Section 5-4: Indirect Reasoning

March 7, 2012

Warm-up

Warm-up: Practice 5-3: p. 58, 1-13

Warm-up

Warm-up

Warm-up

Questions on Homework?

Section 5-4: Indirect Reasoning Objectives: Today you will learn to

use indirect reasoning to prove a statement.

Section 5-4: Negation

Negation – the negation of a statement has the opposite truth value.

Statement: “Benson is the capital of NC.” Negation: “Benson is NOT the capital of NC.”

Statement: “∠ABC is obtuse.” Negation: “∠ABC is not obtuse.”

Section 5-4: Inverse

Inverse – negates both the hypothesis and the conclusion.

Statement: “If a figure is a square, then it is a rectangle.”

Inverse: (negate both): “If a figure is not a square, then it is not a rectangle.”

(Note: Inverses do not always have the same truth value as the original statement.)

Section 5-4: Contrapositive

Contrapositive – switches the hypothesis and conclusion and negates them both.

Statement: “If a figure is a square, then it is a rectangle.”

Contrapositive: (switch and negate both): “If a figure is not a rectangle, then it is not a square.”

(Note: Contrapositives always have the same truth value as the original statement. They are equivalent statements.)

Section 5-4: Indirect Reasoning

Indirect Reasoning: When all possibilities are considered and all

but one are proved false. The remaining possibility must be true.

Section 5-4: Indirect Reasoning

You get home from school and your brother tells you Hannah called.

You know two Hannahs, but your brother doesn’t know which one called.

You remember that one of the Hannahs you know has band practice after school, so it couldn’t have been her.

It must have been the other Hannah.

Section 5-4: Indirect Proofs

Indirect Proof: A proof that uses indirect reasoning. You

know that a statement or its negation is true. Both cannot be true.

Section 5-4: Indirect Proofs

Steps to Writing an Indirect Proof (p. 266)1. Assume the opposite (negation) of what you want

to prove.

2. Show that this assumption leads to a contradiction.

3. Conclude that the assumption must be false and that what you want to prove must be true.

Section 5-4: Indirect Proofs

You get home from school and your brother tells you Hannah called. You know two Hannahs, but your brother doesn’t know which one called.

Assumption: It was Hannah “A”Contradiction: You remember Hannah “A” has band practice after school, so it couldn’t have been her.

Conclusion: It must have been the other Hannah.

First Step: Assume Opposite

Example 1: Prove: “A triangle cannot have two right

angles.”Assume: A triangle does have two right

angles.

Example 2: Prove: m∠A < m∠BAssume: m∠A ≥ m∠B

First Step: Assume Opposite

Example 3: Prove: Quadrilateral QUIZ does not have

four acute angles.Assume: Quadrilateral QUIZ does have four

acute angles.

Example 4:Prove: ΔABC ≅ ΔXYZAssume: ΔABC ≆ ΔXYZ

Second Step: Identify Contradictions

Example 5:

I. ΔABC is an acute triangle.II. ΔABC is a scalene triangle.III. ΔABC is an equilateral triangle.

II and III contradict each other

Second Step: Identify Contradictions

Example 6:

I. P, Q, and R are coplanarII. P, Q, and R are collinearIII. m∠PQR = 60

II and III contradict each other

Second Step: Identify Contradictions

Example 7:

I. ΔABC is scaleneII. m∠A < m∠BIII. m∠A = m∠C

I and III contradict each other

Second Step: Identify Contradictions

Example 8:

I. m∠A - m∠B = 0II. m∠B < m∠AIII. ∠A and ∠B are supplementary

I and II contradict each other

Write an Indirect Proof: Example 9

Prove: ΔABC cannot contain 2 obtuse angles.

1. Assume the opposite of what you want to prove: ΔABC does contain 2 obtuse angles. Let ∠A and ∠B be obtuse.

2. Find contradiction: If ∠A and ∠B are obtuse, m∠A > 90 and m∠B > 90. So, m∠A + m∠B > 180. Since m∠C > 0, then m∠A + m∠B + m∠C > 180. This contradicts the Triangle Angle-Sum Theorem.

3. Conclusion: The assumption in Step 1 must be wrong. So, ΔABC cannot contain 2 obtuse angles

Write an Indirect Proof:

Example 10

Write an Indirect Proof: Example 11

Wrap-up Today you learned to use indirect

reasoning to prove a statement. Tomorrow you’ll learn about triangle

inequalities. Quiz on 5-1 to 5-3 tomorrow!

Homework p. 267-269, # 10-21, 31, 37, 38, 40 p. 272, 3rd column only: 3-33