Chapter 6: Inequalities and Indirect Proofs

6-1 Inequalities6-2 Inverses and Contrapositives

InequalitiesIn this chapter you will work with segments having

unequal lengths and angles having unequal measures.

Complete each conclusion by inserting one of the symbols: <, =, or >.

1. Given: AC > AB; AB > BCConclusion: AC _____ BC.

2. Given: Conclusion: _____ _____

Answers: 1. > 2. >; >

A

C

B

D

C

B

A

Properties of Inequalities

If a > b and c ≥ d, then a + c > b + d

If a > b and c > 0, then ac > bc and

If a > b and c < 0, then ac < bc and

If a > b and b > c, then a > c If a = b + c and c > 0, then a > b

a bc c

a bc c

Theorem 6-1 The Exterior Angle Inequality Theorem

The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.

How can you use the Exterior Angle Theorem and the properties of Inequalities to prove this theorem?

1

2

3 4

Statement

p → qIf p, then q.

 If today is Monday, then tomorrow is Tuesday.

Converse (flip)

q → pIf q, then p.

If tomorrow is Tuesday, then today is Monday.

 

Flip

Review: Conditional statement and its converse.

Statement

p → qIf p, then q.

 If today is Monday, then tomorrow is Tuesday.

 Inverse (negate)

~p → ~qIf not p, then not q.

 If today is not Monday, then tomorrow is not Tuesday.

Negate

Conditional statement and its inverse. (Negate the hypothesis and conclusion.

Statement

p → qIf p, then q.

 If today is Monday, then tomorrow is Tuesday.

 

Converse (flip)

q → pIf q, then p.

  If tomorrow is Tuesday, then today is Monday.

Inverse (negate)

~p → ~qIf not p, then not q.

 If today is not Monday, then tomorrow is not Tuesday.

 

Contrapositive (flip and negate or

negate and flip)

~q → ~pIf not q, then not p.

  If tomorrow is not Tuesday, then today is not Monday.

 

Negate

Flip

Negate

Flip

Use a Venn diagram to represent a conditional.If p, then q. Also represents: if not q, then not p. (logically equivalent)

A statement and its contrapositive are logically equivalent.

If q, then p. Also represents: if not p, then not q. (logically equivalent)

Venn Diagrams

p

q

p

q

Is a statement logically equivalent to its (a) converse or (b) inverse?

p

q(a) No, since you can be in the q and

still not be in the p. (converse)

(b) No, since you can be in the ~p but still be in the q (instead of the ~q. (inverse)

The converse and the inverse of a statement are logically equivalent.

p

q

Statement

p → qIf p, then q.

 

Converse (flip)

q → pIf q, then p.

 

Inverse (negate)

~p → ~qIf not p, then not q.

 

Contrapositive (flip and negate)

~q → ~pIf not q, then not p.

 

Logically Equivalent

Logically Equivalent

Law of Syllogism If p → q and q → r are true

statements, then p → r is true.

Given the following true statements:• If a bird is the fastest bird on land, then it is the

largest of all birds. • If a bird is the largest of all birds, then it is an

ostrich. We can conclude:

• If a bird is the fastest bird on land, then it is an ostrich.

p

qr