Transcript 2.1-2.4 PowerPoint

Using
Inductive
Reasoning
to
Using
Inductive
Reasoning
to
2-1
2-1 Make
Conjectures
Make
Conjectures
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Geometry
Holt
McDougal
Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Example 1A: Identifying a Pattern
Find the next item in the pattern.
January, March, May, ...
Alternating months of the year make up the pattern.
The next month is July.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Example 1B: Identifying a Pattern
Find the next item in the pattern.
7, 14, 21, 28, …
Multiples of 7 make up the pattern.
The next multiple is 35.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
When several examples form a pattern and you
assume the pattern will continue, you are
applying inductive reasoning. Inductive
reasoning is the process of reasoning that a rule
or statement is true because specific cases are
true. You may use inductive reasoning to draw a
conclusion from a pattern. A statement you
believe to be true based on inductive reasoning is
called a conjecture.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Example 2A: Making a Conjecture
Complete the conjecture.
The sum of two positive numbers is
? .
List some examples and look for a pattern.
1 + 1 = 2 3.14 + 0.01 = 3.15
3,900 + 1,000,017 = 1,003,917
The sum of two positive numbers is positive.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Check It Out! Example 2
Complete the conjecture.
The product of two odd numbers is
? .
List some examples and look for a pattern.
11=1
33=9
5  7 = 35
The product of two odd numbers is odd.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
To show that a conjecture is always true, you must
prove it.
To show that a conjecture is false, you have to find
only one example in which the conjecture is not true.
This case is called a counterexample.
A counterexample can be a drawing, a statement, or a
number.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Inductive Reasoning
1. Look for a pattern.
2. Make a conjecture.
3. Prove the conjecture or find a
counterexample.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Example 4A: Finding a Counterexample
Show that the conjecture is false by finding a
counterexample.
For every integer n, n3 is positive.
Pick integers and substitute them into the expression
to see if the conjecture holds.
Let n = 1. Since n3 = 1 and 1 > 0, the conjecture holds.
Let n = –3. Since n3 = –27 and –27  0, the
conjecture is false.
n = –3 is a counterexample.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Example 4B: Finding a Counterexample
Show that the conjecture is false by finding a
counterexample.
Two complementary angles are not congruent.
45° + 45° = 90°
If the two congruent angles both measure 45°, the
conjecture is false.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Check It Out! Example 4b
Show that the conjecture is false by finding a
counterexample.
Supplementary angles are adjacent.
23°
157°
The supplementary angles are not adjacent,
so the conjecture is false.
Holt McDougal Geometry
2-2
2-2 Conditional
ConditionalStatements
Statements
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Holt
McDougal
Geometry
Geometry
2-2 Conditional Statements
By phrasing a conjecture as an if-then statement,
you can quickly identify its hypothesis and
conclusion.
Holt McDougal Geometry
2-2 Conditional Statements
Example 1: Identifying the Parts of a Conditional
Statement
Identify the hypothesis and conclusion of each
conditional.
A. If today is Thanksgiving Day, then today is
Thursday.
Hypothesis: Today is Thanksgiving Day.
Conclusion: Today is Thursday.
B. A number is a rational number if it is an
integer.
Hypothesis: A number is an integer.
Conclusion: The number is a rational number.
Holt McDougal Geometry
2-2 Conditional Statements
Check It Out! Example 1
Identify the hypothesis and conclusion of the
statement.
"A number is divisible by 3 if it is divisible by 6."
Hypothesis: A number is divisible by 6.
Conclusion: A number is divisible by 3.
Holt McDougal Geometry
2-2 Conditional Statements
Many sentences without the words if and then can
be written as conditionals. To do so, identify the
sentence’s hypothesis and conclusion by figuring
out which part of the statement depends on the
other.
Holt McDougal Geometry
2-2 Conditional Statements
Example 2A: Writing a Conditional Statement
Write a conditional statement from the
following.
An obtuse triangle has exactly one obtuse
angle.
An obtuse triangle
has exactly one obtuse angle.
Identify the
hypothesis and the
conclusion.
If a triangle is obtuse, then it has exactly one
obtuse angle.
Holt McDougal Geometry
2-2 Conditional Statements
Check It Out! Example 2
Write a conditional statement from the
sentence “Two angles that are complementary
are acute.”
Two angles that are complementary Identify the
hypothesis
are acute.
and the
conclusion.
If two angles are complementary, then they
are acute.
Holt McDougal Geometry
2-2 Conditional Statements
A conditional statement has a truth value of either
true (T) or false (F). It is false only when the
hypothesis is true and the conclusion is false.
To show that a conditional statement is false, you
need to find only one counterexample where the
hypothesis is true and the conclusion is false.
Holt McDougal Geometry
2-2 Conditional Statements
Example 3A: Analyzing the Truth Value of a
Conditional Statement
Determine if the conditional is true. If false,
give a counterexample.
If this month is August, then next month is
September.
When the hypothesis is true, the conclusion is
also true because September follows August.
So the conditional is true.
Holt McDougal Geometry
2-2 Conditional Statements
Example 3B: Analyzing the Truth Value of a
Conditional Statement
Determine if the conditional is true. If false,
give a counterexample.
If two angles are acute, then they are congruent.
You can have acute angles with measures of
80° and 30°. In this case, the hypothesis is
true, but the conclusion is false.
Since you can find a counterexample, the
conditional is false.
Holt McDougal Geometry
2-2 Conditional Statements
Related Conditionals
Definition
A conditional is a statement
that can be written in the form
"If p, then q."
Holt McDougal Geometry
Symbols
pq
2-2 Conditional Statements
Related Conditionals
Definition
The converse is the statement
formed by exchanging the
hypothesis and conclusion.
Holt McDougal Geometry
Symbols
qp
2-2 Conditional Statements
Related Conditionals
Definition
The inverse is the statement
formed by negating the
hypothesis and conclusion.
Holt McDougal Geometry
Symbols
~p  ~q
2-2 Conditional Statements
Related Conditionals
Definition
The contrapositive is the
statement formed by both
exchanging and negating the
hypothesis and conclusion.
Holt McDougal Geometry
Symbols
~q  ~p
2-2 Conditional Statements
Check It Out! Example 4
Write the converse, inverse, and contrapostive
of the conditional statement “If an animal is a
cat, then it has four paws.” Find the truth value
of each.
If an animal is a cat, then it has four paws.
Holt McDougal Geometry
2-2 Conditional Statements
Check It Out! Example 4
If an animal is a cat, then it has four paws.
Converse: If an animal has 4 paws, then it is a cat.
There are other animals that have 4 paws that are not
cats, so the converse is false.
Inverse: If an animal is not a cat, then it does not
have 4 paws.
There are animals that are not cats that have 4 paws,
so the inverse is false.
Contrapositive: If an animal does not have 4 paws,
then it is not a cat; True.
Cats have 4 paws, so the contrapositive is true.
Holt McDougal Geometry
2-2 Conditional Statements
Related conditional statements that have the same
truth value are called logically equivalent
statements. A conditional and its contrapositive
are logically equivalent, and so are the converse
and inverse.
Holt McDougal Geometry
Using
Deductive
Reasoning
Using
Deductive
Reasoning
2-3
2-3 to Verify Conjectures
to Verify Conjectures
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Geometry
Holt
McDougal
Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Deductive reasoning is the process of using
logic to draw conclusions from given facts,
definitions, and properties.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Example 1A: Media Application
Is the conclusion a result of inductive or
deductive reasoning?
There is a myth that you can balance an egg on
its end only on the spring equinox. A person
was able to balance an egg on July 8,
September 21, and December 19. Therefore this
myth is false.
Since the conclusion is based on a pattern of
observations, it is a result of inductive reasoning.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Example 1B: Media Application
Is the conclusion a result of inductive or
deductive reasoning?
There is a myth that the Great Wall of China is
the only man-made object visible from the
Moon. The Great Wall is barely visible in
photographs taken from 180 miles above Earth.
The Moon is about 237,000 miles from Earth.
Therefore, the myth cannot be true.
The conclusion is based on logical reasoning from
scientific research. It is a result of deductive
reasoning.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
In deductive reasoning, if the given facts are
true and you apply the correct logic, then the
conclusion must be true. The Law of
Detachment is one valid form of deductive
reasoning.
Law of Detachment
If p  q is a true statement and p is true, then
q is true.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Example 2A: Verifying Conjectures by Using the Law
of Detachment
Determine if the conjecture is valid by the Law
of Detachment.
Given: If the side lengths of a triangle are 5 cm,
12 cm, and 13 cm, then the area of the triangle
is 30 cm2. The area of ∆PQR is 30 cm2.
Conjecture: The side lengths of ∆PQR are 5cm,
12 cm, and 13 cm.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Example 2A: Verifying Conjectures by Using the Law
of Detachment Continued
Identify the hypothesis and conclusion in the
given conditional.
If the side lengths of a triangle are 5 cm, 12 cm,
and 13 cm, then the area of the triangle is 30
cm2.
The given statement “The area of ∆PQR is 30 cm2”
matches the conclusion of a true conditional. But this
does not mean the hypothesis is true. The dimensions
of the triangle could be different. So the conjecture is
not valid.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Example 2B: Verifying Conjectures by Using the Law
of Detachment
Determine if the conjecture is valid by the Law
of Detachment.
Given: In the World Series, if a team wins four
games, then the team wins the series. The Red
Sox won four games in the 2004 World Series.
Conjecture: The Red Sox won the 2004 World
Series.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Example 2B: Verifying Conjectures by Using the Law
of Detachment Continued
Identify the hypothesis and conclusion in the given
conditional.
In the World Series, if a team wins four games,
then the team wins the series.
The statement “The Red Sox won four games in the
2004 World Series” matches the hypothesis of a true
conditional. By the Law of Detachment, the Red Sox
won the 2004 World Series. The conjecture is valid.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Another valid form of deductive reasoning is
the Law of Syllogism. It allows you to draw
conclusions from two conditional statements
when the conclusion of one is the hypothesis of
the other.
Law of Syllogism
If p  q and q  r are true statements, then
p  r is a true statement.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Example 3A: Verifying Conjectures by Using the Law
of Syllogism
Determine if the conjecture is valid by the Law
of Syllogism.
Given: If a figure is a kite, then it is a
quadrilateral. If a figure is a quadrilateral, then
it is a polygon.
Conjecture: If a figure is a kite, then it is a
polygon.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Example 3A: Verifying Conjectures by Using the Law
of Syllogism Continued
Let p, q, and r represent the following.
p: A figure is a kite.
q: A figure is a quadrilateral.
r: A figure is a polygon.
You are given that p  q and q  r.
Since q is the conclusion of the first conditional
and the hypothesis of the second conditional, you
can conclude that p  r. The conjecture is valid
by Law of Syllogism.
Holt McDougal Geometry