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

Holt Geometry

2-1 Using Inductive Reasoning to Make Conjectures2-1 Using Inductive Reasoning to Make

Conjectures

Holt Geometry

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

Holt Geometry

2-1 Using Inductive Reasoning to Make Conjectures

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°.

Holt Geometry

2-1 Using Inductive Reasoning to Make Conjectures

TSW use inductive reasoning to identify patterns and make conjectures.

TSW find counterexamples to disprove conjectures.

Objectives

Holt Geometry

2-1 Using Inductive Reasoning to Make Conjectures

Biologists use inductive reasoning to develop theories about migration patterns.

Holt Geometry

2-1 Using Inductive Reasoning to Make Conjectures

inductive reasoningconjecturecounterexample

Vocabulary

Holt Geometry

2-1 Using Inductive Reasoning to Make Conjectures

Find the next item in the pattern.

Example 1: Identifying a Pattern

January, March, May, ...

Holt Geometry

2-1 Using Inductive Reasoning to Make Conjectures

Find the next item in the pattern.

Example 2: Identifying a Pattern

7, 14, 21, 28, …

Holt Geometry

2-1 Using Inductive Reasoning to Make Conjectures

Find the next item in the pattern.

Example 3: Identifying a Pattern

Holt Geometry

2-1 Using Inductive Reasoning to Make Conjectures

Example 4

Find the next item in the pattern 0.4, 0.04, 0.004, …

Holt Geometry

2-1 Using Inductive Reasoning to 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

2-1 Using Inductive Reasoning to Make Conjectures

Complete the conjecture.

Example 5: Making a Conjecture

The sum of two positive numbers is ? .

Holt Geometry

2-1 Using Inductive Reasoning to Make Conjectures

Complete the conjecture.

Example 6: Making a Conjecture

The number of lines formed by 4 points, no three of which are collinear, is ? .

Holt Geometry

2-1 Using Inductive Reasoning to Make Conjectures

Example 7

The product of two odd numbers is ? .

Complete the conjecture.

Holt Geometry

2-1 Using Inductive Reasoning to Make Conjectures

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 8 9

Holt Geometry

2-1 Using Inductive Reasoning to Make Conjectures

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 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 find only one example in which the conjecture is not true. This case is called a counterexample.

To show that a conjecture is always true, you must prove it.

A counterexample can be a drawing, a statement, or a number.

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 a counterexample.

Holt Geometry

2-1 Using Inductive Reasoning to Make Conjectures

Show that the conjecture is false by finding a counterexample.

Example 10: Finding a Counterexample

For every integer n, n3 is positive.

Holt Geometry

2-1 Using Inductive Reasoning to Make Conjectures

Show that the conjecture is false by finding a counterexample.

Example 11: Finding a Counterexample

Two complementary angles are not congruent.

Holt Geometry

2-1 Using Inductive Reasoning to Make Conjectures

Show that the conjecture is false by finding a counterexample.

Example 12: 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, TexasJan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

88 89 97 99 107 109 110 107 106 103 92 89

Holt Geometry

2-1 Using Inductive Reasoning to Make Conjectures

Example 13

For any real number x, x2 ≥ x.

Show that the conjecture is false by finding a counterexample.

Holt Geometry

2-1 Using Inductive Reasoning to Make Conjectures

Example 14

Supplementary angles are adjacent.

Show that the conjecture is false by finding a counterexample.

Holt Geometry

2-1 Using Inductive Reasoning to Make Conjectures

Example 15

The radius of every planet in the solar system is less than 50,000 km.

Show that the conjecture is false by finding a counterexample.

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

Holt Geometry

2-1 Using Inductive Reasoning to Make Conjectures

Holt Geometry

2-1 Using Inductive Reasoning to Make Conjectures

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.

false; 2

true

false; 90° and 90°

true