Geometry Section 2-7 1112

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Proving Segment Relationships

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Geometry Section 2-7 1112

  1. 1. SECTION 2-7 Proving Segment Relationships
  2. 2. ESSENTIAL QUESTIONS How do you write proofs involving segment addition? How do you write proofs involving segment congruence?
  3. 3. POSTULATES & THEOREMS Ruler Postulate: Segment Addition Postulate:
  4. 4. POSTULATES & THEOREMS Rule r P o s t u l a t e : The points on any line or segment can be put into one-to-one correspondence with real numbers Segment Addition Postulate:
  5. 5. POSTULATES & THEOREMS Rule r P o s t u l a t e : The points on any line or segment can be put into one-to-one correspondence with real numbers You can measure the distance between two points Segment Addition Postulate:
  6. 6. POSTULATES & THEOREMS Rule r P o s t u l a t e : The points on any line or segment can be put into one-to-one correspondence with real numbers You can measure the distance between two points Segm e n t A d d i t i o n P o s t u l a t e : If A, B, and C are collinear, then B is between A and C if and only if (IFF) AB + BC = AC
  7. 7. THEOREM 2.2 - PROPERTIES OF SEGMENT CONGRUENCE Reflexive Property of Congruence: Symmetric Property of Congruence: Transitive Property of Congruence:
  8. 8. THEOREM 2.2 - PROPERTIES OF SEGMENT CONGRUENCE Reflexive Property of Congruence: Symmetric Property of Congruence: Transitive Property of Congruence:
  9. 9. THEOREM 2.2 - PROPERTIES OF SEGMENT CONGRUENCE Reflexive Property of Congruence: Symmetric Property of Congruence: If AB ≅ CD, then CD ≅ AB Transitive Property of Congruence:
  10. 10. THEOREM 2.2 - PROPERTIES OF SEGMENT CONGRUENCE Reflexive Property of Congruence: Symmetric Property of Congruence: If AB ≅ CD, then CD ≅ AB Transitive Property of Congruence: If AB ≅ CD and CD ≅ EF, then AB ≅ EF
  11. 11. EXAMPLE 1 Prove that if AB ≅ CD, then AC ≅ BD
  12. 12. EXAMPLE 1 Prove that if AB ≅ CD, then AC ≅ BD 1. AB ≅ CD
  13. 13. EXAMPLE 1 Prove that if AB ≅ CD, then AC ≅ BD 1. AB ≅ CD Given
  14. 14. EXAMPLE 1 Prove that if AB ≅ CD, then AC ≅ BD 1. AB ≅ CD Given 2. AB = CD
  15. 15. EXAMPLE 1 Prove that if AB ≅ CD, then AC ≅ BD 1. AB ≅ CD Given 2. AB = CD Def. of ≅ segments
  16. 16. EXAMPLE 1 Prove that if AB ≅ CD, then AC ≅ BD 1. AB ≅ CD Given 2. AB = CD Def. of ≅ segments 3. BC = BC
  17. 17. EXAMPLE 1 Prove that if AB ≅ CD, then AC ≅ BD 1. AB ≅ CD Given 2. AB = CD Def. of ≅ segments 3. BC = BC Reflexive property of equality
  18. 18. EXAMPLE 1 Prove that if AB ≅ CD, then AC ≅ BD 1. AB ≅ CD Given 2. AB = CD Def. of ≅ segments 3. BC = BC Reflexive property of equality 4. AB + BC = AC
  19. 19. EXAMPLE 1 Prove that if AB ≅ CD, then AC ≅ BD 1. AB ≅ CD Given 2. AB = CD Def. of ≅ segments 3. BC = BC Reflexive property of equality 4. AB + BC = AC Segment Addition
  20. 20. EXAMPLE 1 Prove that if AB ≅ CD, then AC ≅ BD 1. AB ≅ CD Given 2. AB = CD Def. of ≅ segments 3. BC = BC Reflexive property of equality 4. AB + BC = AC Segment Addition 5. CD + BC = AC
  21. 21. EXAMPLE 1 Prove that if AB ≅ CD, then AC ≅ BD 1. AB ≅ CD Given 2. AB = CD Def. of ≅ segments 3. BC = BC Reflexive property of equality 4. AB + BC = AC Segment Addition 5. CD + BC = AC Substitution prop. of equality
  22. 22. EXAMPLE 1 Prove that if AB ≅ CD, then AC ≅ BD
  23. 23. EXAMPLE 1 Prove that if AB ≅ CD, then AC ≅ BD 6. CD + BC = BD
  24. 24. EXAMPLE 1 Prove that if AB ≅ CD, then AC ≅ BD 6. CD + BC = BD Segment Addition
  25. 25. EXAMPLE 1 Prove that if AB ≅ CD, then AC ≅ BD 6. CD + BC = BD Segment Addition 7. AC = BD
  26. 26. EXAMPLE 1 Prove that if AB ≅ CD, then AC ≅ BD 6. CD + BC = BD Segment Addition 7. AC = BD Substitution property of equality
  27. 27. EXAMPLE 1 Prove that if AB ≅ CD, then AC ≅ BD 6. CD + BC = BD Segment Addition 7. AC = BD Substitution property of equality 8. AC ≅ BD
  28. 28. EXAMPLE 1 Prove that if AB ≅ CD, then AC ≅ BD 6. CD + BC = BD Segment Addition 7. AC = BD Substitution property of equality 8. AC ≅ BD Def. of ≅ segments
  29. 29. PROOF The Transitive Property of Congruence If AB ≅ CD and CD ≅ EF, then AB ≅ EF
  30. 30. PROOF The Transitive Property of Congruence If AB ≅ CD and CD ≅ EF, then AB ≅ EF 1. AB ≅ CD and CD ≅ EF
  31. 31. PROOF The Transitive Property of Congruence If AB ≅ CD and CD ≅ EF, then AB ≅ EF 1. AB ≅ CD and CD ≅ EF Given
  32. 32. PROOF The Transitive Property of Congruence If AB ≅ CD and CD ≅ EF, then AB ≅ EF 1. AB ≅ CD and CD ≅ EF Given 2. AB = CD and CD = EF
  33. 33. PROOF The Transitive Property of Congruence If AB ≅ CD and CD ≅ EF, then AB ≅ EF 1. AB ≅ CD and CD ≅ EF Given 2. AB = CD and CD = EF Def. of ≅ segments
  34. 34. PROOF The Transitive Property of Congruence If AB ≅ CD and CD ≅ EF, then AB ≅ EF 1. AB ≅ CD and CD ≅ EF Given 2. AB = CD and CD = EF Def. of ≅ segments 3. AB = EF
  35. 35. PROOF The Transitive Property of Congruence If AB ≅ CD and CD ≅ EF, then AB ≅ EF 1. AB ≅ CD and CD ≅ EF Given 2. AB = CD and CD = EF Def. of ≅ segments 3. AB = EF Transitive property of Equality
  36. 36. PROOF The Transitive Property of Congruence If AB ≅ CD and CD ≅ EF, then AB ≅ EF 1. AB ≅ CD and CD ≅ EF Given 2. AB = CD and CD = EF Def. of ≅ segments 3. AB = EF Transitive property of Equality 4. AB ≅ EF
  37. 37. PROOF The Transitive Property of Congruence If AB ≅ CD and CD ≅ EF, then AB ≅ EF 1. AB ≅ CD and CD ≅ EF Given 2. AB = CD and CD = EF Def. of ≅ segments 3. AB = EF Transitive property of Equality 4. AB ≅ EF Def. of ≅ segments
  38. 38. EXAMPLE 2 Matt Mitarnowski is designing a badge for his club. The length of the top edge of the badge is equal to the length of the left edge of the badge. The top edge of the badge is congruent to the right edge of the badge, and the right edge of the badge is congruent to the bottom edge of the badge. Prove that the bottom edge of the badge is congruent to the left edge of the badge.
  39. 39. EXAMPLE 2 Matt Mitarnowski is designing a badge for his club. The length of the top edge of the badge is equal to the length of the left edge of the badge. The top edge of the badge is congruent to the right edge of the badge, and the right edge of the badge is congruent to the bottom edge of the badge. Prove that the bottom edge of the badge is congruent to the left edge of the badge. A B D C
  40. 40. EXAMPLE 2 Matt Mitarnowski is designing a badge for his club. The length of the top edge of the badge is equal to the length of the left edge of the badge. The top edge of the badge is congruent to the right edge of the badge, and the right edge of the badge is congruent to the bottom edge of the badge. Prove that the bottom edge of the badge is congruent to the left edge of the badge. A B D C Given: AB = AD, AB ≅ BC, and BC ≅ CD
  41. 41. EXAMPLE 2 Matt Mitarnowski is designing a badge for his club. The length of the top edge of the badge is equal to the length of the left edge of the badge. The top edge of the badge is congruent to the right edge of the badge, and the right edge of the badge is congruent to the bottom edge of the badge. Prove that the bottom edge of the badge is congruent to the left edge of the badge. A B D C Given: AB = AD, AB ≅ BC, and BC ≅ CD Prove: AD ≅ CD
  42. 42. EXAMPLE 2 A B D C
  43. 43. EXAMPLE 2 1. AB = AD, AB ≅ BC, and BC ≅ CD A B D C
  44. 44. EXAMPLE 2 1. AB = AD, AB ≅ BC, and BC ≅ CD A B D C Given
  45. 45. EXAMPLE 2 1. AB = AD, AB ≅ BC, and BC ≅ CD Given 2. AB ≅ AD A B D C
  46. 46. EXAMPLE 2 1. AB = AD, AB ≅ BC, and BC ≅ CD A B D C Given 2. AB ≅ AD Def. of ≅ segments
  47. 47. EXAMPLE 2 1. AB = AD, AB ≅ BC, and BC ≅ CD Given 2. AB ≅ AD Def. of ≅ segments 3. AB ≅ CD A B D C
  48. 48. EXAMPLE 2 1. AB = AD, AB ≅ BC, and BC ≅ CD A B D C Given 2. AB ≅ AD Def. of ≅ segments 3. AB ≅ CD Transitive property
  49. 49. EXAMPLE 2 1. AB = AD, AB ≅ BC, and BC ≅ CD A B D C Given 2. AB ≅ AD Def. of ≅ segments 3. AB ≅ CD Transitive property 4. AD ≅ AB
  50. 50. EXAMPLE 2 1. AB = AD, AB ≅ BC, and BC ≅ CD A B D C Given 2. AB ≅ AD Def. of ≅ segments 3. AB ≅ CD Transitive property 4. AD ≅ AB Symmetric property
  51. 51. EXAMPLE 2 1. AB = AD, AB ≅ BC, and BC ≅ CD A B D C Given 2. AB ≅ AD Def. of ≅ segments 3. AB ≅ CD Transitive property 4. AD ≅ AB Symmetric property 5. AD ≅ CD
  52. 52. EXAMPLE 2 1. AB = AD, AB ≅ BC, and BC ≅ CD A B D C Given 2. AB ≅ AD Def. of ≅ segments 3. AB ≅ CD Transitive property 4. AD ≅ AB Symmetric property 5. AD ≅ CD Transitive property
  53. 53. PROBLEM SET
  54. 54. PROBLEM SET p. 145 #1-13, 15, 17, 18 “Trust yourself. Think for yourself. Act for yourself. Speak for yourself. Be yourself. Imitation is suicide.” - Marva Collins

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