Five-Minute Check (over Lesson 5–3)

Then/Now

New Vocabulary

Key Concept: How to Write an Indirect Proof

Example 1: State the Assumption for Starting an Indirect Proof

Example 2: Write an Indirect Algebraic Proof

Example 3: Indirect Algebraic Proof

Example 4: Indirect Proofs in Number Theory

Example 5: Geometry Proof

Over Lesson 5–3

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. RS > ST

B. RS = ST

C. RS < ST

D. no relationship

What is the relationship between the lengths of RS and ST?

______

Over Lesson 5–3

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. RT > ST

B. RT < ST

C. RT = ST

D. no relationship

What is the relationship between the lengths of RT and ST? ___ ___

Over Lesson 5–3

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. mA > mB

B. mA < mB

C. mA = mB

D. cannot determine relationship

What is the relationship between the measures of A and B?

Over Lesson 5–3

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. mB > mC

B. mB < mC

C. mB = mC

D. cannot determine relationship

What is the relationship between the measures of B and C?

Over Lesson 5–3

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. 3

B. 4

C. 6

D. all of the above

Using the Exterior Angle Inequality Theorem, which angle measure is less than m1?

Over Lesson 5–3

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

In ΔTRI, mT = 36, mR = 57, and mI = 87. List the sides in order from shortest to longest.

A. RI, IT, TR

B. IT, RI, TR

C. TR, RI, IT

D. RI, RT, IT

__ __ ___

__ __ ___

__ __ __

__ ___ __

You wrote paragraph, two-column, and flow proofs. (Lessons 2–5, 2–6, and 4–2)

• Write indirect algebraic proofs.

• Write indirect geometric proofs.

• indirect reasoning

• indirect proof

• proof by contradiction

State the Assumption for Starting an Indirect Proof

Answer: is a perpendicular bisector.

A. 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

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A.

B.

C.

D.

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A.

B.

C.

D.

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A.

B. MLH PLH

C.

D.

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 B C D

0% 0%0%0%

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?

A B

0%0%

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

PropertyNow 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 B C D

0% 0%0%0%

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.

A B

0%

0%

http://glencoe.mcgraw-hill.com/sites/0078884845/