Question 556480: How do you set up an indirect proof & solve it?
Found 2 solutions by richard1234, EdenWolf:
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! Assume the opposite of what you are trying to prove is true, find a contradiction.
For example, if you wanted to prove that there are infinitely many prime numbers, first assume that there are a finite number of primes. If you let N = 2*3*5*7*...*p where p is the "largest" prime, then N+1 must either be prime or be the product of two primes. However, N+1 cannot be divisible by any prime less than or equal to p, so if N+1 is composite, a larger prime divides it --> contradiction.
Answer by EdenWolf(517)  (Show Source):
You can put this solution on YOUR website! First, identify the conjecture to be proven. This is called the original conjecture. Let's say that the original conjecture is "The sum of the measures of the angles of a triangle cannot be greater than 180 degrees."
Now, let's make an assumption, which is basically the opposite of the conjecture. For this example, the assumption will be: "The sum of the measures of the angles of a triangle CAN be greater than 180 degrees."
Indirect proofs can be set up as two column proofs or as paragraph proofs. For this example, we will use a paragraph proof.
Let's draw a triangle ABC first.
 .
Draw a line ACD that cointains segment AC. This would actually be the first statement. The reason for that would be that through any two points, there is exactly one line. Each segment is contained on exactly one line.
Angle DCB is an exterior angle because of the definition of an exterior angle.
Angle DCB and angle ACB are a linear pair. Why? Because of the definition of a linear pair.
Now, the measure of angle DCB + the measure of angle ACB = 180 degrees by the Linear Pair Theorem.
The measure of angle ABC + the measure of angle CAB = the measure of angle DCB by the Exterior Angles Theorem.
Our last step: The measure of angle ABC + the measure of angle CAB + the measure of angle ACB = 180 degrees by substitution.
Now we must write a contradiction: "Therefore, the sum of the measures of angles of a triangle CANNOT be greater than 180 degrees."
|
|
|
