Author: Tim Heppner
Last saved by: MHE
Company: MCGRAW-HILL

Geometry	Chapter 5

Lesson 5-3

Example 1  State Assumptions
State the assumption you would make to start an indirect proof of each statement.
a.	m∠ABC = m∠DBG
	m∠ABC ≠ m∠DBG

b.	 ⊥ 
	 is not perpendicular to .

c.	m∠1 = 58
	m∠1 ≠ 58

d.	△ABC is a right triangle.
	△ABC is not a right triangle.

Example 2  Algebraic Proof
Given:	3x + 6 < 15
Prove:	x < 3

Indirect Proof:
Step 1	Assume that x is not less than 3. That is, assume that x > 3 or x = 3.

Step 2	Make a table with several possibilities for x given
that x > 3 or x = 3. 
This is a contradiction because when x > 3 or x = 3
3x + 6 >15.

Step 3	In both cases, the assumption leads to the contradiction 
of a known fact. Therefore, the assumption that x is not less 
	than 3 must be false, which means that x < 3 must be true.

x

3x + 6

| 3 | 15
| 4 | 18
| 5 | 21
| 6 | 24
| 7 | 27

Example 3  Use Indirect Proof
SHOPPING  Frankie bought two shirts for just over $54, before tax. A few weeks 
later, her friend, Tanya asked her how much each shirt cost. Frankie could not 
remember the individual prices. Use inductive reasoning to show that at least one 
of the shirts cost more than $27.
 
Given:	The two shirts cost more than $54.
Prove:	At least one of the shirts cost more than $27.
	That is, if 2x > 54 then x > 27.

Indirect Proof:
Step 1	Assume that neither shirt cost more than $27. That is, x ≤ 27 and y ≤ 27.

Step 2	If x ≤ 27 and y ≤ 27, then x + y ≤ 54. That is a contradiction because we know that the two shirts cost more than $54.

Step 3	The assumption leads to the contradiction of a known fact. Therefore, the assumption that x ≤ 27 and y ≤ 27 must be false. Thus, at least one of the shirts had to have cost more than $27.

Example 4  Geometry Proof
Given:	∠ABC is not a right angle.	
Prove:	m∠ABC ≠ 90

Indirect Proof:
Step 1	Assume m∠ABC = 90.

Step 2	If m∠ABC = 90, then by definition of a right angle, ∠ABC is a right angle. However, this contradicts the given statement.

Step 3	Since the assumption leads to a contradiction, the assumption must be false. Therefore, m∠ABC ≠ 90.