Geometry Section 2-3
- 2. Essential Questions How do you use the Law of Detachment? How do you use the Law of Syllogism?
- 3. Vocabulary 1. Deductive Reasoning: 2. Valid: 3. Law of Detachment:
- 4. Vocabulary 1. Deductive Reasoning: Using facts, rules, definitions, and properties to reach a logical conclusion from given statements 2. Valid: 3. Law of Detachment:
- 5. Vocabulary 1. Deductive Reasoning: Using facts, rules, definitions, and properties to reach a logical conclusion from given statements 2. Valid: A logically correct statement or argument; follows the pattern of a Law 3. Law of Detachment:
- 6. Vocabulary 1. Deductive Reasoning: Using facts, rules, definitions, and properties to reach a logical conclusion from given statements 2. Valid: A logically correct statement or argument; follows the pattern of a Law 3. Law of Detachment: If p q is a true statement and p is true, then q is true.
- 7. Vocabulary 1. Deductive Reasoning: Using facts, rules, definitions, and properties to reach a logical conclusion from given statements 2. Valid: A logically correct statement or argument; follows the pattern of a Law 3. Law of Detachment: If p q is a true statement and p is true, then q is true. Example: Given: If an iPhone has no charge, then it won’t turn on. Statement: Matt’s iPhone has no charge. Conclusion: Matt’s iPhone won’t turn on.
- 8. Vocabulary 4. Law of Syllogism:
- 9. Vocabulary 4. Law of Syllogism:If p q and q r are true statements, then p r is a true statement
- 10. Vocabulary 4. Law of Syllogism: 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. If p q and q r are true statements, then p r is a true statement
- 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. 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. 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. 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. 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. 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. Fails the Laws of Detachment and Syllogism. A quadrilateral can have two acute angles, but it is not a right triangle.
- 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. 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. Valid. Passes the Law of Detachment. A square has four right angles and opposite parallel sides, so it is also a rectangle.
- 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 Square s Valid. Passes the Law of Detachment. A square has four right angles and opposite parallel sides, so it is also a rectangle.
- 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. 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