2-4 Deductive Reasoning Deductive Reasoning: Using facts, rules, definitions, or properties to reach logical conclusions. Law of Detachment: A form of deductive reasoning that is used to draw conclusions from true conditional statements. If p  q is true and p is true, then q is also true. [(p  q) Λ p]  q

Example #1 The statement below is a true conditional. Determine whether each conclusion is valid based on the given information. Explain your reasoning. If segments are parallel, then they do not intersect. Given: AB and CD do not intersect. Conclusion: AB || CD Answer: Invalid; segments may not intersect and may not be parallel.

Law of Syllogism: Law of logic that is similar to the transitive property of equality. If p  q and q  r are true, then p  r is also true. [(p  q) Λ (q  r)]  (p  r) Example: If 2x = 14, then x = 7 and if x = 7, then 1/x = 1/7. Therefore, if 2x = 14, then 1/x = 1/7

Example #2 Use the Law of Syllogism to determine whether a valid conclusion can be reached from each set of statements. A) (1) If you stand in line, then you will get to ride the new roller coaster. (2) If you are at least 48 inches tall, you will get to ride the new roller coaster. No valid Conclusion B) (1) If a polygon has six congruent sides, then it is a regular hexagon. (2) If a regular hexagon has a side length of 3 units, then the perimeter is 3(6) or 18 units. If a polygon has six congruent sides, then the perimeter is x(6) or 6x.

Example #3 Determine whether statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not, write invalid. (1) The length of a side of square A is the same as the length of side of square B. (2) If the lengths of the sides of two squares are the same, then the squares have the same perimeter. (3) Square A and square B have the same perimeter. Valid – Law of Detachment

Homework #12 Practice Worksheet & p. 102 9-25 odd, 31, 34-35