Consider the following conditional statement: If a triangle is a right triangle, then the acute angles are complementary (their sum is 90°). When you use indirect reasoning to prove this statement, what contradiction arises?
A.
You have a right triangle with no right angles.
B.
You have a triangle with more than one right angle.
C.
You have two acute angles that sum to be more than 180°.
D.
You have a triangle with two obtuse angles
2007-05-16 06:46:33 · 5 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
A.
Assume that the angles apart from the known angle y total some amount x which is not equal to 90deg.
x + y = 180deg. (angles of a triangle).
x is not 90deg. (assumption made).
Therefore 180deg - x is not 90deg.
y is therefore not 90deg.
As y is not 90deg, and x is not 90deg then the triangle has no right angles.
2007-05-16 07:02:35 · answer #1 · answered by Anonymous · 0⤊ 0⤋
When you use indirect reasoning, you assume the opposite of what you have to prove and show that leads to a contradiction. In this case, you have to prove that the acute angles of the triangle are complementary, so you first assume that they are not complementary, that is, their sum is not 90 degrees. Then using the fact that the measures of the angles of a triangle must sum to 180 degrees, you see immediately that the third angle cannot be 90 degrees. So you have a right triangle with no right angle.
This question isn't very good, since you can arrive at any contradiction if falsities are assumed to be true.
2007-05-16 14:02:37 · answer #2 · answered by bictor717 3 · 0⤊ 0⤋
If the two acute angles are NOT complimentary and there sum is NOT 90, then the 3rd angle cannot be 90 as the sum is supposed to be 180.
A) You have a right triangle with no right angles.
2007-05-16 14:13:22 · answer #3 · answered by tbolling2 4 · 0⤊ 0⤋
You have to statements:
P = "A triangle has a right angle".
Q = "The other two angles are complementray".
You want to show that P => Q. To show this indirectly is the same as proving ~Q => ~P, i.e., proving that if two angles in a triangle are not complementary, then the third angle is not right.
If it was given up front that the triangle is right, then you arrive at the contradiction A: a right triangle with no right angles.
A. Bogomolny
http://www.cut-the-knot.org
2007-05-16 14:09:34 · answer #4 · answered by Pythagorean Fan 1 · 0⤊ 0⤋
I don't know. Isn't the statement axiomatically true?
2007-05-16 13:56:27 · answer #5 · answered by gebobs 6 · 0⤊ 0⤋