2. Essential Questions
• How do you analyze statements in if-then form?
• How do you write the converse, inverse, and
contrapositive of if-then statements?
Thursday, November 6, 14
4. Vocabulary
1. C o n d i t io n a l S t a t e m e n t : A statement that fits the
if-then form, providing a connection between the
two phrases
2. If-Then Statement:
3. Hypothesis:
4. Conclusion:
Thursday, November 6, 14
5. Vocabulary
1. C o n d i t io n a l S t a t e m e n t : A statement that fits the
if-then form, providing a connection between the
two phrases
2. If - T h e n S t a t e m e n t : Another name for a
conditional statement; in the form of if p, then q
3. Hypothesis:
4. Conclusion:
Thursday, November 6, 14
6. Vocabulary
1. C o n d i t io n a l S t a t e m e n t : A statement that fits the
if-then form, providing a connection between the
two phrases
2. If - T h e n S t a t e m e n t : Another name for a
conditional statement; in the form of if p, then q
p→q
3. Hypothesis:
4. Conclusion:
Thursday, November 6, 14
7. Vocabulary
1. C o n d i t io n a l S t a t e m e n t : A statement that fits the
if-then form, providing a connection between the
two phrases
2. If - T h e n S t a t e m e n t : Another name for a
conditional statement; in the form of if p, then q
p→q
3. H y p o t h e s i s : The phrase that is the “if” part of the
conditional
4. Conclusion:
Thursday, November 6, 14
8. Vocabulary
1. C o n d i t io n a l S t a t e m e n t : A statement that fits the
if-then form, providing a connection between the
two phrases
2. If - T h e n S t a t e m e n t : Another name for a
conditional statement; in the form of if p, then q
p→q
3. H y p o t h e s i s : The phrase that is the “if” part of the
conditional
4. C o n c l u s i o n : The phrase that is the “then” part of
the conditional
Thursday, November 6, 14
9. Vocabulary
5. Related Conditionals:
6. Converse:
7. Inverse:
Thursday, November 6, 14
10. Vocabulary
5. R e l a t e d C o n d i t i o n a ls : Statements that are based
off of a given conditional statement
6. Converse:
7. Inverse:
Thursday, November 6, 14
11. Vocabulary
5. R e l a t e d C o n d i t i o n a ls : Statements that are based
off of a given conditional statement
6. C o n v e r s e : A statement that is created by
switching the hypothesis and conclusion of a
conditional
7. Inverse:
Thursday, November 6, 14
12. Vocabulary
5. R e l a t e d C o n d i t i o n a ls : Statements that are based
off of a given conditional statement
6. C o n v e r s e : A statement that is created by
switching the hypothesis and conclusion of a
conditional
q→ p
7. Inverse:
Thursday, November 6, 14
13. Vocabulary
5. R e l a t e d C o n d i t i o n a ls : Statements that are based
off of a given conditional statement
6. C o n v e r s e : A statement that is created by
switching the hypothesis and conclusion of a
conditional
q→ p
7. I n v e r s e : A statement that is created by negating
the hypothesis and conclusion of a conditional
Thursday, November 6, 14
14. Vocabulary
5. R e l a t e d C o n d i t i o n a ls : Statements that are based
off of a given conditional statement
6. C o n v e r s e : A statement that is created by
switching the hypothesis and conclusion of a
conditional
q→ p
7. I n v e r s e : A statement that is created by negating
the hypothesis and conclusion of a conditional
~ p→~ q
Thursday, November 6, 14
16. Vocabulary
8. C o n t r a p o s i t iv e : A statement that is created by
negating the hypothesis and conclusion of the
converse of the conditional
9. Logically Equivalent:
Thursday, November 6, 14
17. Vocabulary
8. C o n t r a p o s i t iv e : A statement that is created by
negating the hypothesis and conclusion of the
converse of the conditional
~ q→~ p
9. Logically Equivalent:
Thursday, November 6, 14
18. Vocabulary
8. C o n t r a p o s i t iv e : A statement that is created by
negating the hypothesis and conclusion of the
converse of the conditional
~ q→~ p
9. L o g ic a l ly E q u i v a l e n t : Statements with the same
truth values
Thursday, November 6, 14
19. Vocabulary
8. C o n t r a p o s i t iv e : A statement that is created by
negating the hypothesis and conclusion of the
converse of the conditional
~ q→~ p
9. L o g ic a l ly E q u i v a l e n t : Statements with the same
truth values
A conditional and its contrapositive
Thursday, November 6, 14
20. Vocabulary
8. C o n t r a p o s i t iv e : A statement that is created by
negating the hypothesis and conclusion of the
converse of the conditional
~ q→~ p
9. L o g ic a l ly E q u i v a l e n t : Statements with the same
truth values
A conditional and its contrapositive
The converse and inverse of a conditional
Thursday, November 6, 14
21. Example 1
Identify the hypothesis and conclusion of each
statement.
a. If a polygon has eight sides, then it is an octagon.
b. Matt Mitarnowski will advance to the next level if he
completes the Towers of Hanoi in his computer game.
Thursday, November 6, 14
22. Example 1
Identify the hypothesis and conclusion of each
statement.
a. If a polygon has eight sides, then it is an octagon.
b. Matt Mitarnowski will advance to the next level if he
completes the Towers of Hanoi in his computer game.
Thursday, November 6, 14
23. Example 1
Identify the hypothesis and conclusion of each
statement.
a. If a polygon has eight sides, then it is an octagon.
Hypothesis
b. Matt Mitarnowski will advance to the next level if he
completes the Towers of Hanoi in his computer game.
Thursday, November 6, 14
24. Example 1
Identify the hypothesis and conclusion of each
statement.
a. If a polygon has eight sides, then it is an octagon.
Hypothesis
b. Matt Mitarnowski will advance to the next level if he
completes the Towers of Hanoi in his computer game.
Thursday, November 6, 14
25. Example 1
Identify the hypothesis and conclusion of each
statement.
a. If a polygon has eight sides, then it is an octagon.
Hypothesis Conclusion
b. Matt Mitarnowski will advance to the next level if he
completes the Towers of Hanoi in his computer game.
Thursday, November 6, 14
26. Example 1
Identify the hypothesis and conclusion of each
statement.
a. If a polygon has eight sides, then it is an octagon.
Hypothesis Conclusion
b. Matt Mitarnowski will advance to the next level if he
completes the Towers of Hanoi in his computer game.
Hypothesis
Thursday, November 6, 14
27. Example 1
Identify the hypothesis and conclusion of each
statement.
a. If a polygon has eight sides, then it is an octagon.
Hypothesis Conclusion
Conclusion
b. Matt Mitarnowski will advance to the next level if he
completes the Towers of Hanoi in his computer game.
Hypothesis
Thursday, November 6, 14
28. Example 2
Identify the hypothesis and conclusion of each
statement. Then write each statement in the if-then
form.
a. Measured distance is positive.
Thursday, November 6, 14
29. Example 2
Identify the hypothesis and conclusion of each
statement. Then write each statement in the if-then
form.
a. Measured distance is positive.
Hypothesis: A distance is measured
Thursday, November 6, 14
30. Example 2
Identify the hypothesis and conclusion of each
statement. Then write each statement in the if-then
form.
a. Measured distance is positive.
Hypothesis: A distance is measured
Conclusion: It is positive
Thursday, November 6, 14
31. Example 2
Identify the hypothesis and conclusion of each
statement. Then write each statement in the if-then
form.
a. Measured distance is positive.
Hypothesis: A distance is measured
Conclusion: It is positive
If a distance is measured, then it is positive.
Thursday, November 6, 14
32. Example 2
Identify the hypothesis and conclusion of each
statement. Then write each statement in the if-then
form.
b. A six-sided polygon is a hexagon
Thursday, November 6, 14
33. Example 2
Identify the hypothesis and conclusion of each
statement. Then write each statement in the if-then
form.
b. A six-sided polygon is a hexagon
Hypothesis: A polygon has six sides
Thursday, November 6, 14
34. Example 2
Identify the hypothesis and conclusion of each
statement. Then write each statement in the if-then
form.
b. A six-sided polygon is a hexagon
Hypothesis: A polygon has six sides
Conclusion: It is a hexagon
Thursday, November 6, 14
35. Example 2
Identify the hypothesis and conclusion of each
statement. Then write each statement in the if-then
form.
b. A six-sided polygon is a hexagon
Hypothesis: A polygon has six sides
Conclusion: It is a hexagon
If a polygon has six sides, then it is a hexagon.
Thursday, November 6, 14
36. Example 3
Determine the truth value of each conditional
statement. If true, explain your reasoning. If false,
give a counter example.
a. If you subtract a whole number from another whole
number, the result is also a whole number.
Thursday, November 6, 14
37. Example 3
Determine the truth value of each conditional
statement. If true, explain your reasoning. If false,
give a counter example.
a. If you subtract a whole number from another whole
number, the result is also a whole number.
False
Thursday, November 6, 14
38. Example 3
Determine the truth value of each conditional
statement. If true, explain your reasoning. If false,
give a counter example.
a. If you subtract a whole number from another whole
number, the result is also a whole number.
False
5 − 11 = −6
Thursday, November 6, 14
39. Example 3
Determine the truth value of each conditional
statement. If true, explain your reasoning. If false,
give a counter example.
b. If last month was September, then this month is
October.
c. When a rectangle has an obtuse angle, it is a
parallelogram.
Thursday, November 6, 14
40. Example 3
Determine the truth value of each conditional
statement. If true, explain your reasoning. If false,
give a counter example.
b. If last month was September, then this month is
October.
True
c. When a rectangle has an obtuse angle, it is a
parallelogram.
Thursday, November 6, 14
41. Example 3
Determine the truth value of each conditional
statement. If true, explain your reasoning. If false,
give a counter example.
b. If last month was September, then this month is
October.
True
c. When a rectangle has an obtuse angle, it is a
parallelogram.
True
Thursday, November 6, 14
42. Example 3
Determine the truth value of each conditional
statement. If true, explain your reasoning. If false,
give a counter example.
b. If last month was September, then this month is
October.
True
c. When a rectangle has an obtuse angle, it is a
parallelogram.
True A rectangle cannot have an obtuse angle, so we
cannot test this. All rectangles are parallelograms.
Thursday, November 6, 14
43. Example 4
Determine the converse, inverse, and contrapositive for
the following statement. Then determine if the new
statement is true. If false, give a counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Converse:
Thursday, November 6, 14
44. Example 4
Determine the converse, inverse, and contrapositive for
the following statement. Then determine if the new
statement is true. If false, give a counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Converse:
If MN ≅ NO, then N is the midpoint of MO.
Thursday, November 6, 14
45. Example 4
Determine the converse, inverse, and contrapositive for
the following statement. Then determine if the new
statement is true. If false, give a counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Converse:
If MN ≅ NO, then N is the midpoint of MO.
False
Thursday, November 6, 14
46. Example 4
Determine the converse, inverse, and contrapositive for
the following statement. Then determine if the new
statement is true. If false, give a counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Converse:
If MN ≅ NO, then N is the midpoint of MO.
False
M, N, and O might not be collinear
Thursday, November 6, 14
47. Example 4
Determine the converse, inverse, and contrapositive for
the following statement. Then determine if the new
statement is true. If false, give a counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Converse:
If MN ≅ NO, then N is the midpoint of MO.
False
M, N, and O might not be collinear
M N O
Thursday, November 6, 14
48. Example 4
Determine the converse, inverse, and contrapositive for
the following statement. Then determine if the new
statement is true. If false, give a counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Inverse:
Thursday, November 6, 14
49. Example 4
Determine the converse, inverse, and contrapositive for
the following statement. Then determine if the new
statement is true. If false, give a counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Inverse:
If N is not the midpoint of MO, then MN ≅ NO.
Thursday, November 6, 14
50. Example 4
Determine the converse, inverse, and contrapositive for
the following statement. Then determine if the new
statement is true. If false, give a counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Inverse:
False
If N is not the midpoint of MO, then MN ≅ NO.
Thursday, November 6, 14
51. Example 4
Determine the converse, inverse, and contrapositive for
the following statement. Then determine if the new
statement is true. If false, give a counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Inverse:
False
If N is not the midpoint of MO, then MN ≅ NO.
If N is not on MO, then MN could be congruent to NO.
Thursday, November 6, 14
52. Example 4
Determine the converse, inverse, and contrapositive for
the following statement. Then determine if the new
statement is true. If false, give a counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Inverse:
False
If N is not the midpoint of MO, then MN ≅ NO.
If N is not on MO, then MN could be congruent to NO.
M N O
Thursday, November 6, 14
53. Example 4
Determine the converse, inverse, and contrapositive for
the following statement. Then determine if the new
statement is true. If false, give a counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Contrapositive:
Thursday, November 6, 14
54. Example 4
Determine the converse, inverse, and contrapositive for
the following statement. Then determine if the new
statement is true. If false, give a counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Contrapositive:
If MN ≅ NO, then N is not the midpoint of MO.
Thursday, November 6, 14
55. Example 4
Determine the converse, inverse, and contrapositive for
the following statement. Then determine if the new
statement is true. If false, give a counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Contrapositive:
If MN ≅ NO, then N is not the midpoint of MO.
True
Thursday, November 6, 14
57. Problem Set
p. 109 #1-51 odd, 63
“Don’t be discouraged by a failure. It can be a positive experience. Failure is, in a
sense, the highway to success, inasmuch as every discovery of what is false
leads us to seek earnestly after what is true, and every fresh experience points
out some form of error which we shall afterwards carefully avoid.” - John Keats
Thursday, November 6, 14
