Chapter 2 Final Review The Nature of Deductive Reasoning

1. Chapter 2 Final ReviewThe Nature of Deductive Reasoning Geometry (H) Period 2 Mrs. Liedell Elizabeth Wig

2. 2-1: Conditional Statements Consist of two parts: Hypothesis and Conclusion “If it was sunny outside (hypothesis)… Then I would ride my bike to school.” (conclusion) Can be represented as a  b Read as “a, then b” Conclusion is the “then clause” Hypothesis is the “if clause” b a b

3. Euler Diagrams Can be used to represent conditional statements a, the “if” statement, is on the inside b, the “then” statement, is on the outside. “If you are Socrates, then you are a man.” This means that all Socrates’ are Men. This does NOT mean that all Men are Socrates

4. 2-2: Converses and Definitions The converse of a statement is found by interchanging the hypothesis and conclusion. a b becomes b  a “If it’s raining, then I get wet.” This is true. “If I get wet, then it’s raining.” This is not true. Practice: find the converse for “If I am hungry, then I eat food.

5. 2-2.5: Contrapositive and Inverse • This statement is “If I sleep, I breathe.” ab • Converse (Interchange both): “If I breathe, I sleep.” • NOT TRUE! b a • Inverse (Deny both): “If I don’t sleep, I don’t breathe.” • NOT TRUE! not a  not b • Contrapositive: “If I don’t breathe, I don’t sleep.” • TRUE! not b  not a • Contrapositive and Original Statement are true together. • Converse and Inverse are also logically equivalent. b I sleep I breathe

6. 2-2 Ctd: “Only if” and “Iff” Only if statements go in front of the conclusion “It is raining only if I get wet” = “If it is raining, I get wet.” “Iff” is used when a conditional and its converse are true. This is often used for definitions. Example: A polygon is a triangle iff it has three sides. Both the statement and its converse are true.

7. 2-3: Direct Proofs • Syllogisms • a  b, • b c, • so a  c • This is a direct proof • The statements a  b and b c are called the premises of the argument • The statement a  c is called the conclusion of the argument. • Practice: Find the premises and conclusion of the Euler diagram at the right.

8. More 2-3 • Theorem: a statement that is proved by reasoning deductively from already accepted statements. • It is important to make sure that ALL the premises are true. • Practice: Spot the premise that isn’t true! • If a shape is a rhombus, it has four straight sides. • If a shape has four straight sides, it is a triangle. • If a shape is a triangle, then its angles add up to 180 degrees. • SO… if a shape is a rhombus, then its angles add up to 180 degrees. • The second premise is not true!

9. 2-4: Indirect Proofs • With an indirect proof, an assumption is made at the beginning that leads to a contradiction. • This is used to disprove a statement. • Example: The earth is not flat • Assumption: This is the “What if?” • “What if the earth was flat?” • If the earth was flat, the stars would rise at the same time in Tokyo. • Contradiction: Why this statement isn’t true. • The stars don’t rise at the same time in Tokyo as they do here. • Desired Conclusion: • Thus, the earth is not flat.

10. Indirect Proof (Another Example)

11. 2-5: A Deductive System • Postulate: A statement that is assumed to be true without proof. Here are two: • Two points determine a line. • Three noncollinear points determine a plane. • Postulates are necessary to create theorems. Otherwise there are circular definitions.

12. 2-6: Famous Theorem #1 • The Pythagorean Theorem!! • The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides (a2 +b2 = c2) • Practice: Hypotenuse = 30. 1 Leg = 18. What is the other leg?

13. 2-6 Continued • The Triangle Angle Sum Theorem • The sum of the angles of a triangle is 180 degrees. • Practice: Two angles of a triangle are 72 degrees and 59 degrees. What is the other angle?

14. Two Circle Theorems Area of a circle: (r is the radius) Circumference: C = (d is the diameter)