1. Using Deductive Reasoning to Verify Conjectures 2-3
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Geometry
Holt Geometry

2. Warm Up Identify the hypothesis and conclusion of each conditional.
1. A mapping that is a reflection is a type of transformation.
2. The quotient of two negative numbers is positive.
3. Determine if the conditional “If x is a number then |x| > 0” is true. If false, give a counterexample.
H: A mapping is a reflection.
C: The mapping is a transformation.
H: Two numbers are negative.
C: The quotient is positive.
F; x = 0.

3. Objective
Apply the Law of Detachment and the Law of Syllogism in logical reasoning.

4. Deductive reasoning is the process of using logic to draw conclusions from given facts, definitions, and properties.

5. In deductive reasoning, if the given facts are true and you apply the correct logic, then the conclusion must be true. The Law of Detachment is one valid form of deductive reasoning.
Law of Detachment
If p  q is a true statement p.
Conclusion: q is true

6. Example 1: Verifying Conjectures by Using the Law of Detachment
Determine the conclusion
Given: If the side lengths of a triangle are 5 cm, 12 cm, and 13 cm, then the area of the triangle is 30 cm2. The side lengths of ∆PQR are 5cm, 12 cm, and 13 cm.
Conclusion: The area of ∆PQR is 30 cm2.

7. Example 2: Verifying Conjectures by Using the Law of Detachment
Determine the conclusion
Given: If the side lengths of a triangle are 5 cm, 12 cm, and 13 cm, then the area of the triangle is 30 cm2. The area of ∆PQR is 30 cm2.
Conclusion: Not Valid. The side lengths could be anything.

8. Example 3: Verifying Conjectures by Using the Law of Detachment
Determine the conclusion
Given: In the World Series, if a team wins four games, then the team wins the series. The Red Sox won four games in the 2004 World Series.
Conjecture: The Red Sox won the 2004 World Series.

9. Example 4
Determine the conclusion
Given: If there is no school, then I will go out to lunch with my friends. I went to Red Robin for lunch today.
Conclusion: invalid

10. Another valid form of deductive reasoning is the Law of Syllogism
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.
Law of Syllogism
If p  q and q  r are true statements, then p  r is a true statement.
**Looking for a repeated q.
** Reminds us of the transitive Property

11. Example 5: Verifying Conjectures by Using the Law of Syllogism
Determine the conclusion
Given: If a figure is a kite, then it is a quadrilateral. If a figure is a quadrilateral, then it is a polygon.
Conclusion: If a figure is a kite, then it is a polygon.

12. Example 3A: Verifying Conjectures by Using the Law of Syllogism Continued
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.
You 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.

13. Example 6: Verifying Conjectures by Using the Law of Syllogism
Determine the conclusion
Given: If a number is divisible by 2, then it is even. If a number is even, then it is an integer.
Conclusion: If a number is divisible by 2, then it is an integer.

14. Example 7
Determine if the conjecture is valid by the Law of Syllogism.
Given: If an animal is a mammal, then it has hair. If an animal is a dog, then it is a mammal.
Conjecture: If an animal is a dog, then it has hair.
Valid! – rearrange the sentences

15. Example 8 and 9: Applying the Laws of Deductive Reasoning
Draw a conclusion from the given information.
8. Given: If 2y = 4, then z = –1. If x + 3 = 12, then 2y = 4. x + 3 = 12
Conclusion: z = –1.
9. If the sum of the measures of two angles is 180°, then the angles are supplementary. If two angles are supplementary, they are not angles of a triangle. mA= 135°, and mB= 45°.
Conclusion: A and B are not angles of a triangle.

16. Example 10
Draw a conclusion from the given information.
Given: 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 each conclusion.
11. Given: If n is a natural number, then n is an integer. If n is an integer, then n is a rational number is a rational number.
Conclusion:
not valid
12. 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.
Conclusion:
Anna is eligible to vote