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Using Inductive Reasoning to Make Conjectures

Holt Geometry

Warm Up

Lesson Presentation

Lesson Quiz

Holt McDougal Geometry

Holt McDougal Geometry

2-1

Using Inductive Reasoning to

Make Conjectures

Warm Up

Complete each sentence.

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1. ? points are points that lie on the same line.

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2. ? points are points that lie in the same plane.

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3. The sum of the measures of two ? angles is 90°.

Collinear

Coplanar

complementary

Holt McDougal Geometry

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Using Inductive Reasoning to

Make Conjectures

Use inductive reasoning to identify patterns and make conjectures.

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Find counterexamples to disprove conjectures.

Objectives

Holt McDougal Geometry

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Using Inductive Reasoning to

Make Conjectures

inductive reasoning

conjecture

counterexample

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Vocabulary

Holt McDougal Geometry

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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 McDougal Geometry

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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 McDougal Geometry

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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.

Holt McDougal Geometry

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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 next item in the pattern has one more zero after the decimal point.

The next item would have 3 zeros after the decimal point, or 0.0004.

Holt McDougal Geometry

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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 McDougal Geometry

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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 = 2 3.14 + 0.01 = 3.15

3,900 + 1,000,017 = 1,003,917

Holt McDougal Geometry

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Using Inductive Reasoning to

Make Conjectures

Complete the conjecture.

Example 2B: Making a Conjecture

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

Draw four points. Make sure no three points are collinear. Count the number of lines formed:

The number of lines formed by four points, no three of which are collinear, is 6.

Holt McDougal Geometry

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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 McDougal Geometry

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Using Inductive Reasoning to

Make Conjectures

Example 3: 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.

Holt McDougal Geometry

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Using Inductive Reasoning to

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Example 3: Biology Application Continued

The smallest blue-whale blow (24 ft) is almost three times higher than the greatest humpback-whale blow (9 ft). Possible conjectures:

The height of a blue whale’s blow is about three times greater than a humpback whale’s blow.

The height of a blue-whale’s blow is greater than a humpback whale’s blow.

Holt McDougal Geometry

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Using Inductive Reasoning to

Make Conjectures

Check It Out! Example 3

Make a conjecture about the lengths of male and female whales based on the data.

In 5 of the 6 pairs of numbers above the female is longer.

Female whales are longer than male whales.

Holt McDougal Geometry

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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 McDougal Geometry

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Using Inductive Reasoning to

Make Conjectures

Holt McDougal Geometry

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Using Inductive Reasoning to

Make Conjectures

Show that the conjecture is false by finding a counterexample.

Example 4A: 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

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Using Inductive Reasoning to

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Show that the conjecture is false by finding a counterexample.

Example 4B: Finding a Counterexample

Two complementary angles are not congruent.

If the two congruent angles both measure 45°, the conjecture is false.

45° + 45° = 90°

Holt McDougal Geometry

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Using Inductive Reasoning to

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Show that the conjecture is false by finding a counterexample.

Example 4C: Finding a Counterexample

The monthly high temperature in Abilene is never below 90°F for two months in a row.

The monthly high temperatures in January and February were 88°F and 89°F, so the conjecture is false.

Holt McDougal Geometry

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Using Inductive Reasoning to

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Check It Out! Example 4a

For any real number x, x² ≥ x.

Show that the conjecture is false by finding a counterexample.

The conjecture is false.

Holt McDougal Geometry

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Using Inductive Reasoning to

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Check It Out! Example 4b

Supplementary angles are adjacent.

Show that the conjecture is false by finding a counterexample.

The supplementary angles are not adjacent, so the conjecture is false.

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Check It Out! Example 4c

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.

Since the radius is half the diameter, the radius of Jupiter is 71,500 km and the radius of Saturn is 60,500 km. The conjecture is false.

Holt McDougal Geometry

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Using Inductive Reasoning to

Make Conjectures