1. LESSON 5–4
Indirect Proof

2. Five-Minute Check (over Lesson 5–3) TEKS 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
Lesson Menu

3. What is the relationship between the lengths of RS and ST?
___
A. RS > ST
B. RS = ST
C. RS < ST
D. no relationship
5-Minute Check 1

4. What is the relationship between the lengths of RT and ST?
___
A. RT > ST
B. RT < ST
C. RT = ST
D. no relationship
5-Minute Check 2

5. What is the relationship between the measures of A and B?
A. mA > mB
B. mA < mB
C. mA = mB
D. cannot determine relationship
5-Minute Check 3

6. What is the relationship between the measures of B and C?
A. mB > mC
B. mB < mC
C. mB = mC
D. cannot determine relationship
5-Minute Check 4

7. Using the Exterior Angle Inequality Theorem, which angle measure is less than m1?
B. 4
C. 6
D. all of the above
5-Minute Check 5

8. In ΔTRI, mT = 36, mR = 57, and mI = 87
In ΔTRI, mT = 36, mR = 57, and mI = 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 __
___
5-Minute Check 6

9. Mathematical Processes G.1(E), G.1(G)
Targeted TEKS
G.6(D) Verify theorems about the relationships in
triangles, including proof of the Pythagorean
Theorem, the sum of interior angles, base
angles of isosceles triangles, midsegments,
and medians, and apply these relationships to
solve problems.
Mathematical Processes
G.1(E), G.1(G)
TEKS

10. You wrote paragraph, two-column, and flow proofs.
Write indirect algebraic proofs.
Write indirect geometric proofs.
Then/Now

11. proof by contradiction
indirect reasoning
indirect proof
proof by contradiction
Vocabulary

12. Concept

13. Answer: is a perpendicular bisector.
State the Assumption for Starting an Indirect Proof
A. State the assumption you would make to start an indirect proof for the statement is not a perpendicular bisector.
Answer: is a perpendicular bisector.
Example 1

14. State the Assumption for Starting an Indirect Proof
Example 1

15. A. B. C. D.
Example 1

16. A. B. C. D.
Example 1

17. A.
B. MLH  PLH C. D.
Example 1

18. 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.
Example 2

19. When x < 2, –2x + 11 > 7 and when x = 2, –2x + 11 = 7.
Write an Indirect Algebraic Proof
Step 2 Make a table with several possibilities for x assuming x < 2 or x = 2.
When x < 2, –2x + 11 > 7 and when x = 2, –2x + 11 = 7.
Example 2

20. 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.
Example 2

21. Which is the correct order of steps for the following indirect proof?
Given: x + 5 > 18
Prove: 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.
Example 2

22. A. I, II, III
B. I, III, II
C. II, III, I
D. III, II, I
Example 2

23. Indirect Algebraic Proof
EDUCATION Marta signed up for three classes at a community college for a little under $1226. There was an administration fee of $360 and a student life fee of $118. The class costs are equal. How can you show that each class cost less than $250?
Step 1 Given: Marta spent less than $1226 Prove: At least one of the classes x cost less than $250. That is, if 3x < 1226, then x < 250.
Indirect Proof: Assume that none of the classes cost less than $250. That is, x ≥ 250.
Example 3

24. Indirect Algebraic Proof
Step 2 If x ≥ 250, then x + x + x ≥ 1226 or ≥ 1226.
Step 3 This contradicts the statement that the total cost was less than $1226, so the assumption that x ≥ 250 must be false. Therefore, one class must cost less than $250.
Example 3

25. SHOPPING David bought four new sweaters for a little under $135
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.
Example 3

26. Step 1 Given: x is a prime number not equal to 3.
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 not equal to 3.
Prove: is not an integer.
Indirect Proof: Assume is an integer. This means = n for some integer n. __ x 3
Example 4

27. Step 2 = n Substitution of assumption x 3
Indirect Proofs in Number Theory
Step 2 = n Substitution of assumption __ x 3
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
Example 4

28. 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 3
Example 4

29. You can express an even integer as 2k for some integer k
You can express an even integer as 2k for some integer k. How can you express an odd integer?
A. 2k + 1
B. 3k
C. k + 1
D. k + 3
Example 4

30. Given: ΔJKL with side lengths 5, 7, and 8 as shown.
Geometry Proof
Given: ΔJKL with side lengths 5, 7, and 8 as shown.
Prove: mK < mL
Write an indirect proof.
Example 5

31. Indirect Proof: Step 1 Assume that
Geometry Proof
Indirect Proof:
Step 1 Assume that
Step 2 By angle-side relationships, By substitution, This inequality is a false statement.
Step 3 Since the assumption leads to a contradiction, the assumption must be false. Therefore, mK < mL.
Example 5

32. Given: ΔABC with side lengths 8, 10, and 12 as shown.
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: mC > mA
Example 5

33. A. Assume mC ≥ mA + mB. By angle-side relationships, AB > BC + AC. Substituting, ≥ 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.
Example 5

34. LESSON 5–4
Indirect Proof