Apply the properties of inequality to positive numbers

1
Chapter 6

• Apply the properties of inequality to positive
  numbers
• State the inverse and contrapositive of a
  conditional
• Write indirect proofs

2
Lecture 1 (6-1)

• Objectives
• Apply the properties of inequality to segment
  lengths and angle measures
• State and use the Exterior Angle Inequality
  Theorem

3
The Concept of Inequality

• Almost everything we have dealt with in Geometry
  thus far has been based on the properties of
  equality and congruence.
• Now we will deal with lengths of segments and
  measures of angles that are not equal.
• For these sorts of relationships, we need a set
  of properties that determine how to manipulate
  inequality relationships. These are called the

4
Properties of Inequality
5
A Prop. of Ineq.

• If you use any property of inequality as
  justification in a proof, the above abbreviated
  reason covers them all.

6
Example
C
Given ?ABC
1
B
Prove m?1gt m?B m?1gt m?C
A
STATEMENTS REASONS

• 1. m?1 m?B m?C 1. Ext. ? Theorem
• 2. m?1gt m?B 2.
  Prop. of Ineq. m?1gt m?C

7
Ext. ? Inequality Theorem (6-1)

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

R
T
1
2
S
8
Lecture 2 (6-2)

• Objectives
• State the inverse and contrapositive of a
  conditional.
• Learn how a Venn diagram can represent a
  conditional.
• Determine when statements are logically
  equivalent.

9
Review

• A conditional is an If-Then statement. The if
  part is called the hypothesis and the then part
  is the conclusion. Conditionals can be linked,
  one to another, to form a proof.
• The converse of a conditional has the hypothesis
  and conclusion reversed. If the converse has the
  same truth as the conditional, we call them a
  biconditional.

10
The Inverse

• If a conditional is rewritten so that both the
  hypothesis and conclusion are negated (not
  added to each), then the resulting statement is
  called the inverse of the conditional.

Conditional If lines are ?, then they form rt.
?s. Inverse If lines are not ?, then they do
not form rt. ?s.
11
The Contrapositive

• If a conditional is rewritten so that both the
  hypothesis and conclusion are negated (not
  added to each) and reversed, then the resulting
  statement is called the contrapositive of the
  conditional.

Conditional If lines are ?, then they form rt.
?s. Contrapositive If lines do not form rt.
?s, then they are not ?.
12
The Venn Diagram

• This is a simple logical drawing where circles
  represent the phrases of a conditional.

What conditional is implied by this drawing?
If a horse, then a mammal. If not a mammal, then
not a horse.
How are these conditionals related?
13
Logical Equivalence

• Two conditionals are said to be logically
  equivalent if the same Venn diagram can represent
  them both. This means that they have the same
  meaning, not just the same truth.

A conditional and its contrapositive are always
logically equivalent.
A converse and an inverse are also always
logically equivalent.
14
Summary of Conditionals

• Given If p, then q.
• Converse If q, then p.
• Inverse If not p, then not q.
• Contrapositive If not q, then not p.

p
q
q
p
15
Lecture 3 (6-3)

• Objectives
• Learn how to write an indirect proof

16
Direct Proof

• All of the proofs we have done this year have
  been direct, that is, we have proven the
  original, given conditional by logically linking
  its hypothesis to its conclusion.

But sometimes it is difficult or impossible to
reason directly. In these cases, an indirect
method may be used.
Indirect logic is very common, as the example
will show
17
Example

• Suppose you are helping a friend study for a test
  he has in the morning. You see him walk out of
  the test room and he is looking sad. You reason
  that he must have done poorly on the test,
  because if he had done well, he would be smiling.
• To reason this way, you stop considering the
  phrase if he does well, then he will smile and
  instead consider he is not smiling, and see
  where that logically leads.

18
Indirect Proof

• To perform an indirect proof, do the following
  steps in paragraph form

• Assume temporarily that the conclusion is not
  true.
• Reason logically until you run into a
  contradiction of a known (or given) fact.
• Point out that the temporary assumption must be
  false, and that therefor the conclusion is proven.

19

• Prove that the bases of a trapezoid have unequal
  lengths.

• Assume temporarily that the bases of a trapezoid
  are congruent.
• They are already parallel, because of the
  definition of a trapezoid.
• Because they are congruent and parallel, the
  figure must be a parallelogram by Th 5-5.
• But a parallelogram has two pairs of parallel
  sides, and a trapezoid only one by definition.
• Our temporary assumption that the bases could be
  congruent was wrong, and the bases of a trapezoid
  cannot be congruent.

20
Lecture 4 (6-4)

• Objectives
• State and apply the inequality theorems for one
  triangle.

21
Theorem 6-2

• If one side of a triangle is longer than a second
  side, then the angle opposite the first side is
  larger than the angle opposite the second side.

Y
X
Z
22
Theorem 6-3

• If one angle of a triangle is longer than a
  second angle, then the side opposite the first
  angle is larger than the side opposite the second
  angle.

Y
X
Z
23
Corollaries

• The perpendicular segment from a point to a line
  is the shortest segment between the point and the
  line.
• The perpendicular segment from a point to a plane
  is the shortest segment between the point and the
  plane.

24
The Triangle Inequality Th (6-4)

• The sum of the lengths of any two sides of a
  triangle must be greater than the length of the
  third side.

Y
X
Z
25
Lecture 5 (6-5)

• Objectives
• Learn and apply the inequality theorems for two
  triangles.

26
Theorem 6-5

• If two sides of one triangle are congruent to two
  sides of another triangle and the included angle
  of the first triangle is larger than the included
  angle of the second triangle, then the third side
  of the first triangle is longer than the third
  side of the second triangle.

27
X
A
Y
Z
C
B
28
Theorem 6-6

• If two sides of one triangle are congruent to two
  sides of another triangle and the third side of
  the first triangle is longer than the third side
  of the second triangle, then the included angle
  of the first triangle is larger than the included
  angle of the second triangle.

29
X
A
Y
Z
C
B