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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
Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Warm Up
Complete each sentence.
1.
2.
?
?
points are points that lie on the same line.
points are points that lie in the same plane.
3. The sum of the measures of two
Holt Geometry
?
angles is 90°.
Using Inductive Reasoning to
2-1 Make Conjectures
Objectives
Use inductive reasoning to identify
patterns and make conjectures.
Find counterexamples to disprove
conjectures.
Holt Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Vocabulary
inductive reasoning
conjecture
counterexample
Holt Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Example 1A: Identifying a Pattern
Find the next item in the pattern.
January, March, May, ...
Holt 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, …
Holt Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Example 1C: Identifying a Pattern
Find the next item in the pattern.
The next figure is
Holt Geometry
.
Using Inductive Reasoning to
2-1 Make Conjectures
Check It Out! Example 1
Find the next item in the pattern 0.4, 0.04, 0.004, …
Holt 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 Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Example 2A: Making a Conjecture
Complete the conjecture.
The sum of two positive numbers is
Holt Geometry
? .
Using Inductive Reasoning to
2-1 Make Conjectures
Example 2B: Making a Conjecture
Complete the conjecture.
The number of lines formed by 4 points, no
three of which are collinear, is ? .
Holt Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Check It Out! Example 2
Complete the conjecture.
The product of two odd numbers is
Holt Geometry
? .
Using Inductive Reasoning to
2-1 Make Conjectures
Check It Out! Example 3
Make a conjecture about the lengths of male and
female whales based on the data.
Average Whale Lengths
Length of Female (ft)
49
51
50
48
51
47
Length of Male (ft)
47
45
44
46
48
48
Holt 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 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 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.
Holt 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.
Holt Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Example 4C: Finding a Counterexample
Show that the conjecture is false by finding a
counterexample.
The monthly high temperature in Abilene is
never below 90°F for two months in a row.
Monthly High Temperatures (ºF) in Abilene, Texas
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug Sep
88
89
97
99
107
109
110
107 106 103
Holt Geometry
Oct Nov Dec
92
89
Using Inductive Reasoning to
2-1 Make Conjectures
Check It Out! Example 4a
Show that the conjecture is false by finding a
counterexample.
For any real number x, x2 ≥ x.
Holt 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.
Holt Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Check It Out! Example 4c
Show that the conjecture is false by finding a
counterexample.
The radius of every planet in the solar system is
less than 50,000 km.
Planets’ Diameters (km)
Mercury Venus Earth Mars
4880
12,100
Holt Geometry
12,800
6790
Jupiter
Saturn
Uranus
Neptune
Pluto
143,000
121,000
51,100
49,500
2300
Using Inductive Reasoning to
2-1 Make Conjectures
Lesson Quiz
Find the next item in each pattern.
1. 0.7, 0.07, 0.007, … 2.
Determine if each conjecture is true. If false,
give a counterexample.
3. The quotient of two negative numbers is a positive
number.
4. Every prime number is odd.
5. Two supplementary angles are not congruent.
6. The square of an odd integer is odd.
Holt Geometry