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What is the definition of an obtuse angle?
| Type of statement | Statement | True or False? |
|---|---|---|
| Conditional statement | If an angle measure is 167^(∘), then the angle is obtuse. | True |
| Converse statement | If an angle is obtuse, then the angle measure is 167^(∘). | False |
| Inverse statement | If an angle measure is not 167^(∘), then the angle is not obtuse. | False |
| Contrapositive statement | If an angle is not obtuse, then the angle measure is not 167^(∘). | True |
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^(∘)
By the definition of obtuse angles we know that they are between 90^(∘) and 180^(∘), which means this is true.
The converse of a conditional statement, q→ p, exchanges the hypothesis and the conclusion of the conditional statement. If an angle is obtuse, then the angle measure is $167^(∘)$. This is not a true statement, as an obtuse angle can have a different measure than 167^(∘).
The inverse of a conditional statement, ~ p→ ~ q, requires us to negate the hypothesis and the conclusion of the conditional statement. If an angle measure is not $167^(∘)$, then the angle is not obtuse. Again, this is not a true statement, as an obtuse angle can have a different measure than 167^(∘).
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 an angle is not obtuse , then the angle measure is not $ 167^(∘)$. This is true, since an angle that is not obtuse cannot have an angle measure that is in the interval 90≤ v ≤ 180^(∘).