Question

write the negation of the hypothesis and the negation of the conclusion for the conditional: if a polygon’s interior angle measures total 1, and?, then the polygon is a decagon. what is the negation of the hypothesis?
a. if a polygon is a decagon, then the polygon’s interior angle measures total 1, and?
b. if a polygon’s interior angle measures, do not total 1, and? then the polygon is not a decagon
c. if a polygon’s interior angle measures do not total 1, and? then the polygon is a decagon
d. a decagon is a decagon.
e. a polygon’s interior angle measures total 1, and?
f. a polygon’s interior angle measures do not total 1, and?
g. if a polygon’s interior angle measures total 1, and? then the polygon is not a decagon

Explanation:

The original conditional statement is "If a polygon’s interior angle measures total \(1, \text{ and}?\) (assuming it's a typo, likely \(180(n - 2)\) or similar, but the key is the hypothesis is "a polygon’s interior angle measures total [some value]" and conclusion is "the polygon is a decagon"). The negation of the hypothesis (the "if" part) is the statement that the hypothesis is false. So we need to find the option that states the polygon’s interior angle does not measure that total. Looking at the options, option F (assuming the typo is corrected, likely "A polygon’s interior angle measures do not total \(1, \text{ and}?\)" but from the options, F is "A polygon’s interior angle measures do not total \(1, \text{ and}?\)" (matching the negation of the hypothesis: the interior angle total is not as stated). Wait, re - evaluating: the hypothesis is "a polygon’s interior angle measures total [x]", so its negation is "a polygon’s interior angle measures do not total [x]". Among the options, option F (text: "A polygon’s interior angle measures do not total \(1, \text{ and}?\)") is the negation of the hypothesis. Wait, maybe the original problem had a typo, but from the options, the negation of the hypothesis (the "if" part) is the statement that the interior angle total is not what's in the hypothesis. So the correct option for the negation of the hypothesis is F (assuming the option F is "A polygon’s interior angle measures do not total \(1, \text{ and}?\)"). Wait, looking at the options again:

Option E: "A polygon’s interior angle measures total \(1, \text{ and}?\)" – this is the hypothesis, not its negation.

Option F: "A polygon’s interior angle measures do not total \(1, \text{ and}?\)" – this is the negation of the hypothesis (since it's the opposite of the "if" part).

Answer:

F. A polygon’s interior angle measures do not total \(1, \text{ and}?\)