Reasoning, Conditionals, and Postulates Sections 2-1, 2-3, 2-5

Find the next item in the pattern. Identifying a Pattern January, March, May,... The next month is July.

Find the next item in the pattern. Identifying a Pattern 7, 14, 21, 28, … The next multiple is 35.

Reasoning Inductive Reasoning – To draw a conclusion from a pattern. Conjecture – A statement you believe to be true based on inductive reasoning. Counterexample – One example in which the conjecture is not true; proves the conjecture is false. Deductive Reasoning – To draw conclusions from given facts, definitions, and properties.

Show that the conjecture is false by finding a counterexample. Example: Finding a Counterexample For every integer n, n 3 is positive. n = –3 is a counterexample.

Lesson Quiz Find the next item in each pattern , 0.07, 0.007, … Determine if each conjecture is true. If false, give a counterexample. 3. The quotient of two negative numbers is a positive number. 4. Every prime number is odd. 5. Two supplementary angles are not congruent. 6. The square of an odd integer is odd. false; 2 true false; 90° and 90° true

Conditional Statements

Identify the hypothesis and conclusion of each conditional. Example A.If today is Thanksgiving Day, then today is Thursday. B. A number is a rational number if it is an integer. C. A number is divisible by 3 if it is divisible by 6.

Determine if the conditional is true. If false, give a counterexample. Examples: 1. If this month is August, then next month is September. 2. If two angles are acute, then they are congruent.

Related Conditionals: Conditional: p → q (read as “if p then q”) Converse: q → p (switch: “if q then p”) Inverse: ~p → ~q (“if not p then not q”) Contrapositive: ~q → ~p (“if not q then not p”)

Example: Biology Application Inverse: If an animal is not an adult insect, then it does not have six legs. Converse: If an animal has six legs, then it is an adult insect. Conditional : If an animal is an adult insect, then it has six legs. Contrapositive: If an animal does not have six legs, then it is not an adult insect.

Lesson Quiz: Part I Identify the hypothesis and conclusion of each conditional. 1. A triangle with one right angle is a right triangle. 2. All even numbers are divisible by Determine if the statement “If n 2 = 144, then n = 12” is true. If false, give a counterexample. H: A number is even. C: The number is divisible by 2. H: A triangle has one right angle. C: The triangle is a right triangle. False; n = –12.

Lesson Quiz: Part II 4. Write the converse, inverse, and contrapositive of the conditional statement “If Maria’s birthday is February 29, then she was born in a leap year.” Converse: If Maria was born in a leap year, then her birthday is February 29. Inverse: If Maria’s birthday is not February 29, then she was not born in a leap year. Contrapositive: If Maria was not born in a leap year, then her birthday is not February 29.

Postulate - A statement that describes the relationship between basic terms in Geometry. Postulates are accepted as true without proof. Examples of some Postulates: Through any 2 points there is exactly 1 line. Through any 3 noncollinear points there is exactly 1 plane. A line contains at least 2 points. A plane contains at least 3 noncollinear points.

Theorem A conjecture or statement that can be shown to be true. Used like a definition or postulate. Midpoint Theorem - If M is the midpoint of AB, then AM  to MB. ABM

Proof A logical argument in which each statement is supported by a statement that is true (or accepted as true). Supporting evidence in a proof (the reason you can make the statement) are usually postulates, theorems, properties, definitions or given information.