This document discusses inequalities and indirect proofs in chapter 6 of a mathematics textbook. It introduces properties of inequalities, such as if a > b and c > 0, then ac > bc. It presents the exterior angle inequality theorem, which states that the measure of an exterior angle of a triangle is greater than the measure of either remote interior angle. It explores how to use properties of inequalities to prove this theorem indirectly. The document also discusses conditional statements and their converses, inverses, and contrapositives. It explains how to use Venn diagrams to represent conditionals and their logical equivalents. Finally, it introduces the law of syllogism for making indirect logical deductions.

1. Chapter 6: Inequalities
and Indirect Proofs
6-1 Inequalities
6-2 Inverses and Contrapositives

2. Inequalities
In this chapter you will work with segments having
unequal lengths and angles having unequal
measures.
B
A
C
B
C
A
D

3. 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
b
c
a
c
b
c
c

4. 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.
2
1
3
4
How can you use the Exterior Angle Theorem and the
properties of Inequalities to prove this theorem?

5. Review: Conditional statement
and its converse.
Statement
p→q
Converse (flip)
Flip
q→p
If p, then q.
If q, then p.
If today is Monday, then
tomorrow is Tuesday.
If tomorrow is Tuesday,
then today is Monday.

6. 
Statement
p→q
If p, then q.
If today is Monday, then
tomorrow is Tuesday.
Negate
Inverse (negate)
~p → ~q
If not p, then not q.
If today is not Monday, then
tomorrow is not Tuesday.
Conditional
statement and
its inverse.
(Negate the
hypothesis and
conclusion.

7. Statement
p→q
Converse (flip)
q→p
Flip
If p, then q.
If q, then p.
If today is Monday, then
tomorrow is Tuesday.
If tomorrow is Tuesday, then
today is Monday.
Negate
Negate
Inverse (negate)
Contrapositive
(flip and negate or
negate and flip)
~p → ~q
~q → ~p
If not p, then not q.
Flip
If today is not Monday, then
tomorrow is not Tuesday.
If not q, then not p.
If tomorrow is not Tuesday,
then today is not Monday.

8. Venn Diagrams
Use a Venn diagram to represent a conditional.
If p, then q.
q
Also represents: if not q, then not p.
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)
p

q

9. Is a statement logically equivalent to its
(a) converse or (b) inverse?
(a)
(b)
No, since you can be in the q and
still not be in the p. (converse)
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.
q
p
q
p

10. Statement
Converse (flip)
p→q
q→p
If p, then q.
If q, then p.
Logically Equivalent
Inverse (negate)
~p → ~q
If not p, then not q.
Logically Equivalent
Contrapositive
(flip and negate)
~q → ~p
If not q, then not p.

11. Law of Syllogism

If p → q and q → r are true
statements, then p → r is true.
q
p

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.
r