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,


H
F 
G
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,

E

F

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
Conditional Statements
“IF-THEN”
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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
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