5-4 Indirect Proof

1
5-4 Indirect Proof

• What is indirect reasoning?
• Who uses indirect reasoning?

2
You wrote paragraph, two-column, and flow proofs.

• Write indirect algebraic proofs.

• Write indirect geometric proofs.

3
Direct Reasoning

• In direct reasoning, you assume that the
  hypothesis is true and show that the conclusion
  must also be true.
• If it is 3pm on a school day, then academic
  classes at Marian High School are finished for
  the day.

4
Indirect Reasoning

• Indirect reasoning shows that a statement is true
  by proving that it cannot be false.
• Assume the oppositecontradict it.

5
(No Transcript)
6
Indirect Reasoning

• Marks car wont start. He knows that there
  are three likely reasons for this.
• His battery is dead
• His starter doesnt work.
• He is out of gas.
• When a cars starter needs to be replaced, the
  car is silent when you try to start it. If the
  battery is dead, the engine turns over slowly,
  if at all. When Mark tries to start the car, it
  sounds normal. What do you think is wrong with
  his car?

Out of gas!
7
Three Key Steps in Indirect Reasoning.

• Assume that the statement you are trying to prove
  is false.
• Show that this assumption leads to a
  contradiction of something you know is true.
• Conclude that your assumption was incorrect, so
  that the statement you originally wanted to prove
  must be true.

8
What would you assume for indirect reasoning?

• If it rains, then I will wash my car.
• It rains and I do not wash my car.

9
State the Assumption for Starting an Indirect
Proof
10
(No Transcript)
11
Write an indirect proof to show that if 2x 11
lt 7, then x gt 2. Given 2x 11 lt 7 Prove x gt 2
Step 1 Indirect Proof The negation of x gt 2 is x
2. So, assume that x lt 2 or x 2 is true. Step
2 Make a table with several possibilities for x
assuming x lt 2 or x 2.
12
Step 2 Make a table with several possibilities
for x assuming x lt 2 or x 2.
When x lt 2, 2x 11 gt 7 and when x 2, 2x 11
7.
Step 3 In both cases, the assumption leads to a
contradiction of the given information that 2x
11 lt 7. Therefore, the assumption that x 2 must
be false, so the original conclusion that x gt 2
must be true.
13
Which is the correct order of steps for the
following indirect proof? Given x 5 gt
18 Prove x gt 13
I. In both cases, the assumption leads to a
contradiction. Therefore, the assumption x 13
is false, so the original conclusion that x gt 13
is true. II. Assume x 13. III. When x lt 13, x
5 18 and when x lt 13, x 5 lt 18.
A. I, II, III B. I, III, II C. II, III, I D. III,
II, I
14
SHOPPING David bought four new sweaters for a
little under 135. The tax was 7, but the
sweater costs varied.Can David show that at
least one of the sweaters cost less than 32?
A. Yes, he can show by indirect proof that
assuming that every sweater costs 32 or more
leads to a contradiction. B. No, assuming every
sweater costs 32 or more does not lead to a
contradiction.
15
Indirect Proof
Step 3 Since the assumption leads to a
contradiction, the assumption must be false.
Therefore, m?K lt m?L.
16
A. Assume m?C m?A m?B. By angle-side
relationships, AB gt BC AC. Substituting, 12
10 8 or 12 18. This is a false
statement. B. Assume m?C m?A. By angle-side
relationships, AB BC. Substituting, 12 8.
This is a false statement.
17
Who uses Indirect Reasoning?

• Auto mechanics
• Physicians diagnosing diseases
• CSI
• Lawyers
• Eliminating possibilities that contradict a know
  fact can lead to the actual cause of a problem.

18

• What is indirect reasoning?
• In direct reasoning, you assume that the
  hypothesis is true and show that the conclusion
  must also be true.
• Who uses indirect reasoning?
• Auto mechanics, doctors, police, lawyers

19
5-4 Assignment

• Page 358, 11-20