Lesson 5-3

Indirect Proof

5-Minute Check on Lesson 5-25-Minute Check on Lesson 5-25-Minute Check on Lesson 5-25-Minute Check on Lesson 5-2 Transparency 5-3

Determine the relationship between the lengths of thegiven sides.1. RS, ST 2. RT, ST

Determine the relationship between the measuresof the given angles.

3. A, B 4. B, C

Refer to the figure.

5. Use the Exterior Angle Inequality Theorem to list allangles whose measures are less than m1.

6. Which angle has the greatestmeasure?Standardized Test Practice:

A CB D31 2 4

5-Minute Check on Lesson 5-25-Minute Check on Lesson 5-25-Minute Check on Lesson 5-25-Minute Check on Lesson 5-2 Transparency 5-3

Determine the relationship between the lengths of thegiven sides.1. RS, ST RS < ST 2. RT, ST RT > ST

Determine the relationship between the measuresof the given angles.

3. A, B mA < mB 4. B, C mB < mC

Refer to the figure.

5. Use the Exterior Angle Inequality Theorem to list allangles whose measures are less than m1.3, 4, 5, 6

6. Which angle has the greatestmeasure?Standardized Test Practice:

A CB D31 2 4

Objectives

• Use indirect proof with algebra

• Use indirect proof with geometry

Vocabulary

• Indirect reasoning – showing something to be false so that the opposite must be true

• Indirect proof – proving the opposite of what you assume is true

• Proof by contradiction – proving the assumption contradicts some fact, definition or theorem

Key Concept

• Step 1: Assume that the conclusion is false, so then the opposite is true.

• Step 2: Show that this assumption leads to a contradiction of the hypothesis, or some other fact, such as a definition, postulate, theorem or corollary

• Step 3: Point out that because the false conclusion leads to an incorrect statement, the original conclusion must be true (the opposite of what we assumed in step 1)

Algebraic ExampleMartha signed up for 3 classes at Wytheville Community

College for a little under $156. There was an administrative fee of $15, but the class costs varied. How can you show that at least one class cost less than $47?Given: Martha spent less than $156Prove: At least one class cost (x) less than $47

Step 1: Assume x $47Step 2: Then $47 + $47 + $47 + $15 $156Step 3: This contradicts what Martha paid, so the

assumption must be false.Therefore one class must cost less than $47!

Geometric Example

Step 1: Assume mK mLStep 2: By angle-side relationships, JL JKStep 3: This contradicts the given side lengths, so

the assumption must be falseTherefore, mK < mL !

J

K

L

8

7

5

Given: JKL with side lengths as shownProve: mK < mL

Answer: is a perpendicular bisector.

State the assumption you would make to start an indirect proof for the statement is not a perpendicular bisector.

Answer:

State the assumption you would make to start an indirect proof for the statement

State the assumption you would make to start an indirect proof for the statement m1 is less than or equal to m2.

Answer: m1 > m2

If m1 m2 is false, then m1 > m2.

State the assumption you would make to start an indirect proof for the statement If B is the midpoint of

and then is congruent to

The conclusion of the conditional statement is is congruent to The negation of the conclusion is is not congruent to

Answer: is not congruent to

State the assumption you would make to start an indirect proof of each statement.

a. is not an altitude.

b.

Answer:

Answer: is an altitude.

d. If is an angle bisector of MLP, then MLH is congruent to PLH.

c. mABC is greater than or equal to mXYZ.

Answer: mABC < mXYZ

Answer: MLH is not congruent to PLH.

Step 1 Assume that .

1Given: ----------- = 20 2y + 4

Prove: y -2

Indirect Proof:

Write an indirect proof.

Step 2 Substitute –2 for y in the equation

Substitution

Step 3 The assumption leads to a contradiction. Therefore, the assumption that must be false, which means that must be true.

This is a contradiction because the denominator cannot be 0.

Multiply.

Add.

Given: ABC with side lengths 8, 10, and 12 as shown.

Prove: mC > mA

Write an indirect proof.

Step 3 Since the assumption leads to a contradiction, the assumption must be false. Therefore, mC > mA.

Indirect Proof:

Step 1 Assume that

Step 2 By angle-side relationships,By substitution, This inequality is a false statement.

SHOPPING David bought four new sweaters for a little under $135. The tax was $7, but the sweater costs varied. How can you show that at least one of the sweaters cost less than $32?

Answer:

Given: David spent less than $135.

Prove: At least one of the sweaters x cost less than $32. That is,

Step 3 The assumption leads to a contradiction of a known fact. Therefore, the assumption that

must be false. Thus, at least one of the sweaters cost less than $32.

Step 1 Assume that none of the sweaters cost less than $32.

Indirect Proof:

Step 2 then the minimum total amount David spent is However, this is a

contradiction since David spent less than $135.

Summary & Homework

• Summary:– In an indirect proof, the conclusion is

assumed to be false and a contradiction is reached

• Homework: – pg 258-9: 4-6, 13, 14 Proofs: 11, 22