1. Inductive Reasoning and Conjecture “Proofs”

2. Definition Conjecture Educated Guess!!!  Inductive Reasoning
Steps you take to make your guess

3. 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

4. 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

5. 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)

6. Examples Given ∠ 1 and ∠ 2 are supplementary
What is our conjecture?
∠ 2 = ∠ 3

7. 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.

8. 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,    Q P W

9. 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

10. 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

11. 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,
∠ B= 150
∠ A= 30

12. Conditional Statements “IF-THEN”

13. 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”)

14. What is Conditional Statement?
Conditional Statements = “If-Then” statements.
The IF-statement is the hypothesis and the THEN-statement is the conclusion .

15. Ex: Underline the hypothesis & circle the conclusion.
If you are a brunette, then you have brown hair.
hypothesis conclusion

16. Ex: Rewrite the statement in “if-then” form
Vertical angles are congruent.
If there are 2 vertical angles, then they are congruent.
If 2 angles are vertical, then they are congruent.

17. 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.

18. 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

19. 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

20. 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

21. 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 21

22. 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.

23. Abbreviation Form of statement: If hypothesis then conclusion
We say : p  q, where
p is called hypothesis, q is called conclusion

24. 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

25. 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

26. 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.

27. Practice