chapter 6 and 6-2
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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
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