q.
p =& Two angles are supplementary
q =& The measures of the angles sum to180^(∘)
Conditional Statement
We can write the conditional statement, p→ q, in an if-then form.
If two angles are supplementary,
then the measures of the angles sum to 180^(∘).By the definition of supplementary angles we know that they have a sum of 180^(∘), so this is true.
Converse
The converse of a conditional statement, q→ p, exchanges the hypothesis and the conclusion of the conditional statement.
If the measures of two angles sum to 180^(∘),
then the two angles are supplementary.
By the definition of supplementary angles, we know the converse is also true.
Inverse
The inverse of a conditional statement, ~ p→ ~ q, requires us to negate the hypothesis and the conclusion of the conditional statement.
If two angles are not supplementary,
then the measures of the angles
do not sum to180^(∘).
Again, the definition of supplementary angles tells us that the angles have a sum of 180^(∘). Therefore, if two angles do not sum to 180^(∘), they cannot be supplementary. This is a true statement.
Contrapositive
The contrapositive of a conditional statement, ~ q→ ~ p, starts out with the converse of the conditional statement. Then we have to negate the hypothesis and the conclusion.
If the measures of two angles do not sum to 180^(∘),
then the two angles are not supplementary.
By the same logic used to know that the inverse is true, we know that the contrapositive is true.
