geometry section 5-4 1112
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Section 5-4Indirect Proof
Tuesday, March 13, 2012
Essential Questions
How do you write indirect algebraic proofs?
How do you write indirect geometric proofs?
Tuesday, March 13, 2012
Vocabulary
1. Indirect Reasoning:
2. Indirect Proof:
3. Proof by Contradiction:
Tuesday, March 13, 2012
Vocabulary
1. Indirect Reasoning: Assuming that the conclusion of a conditional is false, then showing this assumption leads to a contradiction
2. Indirect Proof:
3. Proof by Contradiction:
Tuesday, March 13, 2012
Vocabulary
1. Indirect Reasoning: Assuming that the conclusion of a conditional is false, then showing this assumption leads to a contradiction
2. Indirect Proof: Prove a statement to be true by assuming that which you are trying to prove to be false, then show that is leads to a contradiction
3. Proof by Contradiction:
Tuesday, March 13, 2012
Vocabulary
1. Indirect Reasoning: Assuming that the conclusion of a conditional is false, then showing this assumption leads to a contradiction
2. Indirect Proof: Prove a statement to be true by assuming that which you are trying to prove to be false, then show that is leads to a contradiction
3. Proof by Contradiction: Another name for indirect proof
Tuesday, March 13, 2012
Steps to Write an Indirect Proof
Tuesday, March 13, 2012
Steps to Write an Indirect Proof1. Identify what you are trying to prove, then assume the conclusion to be false.
Tuesday, March 13, 2012
Steps to Write an Indirect Proof1. Identify what you are trying to prove, then assume the conclusion to be false.
2. Follow a logical argument based on your false assumption to find a contradiction in your new assumption.
Tuesday, March 13, 2012
Steps to Write an Indirect Proof1. Identify what you are trying to prove, then assume the conclusion to be false.
2. Follow a logical argument based on your false assumption to find a contradiction in your new assumption.
3. Identify the contradiction, thus proving that the original conclusion must be true.
Tuesday, March 13, 2012
Example 1State the assumption necessary to start an indirect
proof for each statement.
a. EF is not a perpendicular bisector.
b. 3x = 4y + 1
c. If B is the midpoint of LH and LH = 26, then BH is congruent to LB.
Tuesday, March 13, 2012
Example 1State the assumption necessary to start an indirect
proof for each statement.
a. EF is not a perpendicular bisector.
EF is a perpendicular bisector.
b. 3x = 4y + 1
c. If B is the midpoint of LH and LH = 26, then BH is congruent to LB.
Tuesday, March 13, 2012
Example 1State the assumption necessary to start an indirect
proof for each statement.
a. EF is not a perpendicular bisector.
EF is a perpendicular bisector.
b. 3x = 4y + 1 3x ≠ 4y + 1
c. If B is the midpoint of LH and LH = 26, then BH is congruent to LB.
Tuesday, March 13, 2012
Example 1State the assumption necessary to start an indirect
proof for each statement.
a. EF is not a perpendicular bisector.
EF is a perpendicular bisector.
b. 3x = 4y + 1 3x ≠ 4y + 1
c. If B is the midpoint of LH and LH = 26, then BH is congruent to LB.
BH is not congruent to LB.
Tuesday, March 13, 2012
Example 2Write an indirect proof to show that if −2x + 11 < 7,
then x > 2.
Tuesday, March 13, 2012
Example 2Write an indirect proof to show that if −2x + 11 < 7,
then x > 2.
Assume that x ≤ 2.
Tuesday, March 13, 2012
Example 2Write an indirect proof to show that if −2x + 11 < 7,
then x > 2.
Assume that x ≤ 2.
Then −2(2) + 11 < 7
Tuesday, March 13, 2012
Example 2Write an indirect proof to show that if −2x + 11 < 7,
then x > 2.
Assume that x ≤ 2.
Then −2(2) + 11 < 7
−4 + 11 < 7
Tuesday, March 13, 2012
Example 2Write an indirect proof to show that if −2x + 11 < 7,
then x > 2.
Assume that x ≤ 2.
Then −2(2) + 11 < 7
−4 + 11 < 7
7 < 7
Tuesday, March 13, 2012
Example 2Write an indirect proof to show that if −2x + 11 < 7,
then x > 2.
Assume that x ≤ 2.
Then −2(2) + 11 < 7
−4 + 11 < 7
7 < 7
Not true
Tuesday, March 13, 2012
Example 2Write an indirect proof to show that if −2x + 11 < 7,
then x > 2.
Assume that x ≤ 2.
Then −2(2) + 11 < 7
−4 + 11 < 7
7 < 7
Not true
Also, −2(1) + 11 < 7
Tuesday, March 13, 2012
Example 2Write an indirect proof to show that if −2x + 11 < 7,
then x > 2.
Assume that x ≤ 2.
Then −2(2) + 11 < 7
−4 + 11 < 7
7 < 7
Not true
Also, −2(1) + 11 < 7
−2 + 11 < 7
Tuesday, March 13, 2012
Example 2Write an indirect proof to show that if −2x + 11 < 7,
then x > 2.
Assume that x ≤ 2.
Then −2(2) + 11 < 7
−4 + 11 < 7
7 < 7
Not true
Also, −2(1) + 11 < 7
−2 + 11 < 7
9 < 7
Tuesday, March 13, 2012
Example 2Write an indirect proof to show that if −2x + 11 < 7,
then x > 2.
Assume that x ≤ 2.
Then −2(2) + 11 < 7
−4 + 11 < 7
7 < 7
Not true
Also, −2(1) + 11 < 7
−2 + 11 < 7
9 < 7
Also not true
Tuesday, March 13, 2012
Example 2Write an indirect proof to show that if −2x + 11 < 7,
then x > 2.
Assume that x ≤ 2.
Then −2(2) + 11 < 7
−4 + 11 < 7
7 < 7
Not true
Also, −2(1) + 11 < 7
−2 + 11 < 7
9 < 7
Also not true
Both cases provide a contradiction, so therefore x > 2 must be true.
Tuesday, March 13, 2012
Example 3Write an indirect proof to show that if x is a prime
number, then x/3 is not an integer.
Tuesday, March 13, 2012
Example 3Write an indirect proof to show that if x is a prime
number, then x/3 is not an integer.
Assume x/3 is an integer.
Tuesday, March 13, 2012
Example 3Write an indirect proof to show that if x is a prime
number, then x/3 is not an integer.
Assume x/3 is an integer.
Then,
x
3= n.
Tuesday, March 13, 2012
Example 3Write an indirect proof to show that if x is a prime
number, then x/3 is not an integer.
Assume x/3 is an integer.
Then,
x
3= n.
This means x = 3n.
Tuesday, March 13, 2012
Example 3Write an indirect proof to show that if x is a prime
number, then x/3 is not an integer.
Assume x/3 is an integer.
Then,
x
3= n.
This means x = 3n.
This cannot be true, because if x is prime, then 3 cannot be a factor of x as shown in the equation
x = 3n. Therefore, x/3 is not an integer.
Tuesday, March 13, 2012
Example 4Prove by contradiction.
K
J L
8 5
7Given: ∆JKL with side lengths 5, 7, and 8.
Prove: m∠K < m∠L
Tuesday, March 13, 2012
Example 4Prove by contradiction.
K
J L
8 5
7Given: ∆JKL with side lengths 5, 7, and 8.
Prove: m∠K < m∠L
Assume m∠K ≥ m∠L.
Tuesday, March 13, 2012
Example 4Prove by contradiction.
K
J L
8 5
7Given: ∆JKL with side lengths 5, 7, and 8.
Prove: m∠K < m∠L
Assume m∠K ≥ m∠L.
Then, by angle-side relationships, JL ≥ JK.
Tuesday, March 13, 2012
Example 4Prove by contradiction.
K
J L
8 5
7Given: ∆JKL with side lengths 5, 7, and 8.
Prove: m∠K < m∠L
Assume m∠K ≥ m∠L.
Then, by angle-side relationships, JL ≥ JK.
This mean that 7 ≥ 8, which is not true.
Tuesday, March 13, 2012
Example 4Prove by contradiction.
K
J L
8 5
7Given: ∆JKL with side lengths 5, 7, and 8.
Prove: m∠K < m∠L
Assume m∠K ≥ m∠L.
Then, by angle-side relationships, JL ≥ JK.
This mean that 7 ≥ 8, which is not true.
So the assumption m∠K ≥ m∠L is false, so therefore m∠K < m∠L
Tuesday, March 13, 2012
Check Your Understanding
Explore problems #1-10 on page 354
Tuesday, March 13, 2012
Problem Set
Tuesday, March 13, 2012
Problem Set
p. 355 #11-29 odd, 43, 51, 53
“It's not that I'm so smart, it's just that I stay with problems longer.” – Albert Einstein
Tuesday, March 13, 2012
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