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In a right triangle, the longest side is the hypotenuse.
The aim is to prove that the hypotenuse is the longest side of a right triangle. For this case, consider △ABC, where ∠A is a right angle.
Assume temporarily that the negation of the statement is true.
In a right triangle, the longest side is not its hypotenuse.
From the assumption, at least one of the legs of the right triangle is larger than the hypotenuse. Let AC be the leg whose length is greater than the length of the hypotenuse BC. Therefore, by the Triangle Longer Side Theorem, the measure of the angle opposite AC is greater than the measure of the angle opposite BC.
Since m∠A=90∘ and m∠B is greater than m∠A, the sum of the measures of these two angles is greater than 180∘.p | q | p⇒q | ¬(p⇒q) | ¬q | p∧¬q |
---|---|---|---|---|---|
T | T | T | F | F | F |
T | F | F | T | T | T |
F | T | T | F | F | F |
F | F | T | F | T | F |