Inductive Reasoning and Conjecture

“Proofs”

Definition

• Conjecture

Educated Guess!!!

Inductive Reasoning

Steps you take to make your guess

Examples:• Brenda has just gotten a job as the plumber’s

assistant. Her first task is to open all the water valves to release the pressure on the lines. The first four valves she discovered opened when turning counterclockwise…

• What is her conjecture?

All valves will be open by turning them counterclockwise

Examples:

• Eric was driving his friend to school when his car suddenly stopped two blocks away from school…

• What is his conjecture?

The car run out of gas

The battery cable lost its contact

Example:

• For points A, B and C, AB = 10, BC = 8 and AC = 5…

Summarize:

Given : Points A, B and C

AB = 10, BC= 8, AC = 5

What is our conjecture?

Points A , B and C are noncollinear (not on the same line)

Examples

• Given

∠ 1 and 2 are supplementary∠ ∠ 1 and 3 are supplementary∠What is our conjecture?

∠ 2 = 3∠

Counterexamples:

• Sometimes after we make a conjecture, we realize that the conjecture is FALSE. Its takes only one false example to show that a conjecture is NOT TRUE. The false example is called:

Counterexample.

Counterexample:• Points P, Q and W are collinear. Joe made a

conjecture that Q is between P and W. Determine if this conjecture is true or false?

• Given:

Points P, Q and W are collinear

Joe’s Conjecture:

Q is between P and R

Solution: False,

P W Q

Counterexample:

• Determine of the conjecture is true of false?

Given : FG = GH

Conjecture: G is a midpoint of FH

Is this statement TRUE or FALSE?

Remember one example needed to show FALSE

Solution: False,

F

GH

G is NOT a midpoint, G is a vertex

More Examples:

• Determine if this conjecture is TRUE or FALSE based on the given information.

Given : Collinear Points D, E and F

Conjecture: DE + EF = DF

Solution: FALSE,

FE D

More Examples:

• Determine if this conjecture is TRUE or FALSE based on the given information.

Given : A and B are supplementary∠ ∠Conjecture: A and B are adjacent∠ ∠Conclusion: FALSE,

∠ A= 30 ∠ B= 150

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Conditional StatementsConditional Statements“IF-THEN”“IF-THEN”

If- Then Statements

• If- Then Statements are commonly used in everyday life.

Advertisement might say:

“If you buy our product, then you will be happy".

Notice that “If-Then” statements have two

parts, a hypothesis(the part following “if”)

and a conclusion(the part following “Then”)

What is Conditional Statement?

• Conditional Statements = “If-Then” statements.

• The IF-statement is the hypothesis and the THEN-statement is the conclusion .

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Ex: Underline the hypothesis & circle the conclusion.

• If you are a brunette, then you have brown hair.

hypothesis conclusion

Ex: Rewrite the statement in “if-then” form

1. Vertical angles are congruent.

If there are 2 vertical angles, then they are congruent.

If 2 angles are vertical, then they are congruent.

Identify Hypothesis and Conclusion.

If a polygon has 6 sides, then it is a hexagon.

Hypothesis: A polygon has 6 sides

Conclusion: It is a hexagon.

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Identify Hypothesis and Conclusion• John will advance to the next level of play if

he completes the maze in his computer game.

Hypothesis: John completes the maze in his computer game

Conclusion: He will advance to the next level of play

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Write a Statement in If-Then Form

A five-sided polygon is a pentagon

Hypothesis: A polygon has five sides

Conclusion: It is a pentagon

If a polygon has five sides, then it is a pentagon

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True or False?• “IF-THEN“ statements can be TRUE or FALSE.

Its false when the hypothesis is true and the

conclusion is false.

EX: If you live in Idaho, you live in Boise

False

EX: Not all people who live in Idaho live in Boise

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True or False?EX: If two angles are congruent, then they are

vertical

Make sure to show an example to prove false.

EX: False, We can have two congruent angles that are not vertical

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Ex: Find a counterexample to prove the statement is false.

• If x2=81, then x must equal 9.

counterexample: x could be -9

because (-9)2=81, but x≠9.

Abbreviation

• Form of statement:

If hypothesis then conclusion

We say : p q, where

p is called hypothesis, q is called conclusion

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Some More…

• New Statements can be formed from the original statement.

• Original “If-Then”: p q

• Converse: q p

• Inverse: ~ p ~ q , where “~” means NOT

• Countrapositive: ~ q ~ p

Examples:• Rewrite the following statements in “If-Then” form.

Than write a converse, inverse and contrapositive.

Ex: “All elephants are mammals”

If-Then form: If an animal is an elephant, then it

is a mammal

Converse: If an animal is a mammal, then it is an

elephant

Inverse: If an animal is not an elephant, then it

is not a mammal

Countrapositive: If an animal is not a mammal,

then it is not an elephant

The original conditional statement & its contrapositive will always have the same meaning.

The converse & inverse of a conditional statement will always have the same meaning.

Practice