Warm Up Using the diagram on the board: Identify a pair of vertical angles

1. Warm Up Using the diagram on the board: Identify a pair of vertical angles Identify a pair of opposite rays Identify a pair of adjacent angles with vertex P Given m1 = 42, What is m2? What is m3?

2. Chapter 2.4 Deductive Reasoning Target: SWBAT Apply the Law of Detachment and the Law of Syllogism in logical reasoning.

3. Deductive reasoning(logical reasoning) is the process of using logic to draw conclusions from given facts, definitions, and properties.

4. Law of Detachment If p q is a true statement and p is true, then q is true. In deductive reasoning, if the given facts are true and you apply the correct logic, then the conclusion (or “Conjecture”) must be true. The Law of Detachment is one valid form of deductive reasoning.

5. Example 1: Using the Law of Detachment Determine if the conjecture is valid by the Law of Detachment. Given statements: a. In the World Series, if a team wins four games, then the team wins the series. b. The Red Sox won four games in the 2004 World Series. Conjecture: The Red Sox won the 2004 World Series.

6. Example 1: Using the Law of Detachment Notice the hypothesisand conclusion in the given conditional statement. In the World Series, if a team wins four games, then the team wins the series. The statement “The Red Sox won four games in the 2004 World Series” matches the hypothesis of our conditional statement. By the Law of Detachment, the Red Sox must have won the 2004 World Series. The conjecture is valid.

7. Example 2: Using the Law of Detachment Determine if the conjecture is valid by the Law of Detachment. Given statements: a. If the side lengths of a triangle are 5 cm, 12 cm, and 13 cm, then the area of the triangle is 30 cm2. b. The area of ∆PQR is 30 cm2. Conjecture: The side lengths of ∆PQR are 5cm, 12 cm, and 13 cm.

8. Example 2: Using the Law of Detachment Notice the hypothesisand conclusionin the given conditional statement. If the side lengths of a triangle are 5 cm, 12 cm, and 13 cm,thenthe area of the triangle is 30 cm2. The statement “The area of∆PQR is 30 cm2” matches the conclusion of our conditional statement. This does not mean the hypothesis is true as our conditional is not a bi-conditional statement. The dimensions of the triangle could be different. So our conjecture is not valid.

9. Check It Out! Example 3 Determine if the conjecture is valid by the Law of Detachment. Given statements: If a student passes his classes, the student is eligible to play sports. Ramon passed his classes. Conjecture: Ramon is eligible to play sports.

10. Check It Out! Example 3 Hypothesis: a student passes his classesConclusion: the student is eligible to play sports The statement “Ramon passed his classes” matches the hypothesis of our conditional statement. By the Law of Detachment, Ramon is eligible to play sports. The conjecture is valid.

11. Law of Syllogism If p q and q  r are true statements, then p  r is a true statement. Another valid form of deductive reasoning is the Law of Syllogism. It allows you to draw conclusions from two conditional statements when the conclusion of one is the hypothesis of the other. Think of this as a chain of conditional statements.

12. Example 4: Using the Law of Syllogism Determine if the conjecture is valid by the Law of Syllogism. Given statements: a. If a figure is a kite, then it is a quadrilateral. b. If a figure is a quadrilateral, then it is a polygon. Conjecture: If a figure is a kite, then it is a polygon.

13. Example 4: Using the Law of Syllogism Let p, q, and r represent the following. • p: A figure is a kite. q: A figure is a quadrilateral. • r: A figure is a polygon. • We are given that pq and qr. Since q is the conclusion of the first conditional and the hypothesis of the second conditional, you can conclude that pr. The conjecture is valid by Law of Syllogism.

14. Example 5: Using the Law of Syllogism Determine if the conjecture is valid by the Law of Syllogism. Given Statements: If a number is divisible by 2, then it is even. If a number is even, then it is an integer. Conjecture: If a number is an integer, then it is divisible by 2.

15. Example 5: Using the Law of Syllogism Let x, y, and z represent the following. x: A number is divisible by 2. y: A number is even. z: A number is an integer. • Our conjecture (conclusion) attempts to imply that zx.We are given that xy and yz. The Law of Syllogism cannot be used to support our conjecture. The conclusion is not valid.

16. Check It Out! Example 6 Draw a conclusion from the given information. Given statements: a. If a polygon is a triangle, then it has three sides. If a polygon has three sides, then it is not a quadrilateral. Polygon P is a triangle. Conclusion: Polygon P is not a quadrilateral.

17. Lesson Quiz Determine if each conjecture is valid? 1. Given: If n is a natural number, then n is an integer. If n is an integer, then n is a rational number. 0.875 is a rational number. Conjecture: 0.875 is a natural number. not valid 2.Given: If an American citizen is at least 18 years old, then he or she is eligible to vote. Anna is a 20-year-old American citizen. Conjecture: Anna is eligible to vote. valid

18. Assignment #15 - Pages 110-112 Foundation: 7-17 odd Core: 20-24 even, 26-29 Challenge: 32