Chapter Two: Reasoning and Proof

1. Chapter Two:Reasoning and Proof Section 2-3: Deductive Reasoning

2. Objectives • To use the law of detachment. • To use the law of syllogism.

3. Vocabulary • Deductive Reasoning • Law of Detachment • Law of Syllogism

4. Deductive Reasoning • Deductive reasoning is the process of reasoning logically from given statements to conclusions. • Also called “logical reasoning”

5. Law of Detachment • If a conditional is true and its hypothesis is true, then the conclusion is also true. • Symbolic Form: • If p  q and p are both true, then q is true.

6. Using the Law of Detachment • Given: • If M is the midpoint of a segment, then it divides the segment into two congruent segments. • M is the midpoint of AB. • Conclude: • AM = MB

7. Using the Law of Detachment (continued) • Given: • If two segments have the same length, then they are congruent. • AB = 6 and BC = 6. • Conclude: • ___________________

8. Using the Law of Detachment (continued) • Given: • If two lines are parallel, then they are coplanar. • l and m are coplanar. • Conclude: • No conclusion can be drawn because we don’t know whether the hypothesis is true.

9. Law of Syllogism • The law of syllogism allows you to state a conclusion from two true conditional statements when the conclusion of one statement is the hypothesis of the other statements. • Symbolic Form: • If pq and qr, then pr

10. Using the Law of Syllogism • Conditional Statements: • If a number is prime, then it does not have repeated factors. • If a number does not have repeated factors, then it is not a perfect square. • We can conclude: • If a number is prime, then it is not a perfect square.

11. Using the Law of Syllogism (continued) • Conditional Statements: • If a number ends in 0, then it is divisible by 5. • If a number ends in 0, then it is divisible by 10. • We cannot draw any conclusions here because the hypothesis of one conditional is not the conclusion of the other.