Answer is a perpendicular bisector State the assumption you would make to start an indirect proof for the statement is not a perpendicular bisector Example 1 State the Assumption for Starting an Indirect Proof ID: 643087
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Slide1
ConceptSlide2
Example 1
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.Slide3
Example 1
State the Assumption for Starting an Indirect Proof
B.
State the assumption you would make to start an
indirect proof for the statement
3
x
= 4y
+ 1.
Answer:
3
x
≠
4
y
+ 1Slide4
Example 1
State the Assumption for Starting an Indirect ProofSlide5
A
B
C
D
Example 1
A.
B.
C.
D.
Slide6
A
B
C
D
Example 1
A.
B.
C.
D.
Slide7
A
B
C
D
Example 1
A.
B.
MLH
PLH
C.
D.
Slide8
Example 2
Write an Indirect Algebraic Proof
Write an indirect proof to show that if –2
x
+ 11 < 7, then
x
> 2.
Given: –2
x + 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.Slide9
Example 2
Write an Indirect Algebraic Proof
When
x
< 2, –2
x
+ 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.Slide10
Example 2
Write an Indirect Algebraic Proof
Step 3
In both cases, the assumption leads to a contradiction of the given information that
–2
x + 11 < 7. Therefore, the assumption that
x ≤ 2 must be false, so the original conclusion that x > 2 must be true.Slide11
Example 2
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.Slide12
A
B
C
D
Example 2
A.
I, II, III
B.
I, III, II
C.
II, III, I
D.
III, II, ISlide13
Example 3
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:
3
x
+ 15 < 156
Prove:
x
< 47
Indirect Proof:
Assume that none of the classes cost less than 47. That is,
x
≥ 47.
Slide14
Example 3
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.Slide15
A
B
Example 3
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?Slide16
Example 4
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.
__
x
3
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
.
__
x
3
__
x
3
__
x
3Slide17
Example 4
Indirect Proofs in Number Theory
Step 2
=
n
Substitution of assumption
__
x
3
x
= 3
n
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 primeSlide18
Example 4
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.
__
x
3
__
x
3Slide19
A
B
C
D
Example 4
A.
2
k + 1
B.
3kC.
k
+ 1
D.
k
+ 3
You can express an even integer as 2
k
for some integer
k
. How can you express an odd integer?Slide20
Example 5
Geometry Proof
Given:
Δ
JKL
with side lengths 5, 7, and 8 as shown.Prove: mK < mL
Write an indirect proof.
Slide21
Example 5
Geometry Proof
Step 3
Since the assumption leads to a contradiction, the assumption must be false. Therefore,
m
K < mL.
Indirect Proof:
Step 1
Assume that
Step 2
By angle-side relationships, By substitution, . This inequality is a false statement.Slide22
Example 5
Which statement shows that the assumption leads to a contradiction for this indirect proof?
Given:
Δ
ABC
with side lengths 8, 10, and 12 as shown.
Prove:
m
C >
m
A
Slide23
A
B
Example 5
A.
Assume
m
C
≥
m
A
+
m
B.
By angle-side relationships,
AB
>
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.
Prove:
m
C >
m
A