Geometry

1. Geometry Chapter 2: Reasoning and Proof

2. What you will learn today: • Make conjectures based on inductive reasoning • Find counterexamples • Create conjunctions and disjunctions • Determine truth values of conjunctions and disjunctions

3. 2.1: Inductive Reasoning and Conjecture • A conjecture is an educated guess based on known information. • Example: • Inductive reasoning is reasoning that uses a number of specific examples to arrive at a generalization or prediction.

4. Example One: • For points P, Q, and R, PQ = 9, QR = 15, and PR = 12. Make a conjecture and draw a figure to illustrate your conjecture.

5. You Do It: • For points L, M, and N, LM = 20, MN = 6, and LN = 14. Make a conjecture and draw a figure to illustrate your conjecture.

6. Counterexample • Conjectures are based on several observations that are mostly true. • It only takes one false example to prove a conjecture is not true. • The false example is called a counterexample.

7. Example Two: • Determine whether each conjecture is true or false. Give a counter example for any false conjecture. • Given: m + y ≥ 10, y ≥ 4 Conclusion: m ≤ 6 • Given: noncollinear points R, S, and T Conclusion:

8. You Do It • Determine whether each conjecture is true or false. Give a counterexample for any specific false conjecture. • Given: WXYZ is a rectangle Conclusion WX = YZ and WZ = XY • Given: JK = KL = LM = JM Conclusion: JKLM is a square

9. 2.2: Logic • A statement is any sentence that is either true or false, but not both. • Example: p: Today is Friday • Where a statement is true or false is its truth value. • Example: p is true

10. Logic • The negation of a statement has the opposite meaning as well as an opposite truth value. • Example: ~p: Today is not Friday • Truth value of ~p is false • Two or more statements can be joined to form a compound statements.

11. Compound Statements • A conjunction is a compound statement formed by joining two or more statements with the word and. • Symbols: p and q  p ˄ q • Both statements have to be true for the conjunction to be true. • Example: • p: Raleigh is a city in NC. • q: Raleigh is the capital of NC. • p ˄ q: Raleigh is a city in NC and Raleigh is the capital of NC.

12. Example One • Use the following statements to write a compound statement for each conjunction. Then find its truth value. • p: January 1 is the first day of the year • q: -5 + 11 = -6 • r: A triangle has three sides • p ˄ q • ~r ˄ q • ~q and r

13. You Do It • Use the following statements to write a compound statement for each conjunction. Then find its truth value. • p: One foot is 14 inches • q: September has 30 days • r: A plane is defined by three noncollinear points • p ˄ q • r ˄ ~p • ~q ˄ r

14. Disjunction • A disjunction is a compound statement formed by joining two or more statements with word or. • Symbols: p or q  p v q • Only one statement has to be true for the disjunction to be true • Example: • p: Raleigh is a city in NC • q: Raleigh is the capital of NC • p v q: Raleigh is a city in NC or Raleigh is the capital of NC

15. Example Two • Use the following statements to write a compound statement for each disjunction. Then find its truth value. • p: 100 ÷ 5 = 20 • q: The length of a radius of a circle is twice the length of its diameter • p or q • ~p v ~q

16. You Do It • Use the following statements to write a compound statement for each disjunction. Then find its truth value. • p: is proper notation for “line AB” • q: centimeters are metric units • r: 9 is a prime number • p v q • ~q v r

17. Venn Diagrams • Conjunctions can be illustrated with Venn diagrams

18. Example Three

19. You Do It

20. Classwork • Complete the following assignment and turn it in when you are finished (this way you don’t have homework over the weekend ) • Worksheet • Both sides – all problems • Will be graded for accuracy

22. Warm - Up • Make a conjecture about the following: • A, B, and C are points. AB = 2, BC = 4, and AC = 3 • Determine whether the following is true or false. Give a counterexample if the statement is false • Given: Points A, B, and C are collinear. Conclusion: AB + BC = AC • Given: Conclusion: • Create the compound statement and determine the truth value. • p: 10 + 8 = 18 • q: A rectangle has 3 sides • p ˄ ~q • ~p v ~q

23. 2.3: Conditional Statements • Analyze statements in the if – then form. • Write the converse, inverse, and contrapositive of if – then statements. • Write and understand biconditional statements. • Get $1500 cash back when you buy a new car. • Free cell phone with every one – year service enrollment.

24. The statements on the previous slide are examples of conditional statements. • A conditional statement is a statement that can be written in the if-then form. • Example: • Get $1500 cash back when you buy a new car. • If you buy a car, then you get $1500 cash back.

25. If – then statement • An if – then statement is written in the for if p, then q. • The phrase immediately following the word if is called the hypothesis • The phrase immediately following the word then is called the conclusion • p → q, read if p then q, or p implies q.

26. Example 1 • Identify the hypothesis and conclusion for each statement. • If points A, B, and C lie on line m, then they are collinear. • The Tigers will play in the tournament if they win their next game.

27. Your Turn • Identify the hypothesis and conclusion of each statement. • If a polygon has 6 sides, then it is a hexagon. • Tamika will advance to the next level of play if she completes the maze in her computer game.

28. Writing statements in if – then form • Some statements are conditional but are not in if – then form. • It is easier to identify the hypothesis and conclusion before writing the sentence in if – then form • Example: • All apes love bananas • Hypothesis: An animal is an ape • Conclusion: It loves bananas • If – then: If an animal is an ape, then it loves bananas

29. Example 2 • Identify the hypothesis and conclusion of each statement. Then write each statement in the if – then form. • An angle with a measure greater than 90 is an obtuse angle. • Perpendicular lines intersect.

30. Your Turn • Identify the hypothesis and conclusion of each statement. Then write each statement in the if – then form. • Distance is positive. • A five – sided polygon is a pentagon.

31. Truth Value • All cases of conditional statements are true except where the hypothesis is true and the conclusion is false.

32. Example 3 • Determine the truth value of the following statement for each set of conditions If you get 100% on your test, then your teacher will give you an A. • You get 100%; your teacher gives you an A • True • You get 100%; your teacher gives you a B • False • You get 98%: your teacher gives you an A • True • You get 85%; your teacher gives you a B • True

33. Your Turn • Determine the truth value of the following statement for each set of conditions If Parker rests for 10 days, his ankle will heal. • Parker rests for 10 days, and he still has a hurt ankle • False • Parker rests for 3 days, and he still has a hurt ankle • True • Parker rests for 10 days, and he does not have a hurt ankle anymore • True • Parker rests for 7 days, and he does not have a hurt ankle anymore • True

34. Related Conditionals

35. You can not determine any relationship between a conditional and the converse and inverse as far as truth value. • However, the following is true: • The truth value of the conditional and contrapositive will always be the same • The truth value of the converse and the inverse will always be the same • Statements with the same truth values are said to be logically equivalent • Conditional and Contrapositive are logically equivalent • Converse and Inverse are logically equivalent

36. Example 3 • Write the conditional, converse, inverse, and contrapositive of the statement: Linear pairs of angles are supplementary. Determine whether each statement is true or false. If a statement is false, give a counterexample.

37. Your Turn • Write the conditional, converse, inverse, and contrapositive of the statement: All squares are rectangles. Determine whether each statement is true or false. If the statement is false, give a counterexample.

38. Biconditional Statement • A biconditional statement is the conjunction of a conditional statement and its converse. • (p → q) ^ (q → p) is written p ↔ q, and read p if and only if q, can be abbreviated iff • Both the conditional and the converse must be true for a biconditional to be true.

39. Example 4 • Write each biconditional as a conditional and its converse. Then determine whether the biconditional is true or false. If false, give a counterexample. • Two angles measures are complements if and only if their sum is 90. • x > 9 iff x > 0

40. Your Turn • Write each biconditional as a conditional and its converse. Then determine whether the biconditional is true or false. If false, give a counterexample. • A calculator will run if and only if it has batteries. • 3x – 4 = 30 iff x = 7

41. Homework • Workbook • Section 2.3 • 1 – 10 (all) • You do not have any practice on the biconditional statement. Make sure you know how to create statements and know the truth value. It will be covered on your quiz tomorrow!

43. Warm - Up • Identify the hypothesis and conclusion of each statement • If 2x + 6 = 10, then x = 2 • Write each statement in if – then form • Get a free visit with a one – year fitness plan • Vertical angles are congruent • Write the converse, inverse, and contrapositive of each conditional statement. Determine whether each related conditional is true or false. • All rectangles are quadrilaterals • If you live in Dallas, then you live in Texas.

44. Homework • H: 3x + 4 = -5; C : x = -3 • H: you take a class in television broadcasting; C = you will film a sporting event • If you do not remember the past, then you are condemned to repeat it. • If two angles are adjacent, then they share a common vertex and a common side. • True • True • True • Converse: If -8 > 0 then (-8)2 > 0; true Inverse: If (-8)2 ≤ 0, then -8 ≤ 0 true Contrapositive: If – 8 ≤ 0, then (-8)2 ≤ 0; false • If you are a junior, then you wait on tables • If you wait on tables, then you are a junior

45. Quiz Time • Please clear off your desk • You will have plenty of time to complete your quiz • When you are finished, please remain quiet until everyone else has finished. • We will begin 2.4: Deductive Reasoning with the Law of Detachment and Law of Syllogism

46. 2.4: Deductive Reasoning • Use the Law of Detachment • Use the Law of Syllogism • When you are ill, your doctor may prescribe an antibiotic to help you get better. Doctors may use a dose chart to determine the correct amount of medicine based on your weight.

47. The process that the doctors use to determine the amount of medicine a patient should take is called deductive reasoning. • Deductive reasoning uses facts, rules, definitions, or properties to reach a logical conclusion. • One way to do this is the Law of Detachment.

48. Law of Detachment • If p → q is true and p is true, then q is also true. • Example:

49. Example 1 • The following is a true conditional. Determine whether each conclusion is a valid based on the given information. Explain your reasoning. If a ray is an angle bisector, then it divides the angle into two congruent angles.

50. Your Turn • The following is a true conditional. determine whether each conclusion is a valid based on the given information. Explain your reasoning. If two segments are congruent and the second segment is congruent to a third segment, then the first segment is also congruent to the third. • Given: WX = UV and UV = RT. Conclusion: WX = RT. • Given: UV and WX = RT. Conclusion: WX = UV and UV = RT