concept. example 1 state the assumption for starting an indirect proof answer: is a perpendicular...
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State the Assumption for Starting an Indirect Proof
Answer: is a perpendicular bisector.
State the assumption you would make to start an indirect proof for the statement
is not a perpendicular bisector.
State the Assumption for Starting an Indirect Proof
B. State the assumption you would make to start an
indirect proof for the statement
3x = 4y + 1.
Answer: 3x ≠ 4y + 1
State the Assumption for Starting an Indirect Proof
Write an Indirect Algebraic Proof
Write an indirect proof to show that if –2x + 11 < 7, then x > 2.
Given: –2x + 11 < 7
Prove: x > 2
Step 1 Indirect Proof:
The negation of x > 2 is x ≤ 2. So, assume that x < 2 or x = 2 is true.
Step 2 Make a table with several possibilities for x assuming x < 2 or x = 2.
Write an Indirect Algebraic Proof
When x < 2, –2x + 11 > 7 and when x = 2, –2x + 11 = 7.
Step 2 Make a table with several possibilities for x assuming x < 2 or x = 2.
Write an Indirect Algebraic Proof
Step 3 In both cases, the assumption leads to a contradiction of the given information that
–2x + 11 < 7. Therefore, the assumption that x ≤ 2 must be false, so the original conclusion that x > 2 must be true.
Which is the correct order of steps for the following indirect proof.
Given: x + 5 > 18
Proof: x > 13
I. In both cases, the assumption leads to a contradiction. Therefore, the assumption x ≤ 13 is false, so the original conclusion that x > 13 is true.
II. Assume x ≤ 13.
III. When x < 13, x + 5 = 18 and when x < 13, x + 5 < 18.
A. A
B. B
C. C
D. D
A. I, II, III
B. I, III, II
C. II, III, I
D. III, II, I
Indirect Algebraic Proof
EDUCATION Marta signed up for three classes at a community college for a little under $156. There was an administration fee of $15, and the class costs are equal. How can you show that each class cost less than $47?
Let x be the costs of the three classes.
Step 1 Given: 3x + 15 < 156
Prove: x < 47Indirect Proof:Assume that none of the classes cost less than 47. That is, x ≥ 47.
Indirect Algebraic Proof
Step 2 If x ≥ 47 then x + x + x + 15 ≥ 47 + 47 + 47 + 15 or x + x + x + 15 ≥ 156.
Step 3 This contradicts the statement that the total cost was less than $156, so the assumption that x ≥ 47 must be false. Therefore, one class must cost less than 47.
A. A
B. B
A. Yes, he can show by indirect proof that assuming that a sweater costs $32 or more leads to a contradiction.
B. No, assuming a sweater costs $32 or more does not lead to a contradiction.
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?
Indirect Proofs in Number Theory
Write an indirect proof to show that if x is a prime
number not equal to 3, then is not an integer.__x3
Step 1 Given: x is a prime number.
Prove: is not an integer.
Indirect Proof: Assume is an integer.
This means = n for some integer n.
__x3
__x3
__x3
Indirect Proofs in Number Theory
Step 2 = n Substitution of assumption__x3
x = 3n Multiplication Property
Now determine whether x is a prime number. Since x ≠ 3, n ≠ 1. So x is a product of two factors, 3 and some number other than 1.
Therefore, x is not a prime
Indirect Proofs in Number Theory
Step 3 Since the assumption that is an integer
leads to a contradiction of the given
statement, the original conclusion that
is not an integer must be true.
__x3
__x3
A. A
B. B
C. C
D. D
A. 2k + 1
B. 3k
C. k + 1
D. k + 3
You can express an even integer as 2k for some integer k. How can you express an odd integer?
Geometry Proof
Given: ΔJKL with side lengths 5, 7, and 8 as shown.
Prove: mK < mL
Write an indirect proof.
Geometry Proof
Step 3 Since the assumption leads to a contradiction, the assumption must be false. Therefore, mK < mL.
Indirect Proof:
Step 1 Assume that
Step 2 By angle-side relationships, By substitution, . This inequality is a false statement.
Which statement shows that the assumption leads to a contradiction for this indirect proof?
Given: ΔABC with side lengths 8, 10, and 12 as shown.
Prove: mC > mA
A. A
B. B
A. Assume mC ≥ mA + mB. By angle-side relationships, AB > BC + AC. Substituting, 12 ≥ 10 + 8 or 12 ≥ 18. This is a false statement.
B. Assume mC ≤ mA. By angle-side relationships, AB ≤ BC. Substituting, 12 ≤ 8. This is a false statement.
Prove: mC > mA