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# Geometry Section 2-4 1112

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Deductive Reasoning

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### Geometry Section 2-4 1112

1. 1. Section 2-4 Deductive Reasoning
2. 2. Essential Questions How do you use the Law of Detachment? How do you use the Law of Syllogism?
3. 3. Vocabulary 1. Deductive Reasoning: 2. Valid: 3. Law of Detachment:
4. 4. Vocabulary 1. D e d u c t i v e R e a s o n i n g : Using facts, rules, definitions, and properties to reach a logical conclusion from given statements 2. Valid: 3. Law of Detachment:
5. 5. Vocabulary 1. D e d u c t i v e R e a s o n i n g : Using facts, rules, definitions, and properties to reach a logical conclusion from given statements 2. V a l i d : A logically correct statement or argument; follows the pattern of a Law 3. Law of Detachment:
6. 6. Vocabulary 1. D e d u c t i v e R e a s o n i n g : Using facts, rules, definitions, and properties to reach a logical conclusion from given statements 2. V a l i d : A logically correct statement or argument; follows the pattern of a Law 3. L a w o f D e t a c h m e n t : If p q is a true statement and p is true, then q is true.
7. 7. Vocabulary 1. D e d u c t i v e R e a s o n i n g : Using facts, rules, definitions, and properties to reach a logical conclusion from given statements 2. V a l i d : A logically correct statement or argument; follows the pattern of a Law 3. L a w o f D e t a c h m e n t : If p q is a true statement and p is true, then q is true. Example: Given: If a computer has no battery, then it won’t turn on. Statement: Matt’s computer has no battery. Conclusion: Matt’s computer won’t turn on.
8. 8. Vocabulary 4. Law of Syllogism:
9. 9. Vocabulary 4. L a w o f S y l l o g i s m : If p q and q r are true statements, then p r is a true statement
10. 10. Vocabulary 4. Law of Syllogism: If p q and q r are true statements, then p r is a true statement Example: Given: If you buy a ticket, then you can get in the arena. If you get in the arena, then you can watch the game. Conclusion: If you buy a ticket, then you can watch the game.
11. 11. Example 1 Determine whether each conclusion is based on inductive or deductive reasoning. Explain your choice. a. In Matt Mitarnowski’s town, the month of March had the most rain for the past 6 years. He thinks March will have the most rain again this year.
12. 12. Example 1 Determine whether each conclusion is based on inductive or deductive reasoning. Explain your choice. a. In Matt Mitarnowski’s town, the month of March had the most rain for the past 6 years. He thinks March will have the most rain again this year. This is inductive reasoning, as it is using past observations to come to the conclusion.
13. 13. Example 1 Determine whether each conclusion is based on inductive or deductive reasoning. Explain your choice. b. Maggie Brann learned that if it is cloudy at night it will not be as cold in the morning than if there are no clouds at night. Maggie knows it will be cloudy tonight, so she believes it will not be cold tomorrow.
14. 14. Example 1 Determine whether each conclusion is based on inductive or deductive reasoning. Explain your choice. b. Maggie Brann learned that if it is cloudy at night it will not be as cold in the morning than if there are no clouds at night. Maggie knows it will be cloudy tonight, so she believes it will not be cold tomorrow. This is deductive reasoning, as Maggie is using learned information and facts to reach the conclusion.
15. 15. Example 2 Determine whether the conclusion is valid based on the given information. If not, write invalid. Explain your reasoning. Given: If a figure is a right triangle, then it has two acute angles. Statement: A figure has two acute angles. Conclusion: The figure is a right triangle.
16. 16. Example 2 Determine whether the conclusion is valid based on the given information. If not, write invalid. Explain your reasoning. Given: If a figure is a right triangle, then it has two acute angles. Statement: A figure has two acute angles. Conclusion: The figure is a right triangle. Invalid conclusion. A quadrilateral can have two acute angles, but it is not a right triangle.
17. 17. Example 3 Determine whether the conclusion is valid based on the given information. If not, write invalid. Explain your reasoning with a Venn diagram. Given: If a figure is a square, it is also a rectangle. Statement: The figure is a square. Conclusion: The square is also a rectangle.
18. 18. Example 3 Determine whether the conclusion is valid based on the given information. If not, write invalid. Explain your reasoning with a Venn diagram. Given: If a figure is a square, it is also a rectangle. Statement: The figure is a square. Conclusion: The square is also a rectangle. A square has four right angles and opposite parallel sides, so it is also a rectangle
19. 19. Example 3 Determine whether the conclusion is valid based on the given information. If not, write invalid. Explain your reasoning with a Venn diagram. Given: If a figure is a square, it is also a rectangle. Statement: The figure is a square. Conclusion: The square is also a rectangle. Rectangles Squares A square has four right angles and opposite parallel sides, so it is also a rectangle
20. 20. Example 4 Draw a valid conclusion from the given statements, if possible. If no valid conclusion can be drawn, write no conclusion and explain your reasoning. Given: An angle bisector divides an angle into two congruent angles. If two angles are congruent, then their measures are equal. Statement: BD bisects ∠ABC.
21. 21. Example 4 Draw a valid conclusion from the given statements, if possible. If no valid conclusion can be drawn, write no conclusion and explain your reasoning. Given: An angle bisector divides an angle into two congruent angles. If two angles are congruent, then their measures are equal. Statement: BD bisects ∠ABC. Conclusion: m∠ABD = m∠DBC
22. 22. Problem Set
23. 23. Problem Set p. 119 #1-39 odd, 45 “Success does not consist in never making blunders, but in never making the same one a second time.” - Josh Billings