Geometry Section 2-5 1112

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Geometry Section 2-5 1112

  1. 1. Section 2-5 Postulates and Paragraph Proofs Thursday, November 6, 14
  2. 2. Essential Questions • How do you identify and use basic postulates about points, lines, and planes? • How do you write paragraph proofs? Thursday, November 6, 14
  3. 3. Vocabulary 1. Postulate: 2. Axiom: 3. Proof: 4. Theorem: 5. Deductive Argument: Thursday, November 6, 14
  4. 4. Vocabulary 1. P o s t u la t e : A statement that is accepted to be true without proof 2. Axiom: 3. Proof: 4. Theorem: 5. Deductive Argument: Thursday, November 6, 14
  5. 5. Vocabulary 1. P o s t u la t e : A statement that is accepted to be true without proof 2. Axiom: Another name for a postulate 3. Proof: 4. Theorem: 5. Deductive Argument: Thursday, November 6, 14
  6. 6. Vocabulary 1. P o s t u la t e : A statement that is accepted to be true without proof 2. Axiom: Another name for a postulate 3. P r o o f: A logical argument made up of statements that are supported by another statement that is accepted as true 4. Theorem: 5. Deductive Argument: Thursday, November 6, 14
  7. 7. Vocabulary 1. P o s t u la t e : A statement that is accepted to be true without proof 2. Axiom: Another name for a postulate 3. P r o o f: A logical argument made up of statements that are supported by another statement that is accepted as true 4. T h e o r e m : A statement or conjecture that has been proven true 5. Deductive Argument: Thursday, November 6, 14
  8. 8. Vocabulary 1. P o s t u la t e : A statement that is accepted to be true without proof 2. Axiom: Another name for a postulate 3. P r o o f: A logical argument made up of statements that are supported by another statement that is accepted as true 4. Theorem: A statement or conjecture that has been A logical chain of statements proven true 5. Deductive Argument: that link the given to what you are trying to prove Thursday, November 6, 14
  9. 9. Vocabulary 6. Paragraph Proof: 7. Informal Proof: Thursday, November 6, 14
  10. 10. Vocabulary 6. P a r a g r a p h P r o o f : When a paragraph is written to logically explain why a given conjecture is true 7. Informal Proof: Thursday, November 6, 14
  11. 11. Vocabulary 6. P a r a g r a p h P r o o f : When a paragraph is written to logically explain why a given conjecture is true 7. I n fo r m a l P r o o f : Another name for a paragraph proof as it allows for free writing to provide the logical explanation Thursday, November 6, 14
  12. 12. Harkening back to Chapter 1 Old ideas about points, lines, and planes are now postulates! Thursday, November 6, 14
  13. 13. Harkening back to Chapter 1 Old ideas about points, lines, and planes are now postulates! 2.1: Through any two points, there is exactly one line. Thursday, November 6, 14
  14. 14. Harkening back to Chapter 1 Old ideas about points, lines, and planes are now postulates! 2.1: Through any two points, there is exactly one line. 2.2: Through any three noncollinear points, there is exactly one plane. Thursday, November 6, 14
  15. 15. Harkening back to Chapter 1 Old ideas about points, lines, and planes are now postulates! 2.1: Through any two points, there is exactly one line. 2.2: Through any three noncollinear points, there is exactly one plane. 2.3: A line contains at least two points. Thursday, November 6, 14
  16. 16. Harkening back to Chapter 1 Old ideas about points, lines, and planes are now postulates! 2.1: Through any two points, there is exactly one line. 2.2: Through any three noncollinear points, there is exactly one plane. 2.3: A line contains at least two points. 2.4: A plane contains at least three noncollinear points. Thursday, November 6, 14
  17. 17. Harkening back to Chapter 1 Old ideas about points, lines, and planes are now postulates! 2.1: Through any two points, there is exactly one line. 2.2: Through any three noncollinear points, there is exactly one plane. 2.3: A line contains at least two points. 2.4: A plane contains at least three noncollinear points. 2.5: If two points lie in a plane, then the entire line containing those points lies in the plane. Thursday, November 6, 14
  18. 18. Harkening back to Chapter 1 Old ideas about points, lines, and planes are now postulates! Thursday, November 6, 14
  19. 19. Harkening back to Chapter 1 Old ideas about points, lines, and planes are now postulates! 2.6: If two lines intersect, then their intersection is exactly one point. Thursday, November 6, 14
  20. 20. Harkening back to Chapter 1 Old ideas about points, lines, and planes are now postulates! 2.6: If two lines intersect, then their intersection is exactly one point. 2.7: If two planes intersect, then their intersection is a line. Thursday, November 6, 14
  21. 21. Example 1 Determine whether the statement is always, sometimes, or never true. a. Points E and F are contained by exactly one line. b. There is exactly one plane that contains points A, B, and C. Thursday, November 6, 14
  22. 22. Example 1 Determine whether the statement is always, sometimes, or never true. a. Points E and F are contained by exactly one line. Always true b. There is exactly one plane that contains points A, B, and C. Thursday, November 6, 14
  23. 23. Example 1 Determine whether the statement is always, sometimes, or never true. a. Points E and F are contained by exactly one line. Always true Only one line can be drawn through any two points b. There is exactly one plane that contains points A, B, and C. Thursday, November 6, 14
  24. 24. Example 1 Determine whether the statement is always, sometimes, or never true. a. Points E and F are contained by exactly one line. Always true Only one line can be drawn through any two points b. There is exactly one plane that contains points A, B, and C. Sometimes true Thursday, November 6, 14
  25. 25. Example 1 Determine whether the statement is always, sometimes, or never true. a. Points E and F are contained by exactly one line. Always true Only one line can be drawn through any two points b. There is exactly one plane that contains points A, B, and C. Sometimes true If the three points are collinear, then an infinite number planes can be drawn. If they are noncollinear, then it is true. Thursday, November 6, 14
  26. 26. Example 1 Determine whether the statement is always, sometimes, or never true. c. Planes R and T intersect at point P. Thursday, November 6, 14
  27. 27. Example 1 Determine whether the statement is always, sometimes, or never true. c. Planes R and T intersect at point P. Never true Thursday, November 6, 14
  28. 28. Example 1 Determine whether the statement is always, sometimes, or never true. c. Planes R and T intersect at point P. Never true Two planes intersect in a line Thursday, November 6, 14
  29. 29. Example 2 Given that AC intersects CD, write a paragraph proof to show that A, C, and D determine a plane. Thursday, November 6, 14
  30. 30. Example 2 Given that AC intersects CD, write a paragraph proof to show that A, C, and D determine a plane. Since the two lines intersect, they must intersect at point C as two lines intersect in exactly one point. Thursday, November 6, 14
  31. 31. Example 2 Given that AC intersects CD, write a paragraph proof to show that A, C, and D determine a plane. Since the two lines intersect, they must intersect at point C as two lines intersect in exactly one point. Points A and D are on different lines, so A, C, and D are noncollinear by definition of noncollinear. Thursday, November 6, 14
  32. 32. Example 2 Given that AC intersects CD, write a paragraph proof to show that A, C, and D determine a plane. Since the two lines intersect, they must intersect at point C as two lines intersect in exactly one point. Points A and D are on different lines, so A, C, and D are noncollinear by definition of noncollinear. Since three noncollinear points determine exactly one plane, points A, C, and D determine a plane. Thursday, November 6, 14
  33. 33. Example 3 Given that M is the midpoint of XY, write a paragraph proof to show that XM ≅ MY. Thursday, November 6, 14
  34. 34. Example 3 Given that M is the midpoint of XY, write a paragraph proof to show that XM ≅ MY. If M is the midpoint of XY, then by the definition of midpoint, XM = MY. Since they have the same measure, we know that, by the definition of congruence, XM ≅ MY. Thursday, November 6, 14
  35. 35. Example 3 Given that M is the midpoint of XY, write a paragraph proof to show that XM ≅ MY. If M is the midpoint of XY, then by the definition of midpoint, XM = MY. Since they have the same measure, we know that, by the definition of congruence, XM ≅ MY. Theorem 2.1 (Midpoint Theorem): Thursday, November 6, 14
  36. 36. Example 3 Given that M is the midpoint of XY, write a paragraph proof to show that XM ≅ MY. If M is the midpoint of XY, then by the definition of midpoint, XM = MY. Since they have the same measure, we know that, by the definition of congruence, XM ≅ MY. Theo r e m 2 . 1 ( M id p o i n t T h e o r e m ) : If M is the midpoint of XY, then XM ≅ MY. Thursday, November 6, 14
  37. 37. Problem Set Thursday, November 6, 14
  38. 38. Problem Set p. 128 #1-41 odd “The first precept was never to accept a thing as true until I knew it as such without a single doubt.” - Rene Descartes Thursday, November 6, 14

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