1. Using Inductive Reasoning to Make Conjectures 2-1
Warm Up
Lesson Presentation
Lesson Quiz
Holt Geometry

2. Complete each sentence.
Are you ready?
Complete each sentence.
1. ? points are points that lie on the same line.
2. ? points are points that lie in the same plane.
3. The sum of the measures of two ? angles is 90°.

3. Objectives
TSW use inductive reasoning to identify patterns and make conjectures.
TSW find counterexamples to disprove conjectures.

4. Biologists use inductive reasoning to develop theories about migration
patterns.

5. Vocabulary
inductive reasoning
conjecture
counterexample

6. Example 1: Identifying a Pattern
Find the next item in the pattern.
January, March, May, ...

7. Example 2: Identifying a Pattern
Find the next item in the pattern.
7, 14, 21, 28, …

8. Example 3: Identifying a Pattern
Find the next item in the pattern.

9. Example 4
Find the next item in the pattern 0.4, 0.04, 0.004, …

10. 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.

11. Example 5: Making a Conjecture
Complete the conjecture.
The sum of two positive numbers is ? .

12. Example 6: Making a Conjecture
Complete the conjecture.
The number of lines formed by 4 points, no three of which are collinear, is ? .

13. Example 7
Complete the conjecture.
The product of two odd numbers is ? .

14. Example 8: Biology Application
The cloud of water leaving a whale’s blowhole when it exhales is called its blow. A biologist observed blue-whale blows of 25 ft, 29 ft, 27 ft, and 24 ft. Another biologist recorded humpback-whale blows of 8 ft, 7 ft, 8 ft, and 9 ft. Make a conjecture based on the data.
Heights of Whale Blows
Height of Blue-whale Blows 25 29 27 24
Height of Humpback-whale Blows 8 7 9

15. Example 9
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 47
Length of Male (ft) 45 44 46

16. 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.

17. Inductive Reasoning
1. Look for a pattern.
2. Make a conjecture.
3. Prove the conjecture or find a counterexample.

18. Example 10: Finding a Counterexample
Show that the conjecture is false by finding a counterexample.
For every integer n, n3 is positive.

19. Example 11: Finding a Counterexample
Show that the conjecture is false by finding a counterexample.
Two complementary angles are not congruent.

20. Monthly High Temperatures (ºF) in Abilene, Texas
Example 12: 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
Oct
Nov
Dec 88 89 97 99
107
109
110
106
103 92

21. Example 13
Show that the conjecture is false by finding a counterexample.
For any real number x, x2 ≥ x.

22. Example 14
Show that the conjecture is false by finding a counterexample.
Supplementary angles are adjacent.

23. Planets’ Diameters (km)
Example 15
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
Jupiter
Saturn
Uranus
Neptune
Pluto
4880
12,100
12,800
6790
143,000
121,000
51,100
49,500
2300

25. Lesson Quiz
Find the next item in each pattern.
1. 0.7, 0.07, 0.007, … 2.
0.0007
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.
true
false; 2
false; 90° and 90°
true