Geometry Chapter 2

1. GeometryChapter 2

2. Chapter 2-1Conditional Statements Objectives: •To recognize conditional statements •to write converse of conditional statements

3. Conditional A conditional is another name for if-then statements The “if” part is the hypothesis The “then” part is the conclusion

4. Conditional Identify the hypothesis and the conclusion of: If Texas won the 2006 Rose Bowl football game, then Texas was college football’s national champion.

5. Hypothesis Texas won the 2006 Rose Bowl football game Conclusion: Texas was college football’s national champion

6. Write a Conditional Write each sentence as a conditional A rectangle has four right angles.

7. If a figure is a rectangle, then it has four right angles.

8. A conditional can have a truth value, true or false. To show that a conditional is true, show that every time the hypothesis is true, the conclusion is also true. To show that a conditional statement is false, you need to find only one counterexample for which the hypothesis is true and the conclusion is false.

9. Finding a Counterexample Show that this conditional is false by finding a counterexample: If it is February, then there are only 28 days in the month.

10. Answer…. Counterexample: Because 2008 was a leap year, February had 29 days. February 2008 is a counterexample. The conditional is false because 2008 is a counterexample.

11. Converse The converse of a conditional switches the hypothesis and the conclusion. Write the converse of the following Conditional: If two lines intersect to form right angles, then they are perpendicular.

12. Answer… If two lines are perpendicular, then they intersect to form right angles.

13. Venn Diagrams Draw a Venn Diagram to illustrate this conditional: If you live in Chicago, then you live in Illinois.

14. The set of things that satisfy the hypothesis lies inside the set of things that satisfy the conclusion Residents Of Illinois Residents Of Chicago

15. Chapter 2-2Biconditionals and Definitions Objectives: •To write biconditionals •To recognize good definitions

16. Writing Biconditionals When a conditional and its converse are true, you can combine them as a true biconditional. Consider… If two angles have the same measure, then the angles are congruent.

17. Conditional: If two angles have the same measure, then the angles are congruent. Converse: If two angles are congruent, then the angles have the same measure. Since both the conditional and converse are true, you can combine them in a true biconditional by using if and only if.

18. if and only if Biconditional: two angles have the same measureif and only if the angles are congruent.

19. Separating a biconditional You can write a biconditional as two conditionals that are converses of each other.

20. Biconditional Statements Example: An angle is a straight angle if and only if its measure is 180˚. Conditional: If an angle is a straight angle, then its measure is 180˚.

21. If, then If pqIf an angle is a straight angle, then its measure is 180˚. (if p, then q) If qpIf an angle measure is 180˚, then it is a straight angle. (if q, then p) Symbolic: You read it: pqp if and only if q An angle is a straight angle if and only if its measure is 180˚.

22. Definition One way to show that a statement is not a good definition is to find a counterexample.

23. 2-3Deductive Reasoning Objectives: To use the Law of Detachment To use the Law of Syllogism

24. Deductive Reasoning Or logical reasoning, is the process of reasoning logically from given statements to a conclusion. If the given statements are true, deductive reasoning produces a true conclusion. For example, a physician uses deductive reasoning to diagnose a patient’s illness.

25. Law of Detachment If a conditional is true and its hypothesis is true, then its conclusion is true In symbolic form: If p  q is a true statement and p is true, Then q is true.

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

27. Law of Syllogism If a number ends in 0, then it is divisible by 10 If a number is divisible by 10, then it is divisible by 5. Conclusion?

28. Law of Syllogism Conclusion: If a number ends in 0, then it is divisible by 5.

29. Mathematical Systems A mathematical system is made up of a __________ and a collection of ____________ in an accepted framework of reasoning. The vocabulary of a mathematical system is made up of ________ and __________ The statements of a mathematical system are ________ and ___________

30. 2-4Reasoning In Algebra Objectives: to connect algebra and geometry In geometry you accept postulates and properties as true. You use deductive reasoning to prove other statements. Some of the properties that you accept as true are the properties of equality from algebra.

31. Properties of Equality Addition Prop: Subtraction: Multiplication Division Reflexsive Symmetric Transitive Substitution

32. The Distributive Property

33. Properties of Congruence Reflexsive Symmetric Transitive

34. 2-5Proving Angles Congruent Objectives:

35. Theorem

36. Vertical Angles Theorem Theorem 2-1:

37. Congruent Supplements Theorem Theorem 2-2:

38. Paragraph Proof

39. Congruent Complements Theorem Theorem 2-3: Theorem 2-4: Theorem 2-5:

40. Chapter 2 Review Reflexsive = Hypothesis= Transitive= Biconditional= Converse= Symmetric= Conclusion= Truth value= Deductive Reasoning= Theorem=