Holt Geometry
2-1
Using Inductive Reasoning to
Make Conjectures
Warm Up
Boxed In
Three boxes contain two coins each.One contains two nickels, one containstwo dimes, and one contains a dimeand a nickel. All three boxes aremislabeled. (Each contains the label ofone of the other two boxes.) If youare permitted to take out only onecoin at a time, how many must youtake out in order to be able to label allthree boxes correctly?
Holt Geometry
2-1
Using Inductive Reasoning to
Make Conjectures
Use inductive reasoning to identifypatterns and make conjectures.
Find counterexamples to disproveconjectures.
Objectives
Holt Geometry
2-1
Using Inductive Reasoning to
Make Conjectures
inductive reasoning
conjecture
counterexample
Vocabulary
Holt Geometry
2-1
Using Inductive Reasoning to
Make Conjectures
Find the next item in the pattern.
Example 1A: Identifying a Pattern
January, March, May, ...
The next month is July.
Alternating months of the year make up the pattern.
Holt Geometry
2-1
Using Inductive Reasoning to
Make Conjectures
Find the next item in the pattern.
Example 1B: Identifying a Pattern
7, 14, 21, 28, …
The next multiple is 35.
Multiples of 7 make up the pattern.
Holt Geometry
2-1
Using Inductive Reasoning to
Make Conjectures
Find the next item in the pattern.
Example 1C: Identifying a Pattern
In this pattern, the figure rotates 90° counter-clockwise each time.
The next figure is .
Holt Geometry
2-1
Using Inductive Reasoning to
Make Conjectures
Check It Out! Example 1
Find the next item in the pattern 0.4, 0.04, 0.004, …
When reading the pattern from left to right, the nextitem in the pattern has one more zero after thedecimal point.
The next item would have 3 zeros after the decimalpoint, or 0.0004.
Holt Geometry
2-1
Using Inductive Reasoning to
Make Conjectures
When several examples form a pattern and youassume the pattern will continue, you areapplying inductive reasoning. Inductivereasoning is the process of reasoning that a ruleor statement is true because specific cases aretrue. You may use inductive reasoning to draw aconclusion from a pattern. A statement youbelieve to be true based on inductive reasoning iscalled a conjecture.
Holt Geometry
2-1
Using Inductive Reasoning to
Make Conjectures
Complete the conjecture.
Example 2A: Making a Conjecture
The sum of two positive numbers is ? .
The sum of two positive numbers is positive.
List some examples and look for a pattern.
1 + 1 = 23.14 + 0.01 = 3.15
3,900 + 1,000,017 = 1,003,917
Holt Geometry
2-1
Using Inductive Reasoning to
Make Conjectures
Complete the conjecture.
Example 2B: Making a Conjecture
The number of lines formed by 4 points, nothree of which are collinear, is ? .
Draw four points. Make sure no three points arecollinear. Count the number of lines formed:
The number of lines formed by four points, nothree of which are collinear, is 6.
Holt Geometry
2-1
Using Inductive Reasoning to
Make Conjectures
Check It Out! Example 2
The product of two odd numbers is ? .
Complete the conjecture.
The product of two odd numbers is odd.
List some examples and look for a pattern.
1 1 = 1 3 3 = 9 5 7 = 35
Holt Geometry
2-1
Using Inductive Reasoning to
Make Conjectures
Example 3: Biology Application
The cloud of water leaving a whale’s blowholewhen it exhales is called its blow. A biologistobserved 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 aconjecture based on the data.
Heights of Whale Blows
Height of Blue-whale Blows
25
29
27
24
Height of Humpback-whaleBlows
8
7
8
9
Holt Geometry
2-1
Using Inductive Reasoning to
Make Conjectures
Example 3: Biology Application Continued
The smallest blue-whale blow (24 ft) is almostthree times higher than the greatest humpback-whale blow (9 ft). Possible conjectures:
The height of a blue whale’s blow is about threetimes greater than a humpback whale’s blow.
The height of a blue-whale’s blow is greater thana humpback whale’s blow.
Holt Geometry
2-1
Using Inductive Reasoning to
Make Conjectures
Check It Out! Example 3
Make a conjecture about the lengths of male andfemale whales based on the data.
In 5 of the 6 pairs of numbers above the female islonger.
Female whales are longer than male whales.
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
2-1
Using Inductive Reasoning to
Make Conjectures
To show that a conjecture is false, you have to findonly one example in which the conjecture is not true.This case is called a counterexample.
To show that a conjecture is always true, you mustprove it.
A counterexample can be a drawing, a statement, or anumber.
Holt Geometry
2-1
Using Inductive Reasoning to
Make Conjectures
Inductive Reasoning
1. Look for a pattern.
2. Make a conjecture.
3. Prove the conjecture or find acounterexample.
Holt Geometry
2-1
Using Inductive Reasoning to
Make Conjectures
Show that the conjecture is false by finding acounterexample.
Example 4A: Finding a Counterexample
For every integer n, n3 is positive.
Pick integers and substitute them into the expressionto 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, theconjecture is false.
n = –3 is a counterexample.
Holt Geometry
2-1
Using Inductive Reasoning to
Make Conjectures
Show that the conjecture is false by finding acounterexample.
Example 4B: Finding a Counterexample
Two complementary angles are not congruent.
If the two congruent angles both measure 45°, theconjecture is false.
45° + 45° = 90°
Holt Geometry
2-1
Using Inductive Reasoning to
Make Conjectures
Show that the conjecture is false by finding acounterexample.
Example 4C: Finding a Counterexample
The monthly high temperature in Abilene isnever 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
107
106
103
92
89
The monthly high temperatures in January and Februarywere 88°F and 89°F, so the conjecture is false.
Holt Geometry
2-1
Using Inductive Reasoning to
Make Conjectures
Check It Out! Example 4a
For any real number x, x2 ≥ x.
Show that the conjecture is false by finding acounterexample.
Let x = .
12
The conjecture is false.
Since = , ≥ .
12
2
12
14
14
Holt Geometry
2-1
Using Inductive Reasoning to
Make Conjectures
Check It Out! Example 4b
Supplementary angles are adjacent.
Show that the conjecture is false by finding acounterexample.
The supplementary angles are not adjacent,so the conjecture is false.
23°
157°
Holt Geometry
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Using Inductive Reasoning to
Make Conjectures
Check It Out! Example 4c
The radius of every planet in the solar system isless than 50,000 km.
Show that the conjecture is false by finding acounterexample.
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
Since the radius is half the diameter, the radius ofJupiter is 71,500 km and the radius of Saturn is60,500 km. The conjecture is false.
Holt Geometry
2-1
Using Inductive Reasoning to
Make Conjectures
Homework p.77-79#11, 12, 14, 16-20,24-26, 30, 31, 33,34, 40, 43, 44-51