Pearson Unit 1 Topic 2: Reasoning and Proof 2-4: Deductive Reasoning Pearson Texas Geometry ©2016 Holt Geometry Texas ©2007

TEKS Focus: Foundation to TEKS (6) Use the process skills with deductive reasoning to prove and apply theorems by using a variety of methods such as coordinate, transformational, and axiomatic and formats such as two-column, paragraph, and flow chart. (1)(G) Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication. (1)(A) Apply mathematics to problems arising in everyday life, society, and the workplace. (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate. (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas.

Reasoning: Inductive vs Deductive

Reasoning: Inductive vs Deductive Recall: Inductive reasoning is the process of arriving at a conclusion based on a set of observations. Your conclusion is only a hypothesis because it is based on only a few observations. This hypothesis may be valid-(follows the rules of logic) or invalid-(does NOT follow the rules of logic) Deductive reasoning is the process of using logic to draw conclusions from given facts, definitions, and properties.

Deductive Reasoning 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.

If p, then q and If q, then r  If p, then r 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. If p, then q and If q, then r  If p, then r

It is not safe to be out in the open. No conclusion is possible. We don’t know if the hypothesis is true. Figure ABCD could be a rhombus.

Example: 2 Determine if the conjecture is VALID (follows the Law of Detachment) 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. VALID p q p is true  q is true Yes, it is valid by the Law of Detachment.

Example: 3 Determine if the conjecture is VALID by the Law of Detachment. Given: If you go on a field trip, you must have a signed permission slip. Lola has a signed permission slip. Conjecture: Lola wants to go on a field trip. INVALID p q q is true  p is true Invalid; her parents could be making her go on the field trip.

Example 4 If a whole number ends in 0, then it is divisible by 5. Valid by Law of Syllogism: if p, then q and if q, then r. Therefore, if p, then r. No conclusion is possible by the Law of Syllogism. There is a p statement and a q statement, but no r statement.

Example: 5 Determine if the conjecture is valid by the Law of Syllogism. Given: 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. Let p, q, and r represent the following. p: A number is divisible by 2. q: A number is even. r: A number is an integer. Answer: You are given that p  q and q  r. The Law of Syllogism cannot be used to deduce that r  p. The conclusion is invalid.

Example: 6 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. Let x, y, and z represent the following. x: An animal is a mammal. y: An animal has hair. z: An animal is a dog. You are given that x  y and z  x. Answer: Since x is the conclusion of the second conditional and the hypothesis of the first conditional, you can conclude that z  y. The conjecture is valid by Law of Syllogism.