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Geometry Section 2-3 1112

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  • 1. Section 2-3 Conditional Statements Thursday, November 6, 14
  • 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
  • 3. Vocabulary 1. Conditional Statement: 2. If-Then Statement: 3. Hypothesis: 4. Conclusion: 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
  • 15. Vocabulary 8. Contrapositive: 9. Logically Equivalent: 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
  • 56. Problem Set 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