5.5 Indirect Proofs

5.5 Indirect Proofs
Learning Objective: I can use indirect reasoning to write proofs.

Draw

The goal of this game is to fill in the empty squares with numbers. The numbers 1, 2, 3, and 4 must appear once in each row and once in each column.

Draw

The goal of this game is to fill in the empty squares with numbers. The numbers 1, 2, 3, and 4 must appear once in each row and once in each column.

A proof involving indirect reasoning.
- Often a statement and its negation are the only possibilities.

Indirect Proofs

All possibilities are considered and then all but one possibility are proved false. The remaining possibility must be true.

Indirect Reasoning

Important Vocabulary

​Steps to Writing an Indirect Proof

Step 1: State as a temporary assumption the opposite (negation) of what you want to prove.
Step 2: Show that this temporary assumption leads to a contradiction.
Step 3: Conclude that the temporary assumption must be false and that what you want to prove must be true.

• The opposite of "do not have" is "do have."

• Assume temporarily that you do have soccer practice today.

B) You do not have soccer practice today.

• The opposite of "is divisible by" is "is not divisible by."

• Assume temporarily that n is not divisible by 5.

​A) An integer n is divisble by 5.

Writing the First Step of an Indirect Proof:
Suppose you want to write an indirect proof of each statement. As the first step of the proof, what would you assume?

Multiple Choice

Suppose you want to write an indirect proof of each statement. As the first step of the proof, what would you assume?

Triangle BOX is not acute.

• Assume temporarily that triangle BOX is acute.

• Assume temporarily that triangle BOX isn't acute.

Multiple Choice

Suppose you want to write an indirect proof of each statement. As the first step of the proof, what would you assume?

At least one pair of shoes you bought cost more than $25..

• Assume temporarily that no pair of shoes you bought cost more than $25.

• Assume temporarily that one pair of shoes you bought cost more than $25.

Multiple Choice

What assumption would you make to start the indirect proof of ∆ABC≅∆DEF?

• ∆ABC≠∆DEF
• ∆ABC≇∆DEF
• <ABC≠<DEF
• AB≅DE

Multiple Choice

What assumption would you make to start the indirect proof of AB≅BC?

• AB≠BC
• BA≅CB
• B is the midpoint of AC
• B is the perpendicular bisector oa AC

Multiple Choice

What assumption would you make to start the indirect proof of if 3x+7>13, then x>2?

• x≤2
• x<2
• x>2
• x=2

Identifying Contradictions

Which two statements contradict each other?

1) FG ∥ KL
2) FG ≅ KL
3) FG ⊥ KL

Identifying Contradictions

Which two statements contradict each other?

1) FG ∥ KL
2) FG ≅ KL
3) FG ⊥ KL

• Segments can be ∥ and ≅. Statements I and II do not contradict each other.

• Segments can be ≅ and ⊥. Statements II and III do not contradict each other.

• ∥ segments do not intersect, so they cannot be ⊥. Statements I and III contradict each other.

Multiple Select

Which TWO statements below contradict each other.

• Quadrilateral is a square.

• Quadrilateral has four equal sides.

• Quadrilateral has no right angles.

Multiple Choice

Which two statements contradict each other?

I. Triangle XYZ is acute.

II. Triangle XYZ is scalene.

III. Triangle XYZ is equiangular.

• I and II

• I and III

• II and III

Multiple Select

Which TWO statements below are contradictory for triangle ABC.

• <B = <C

• AB = BC

• < B = 90^o

Writing an Indirect Proof

Given: ΔABC is scalene
Prove: ∠A, ∠B, and ∠C are all different measures

Steps to Writing an Indirect Proof:
1) Temporarily assume the opposite of what you want to prove.


2) Show this assumption leads to a contradiction.


3) Conclude assumption is false and that what you want to prove is true.

Open Ended

Write an Indirect Proof:

Given: 7(x+y) = 70 and x ≠ 4

Prove: y ≠ 6