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Chapter 2 Student Notes

Chapter 2 Student Notes. Friday, 2/3/12 Dress for Success for Extra Credit. 2.1 Inductive Reasoning and Conjecture. Conjecture -. Make a conjecture from the given statement. Given: The toast is burnt. Conjecture: ___________________________ Given: It is winter.

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Chapter 2 Student Notes

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  1. Chapter 2 Student Notes Friday, 2/3/12 Dress for Success for Extra Credit

  2. 2.1Inductive Reasoning and Conjecture

  3. Conjecture - Make a conjecture from the given statement. Given: The toast is burnt. Conjecture: ___________________________ Given: It is winter. Conjecture: ___________________________ Given: Angle A is a right angle. Conjecture: ___________________________

  4. Counterexample - Write a counterexample for each conjecture. Conjecture: The sky is blue. Counterexample: ________________________________ Conjecture: Angle 1 and Angle 2 are congruent. Counterexample: ________________________________ Conjecture: l and m are parallel Counterexample: ________________________________

  5. Determine if each conjecture is true or false. Give a counterexample for any false conjecture. 1. Given: A, B, C are collinear. Conjecture: A, B, C are on the same line. 2. Given: 1 is a right angle. Conjecture: m1 = 90. 3. Given: AB = BC. Conjecture: B is the midpoint of AC. T / F T / F T / F

  6. Determine if each conjecture is true or false. Give a counterexample for any false conjecture. Given: The dog is brown. Conjecture: It is a chocolate lab. Given: 3 and 4 form a linear pair. Conjecture: 3  4 Given: 1 and 2 are complementary Conjecture: m1 = 45, m2 = 45. T / F T / F T / F m1 = 48, m2 = 42

  7. 2.2Logic

  8. Statement - • Truth Value – • Negation - • Compound Statement –

  9. Conjunction - Symbol for And: Disjunction - • Symbol for Or:

  10. Circle the statement that is true. p: Angle A is a right angle. r: Angle A is an obtuse angle. r: Angle A is an acute angle. 1. p q r 2. p q r > > > >

  11. Truth Table - • Examples of Truth Tables. > >

  12. 2.3Conditional Statements

  13. Conditional Statement - Statement: A right angle has a measure of 90 degrees. If-then: Statement: A car has four wheels. If-then: Statement: A triangle has 3 sides. If-then:

  14. Parts of a Conditional Statement If it is a car, then it has four wheels. • Hypothesis • Conclusion

  15. Converse - Conditional: If it is a car then it has 4 wheels. Converse: Conditional: If it is a pig, then it can fly. Converse: Conditional: If it is a right angle, then it measure 90. Converse:

  16. Inverse - Conditional: If it is a car then it has 4 wheels. Inverse: Conditional: If it is a pig, then it can fly. Inverse: Conditional: If it is a right angle, then it measure 90. Inverse:

  17. Contrapositive - Conditional: If it is a car, then it has 4 wheels. Contrapositive: Conditional: If it is a pig, then it can fly. Contrapositive: Conditional: If it is a right angle, then it measure 90. Contrapositive:

  18. Identify the converse, inverse and contrapositive of each conditional statement. Determine if each statement is true or false. T / F If you go to WMHS, then you are a hornet. T / F Converse: __________________________ __________________________ T / F Inverse: __________________________ __________________________ T / F Contrapositive: ______________________ ______________________

  19. Identify the converse, inverse and contrapositive of each conditional statement. Determine if each statement is true or false. T / F If it is a right angle, then it measures 90. T / F Converse: __________________________ __________________________ T / F Inverse: __________________________ __________________________ T / F Contrapositive: ______________________ ______________________

  20. 2.4Deductive Reasoning

  21. Deductive Reasoning -

  22. Law of Detachment

  23. Law of Syllogism

  24. Examples of the Laws of Detachment and Syllogism. Detachment If it is a triangle, then it has 3 sides. Syllogism If it is a Jeep, then it has 4 wheel drive.

  25. Determine whether the 3rd statement is valid based on the given information. If not, write invalid. If it is a dog, then it has 4 legs. Rover is a dog. Rover has 4 legs. Is it valid? Does it follow one of our Laws?

  26. Determine whether the 3rd statement is valid based on the given information. If not, write invalid. If you are 18 or older, then you are an adult. If you are an adult, then you can vote. If you are 18 or older, then you can vote. Is it valid? Does it follow one of our Laws?

  27. Use the Law of Detachment or the Law of Syllogism to determine if a valid conclusion can be reached. If it can, state it and the law used. If not, write no conclusion. If it is a car, then it has 4 wheels. A Ferrari is a car. __________________________ • If you go to the store, then you will go to the post office. • If you go to the post office, then you will buy stamps. • _________________________________________

  28. Use the Law of Detachment or the Law of Syllogism to determine if a valid conclusion can be reached. If it can, state it and the law used. If not, write no conclusion. If you are in college, then you are at least 18. Pete is in college. ______________________________ • Right angles are congruent. • Angle 1 and Angle 2 are congruent. • _______________________________

  29. 2.5Postulates Postulate – Statement that is accepted without proof.

  30. Postulate 2.1 - A B

  31. Postulate 2.2 - A P C B Plane P Plane ABC

  32. Postulate 2.3 _________________________ _____________________________________ Postulate 2.4 _________________________ _____________________________________ _____________________________________ Postulate 2.5 ________________________ ____________________________________ ____________________________________ ____________________________________

  33. Postulate 2.6 _________________________ _____________________________________ Postulate 2.7 _________________________ _____________________________________ R P

  34. Midpoint Theorem - A M B

  35. Determine if each statement is always, sometimes or never true. A, B, and C are collinear. A, B, and C, are coplanar. RST is a right angle. Two planes intersect to form a line. If AB = BC, the B is the midpoint of AC. Vertical angles are adjacent. If B is the midpoint of AC, then AB = BC.

  36. Determine the number of segments that can be drawn connecting each pair of points. 1. 2.

  37. 2.6Algebraic Proof

  38. Properties Reflexive: Symmetric Transitive Substitution

  39. Properties Distribution Addition / Subtraction Multiplication / Division

  40. Identify each property that justifies each statement. If 7 = x, then x = 7. If x + 5 = 7, then x = 2 If x = 7 and 7 = y, then x = y. If m1 + m2 = 180 and m2 = m3, then m1 + m3 = 180.

  41. Identify each property that justifies each statement. 2x + 1 = 2x + 1 If x – 6 = 7, then x = 13 If 2(x + 3) = 7, then 2x + 6 = 7. If 2x = 16, then x = 8.

  42. Given: 2x – 5 = 13Prove: x = 9 Statements _____________ _____________ _____________ Reasons 1. ___________ 2. ___________ 3. ___________

  43. 2 3 Given: 8 – n = 4 – n Prove: n = 12 Reasons ______________ ______________ ______________ ______________ ______________ Statements ___________________ ___________________ ___________________ ___________________ ___________________

  44. Given: 2x + 1 3 Prove: x = 10 = 7 Reasons 1. Given _________________ _________________ _________________ Statements _________________ _________________ _________________ 2x + 1 3 = 7

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