Geometry Section 6-5 1112

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Rhombi and Squares

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Geometry Section 6-5 1112

  1. 1. Section 6-5 Rhombi and Squares Tuesday, April 29, 14
  2. 2. Essential Questions • How do you recognize and apply the properties of rhombi and squares? • How do you determine whether quadrilaterals are rectangles, rhombi, or squares? Tuesday, April 29, 14
  3. 3. Vocabulary 1. Rhombus: Properties: 2. Square: Properties: Tuesday, April 29, 14
  4. 4. Vocabulary 1. Rhombus: A parallelogram with four congruent sides Properties: 2. Square: Properties: Tuesday, April 29, 14
  5. 5. Vocabulary 1. Rhombus: A parallelogram with four congruent sides Properties: All properties of parallelograms plus four congruent sides, perpendicular diagonals, and angles that are bisected by the diagonals 2. Square: Properties: Tuesday, April 29, 14
  6. 6. Vocabulary 1. Rhombus: A parallelogram with four congruent sides Properties: All properties of parallelograms plus four congruent sides, perpendicular diagonals, and angles that are bisected by the diagonals 2. Square: A parallelogram with four congruent sides and four right angles Properties: Tuesday, April 29, 14
  7. 7. Vocabulary 1. Rhombus: A parallelogram with four congruent sides Properties: All properties of parallelograms plus four congruent sides, perpendicular diagonals, and angles that are bisected by the diagonals 2. Square: A parallelogram with four congruent sides and four right angles Properties: All properties of parallelograms, rectangles, and rhombi Tuesday, April 29, 14
  8. 8. Theorems Diagonals of a Rhombus 6.15: 6.16: Tuesday, April 29, 14
  9. 9. Theorems Diagonals of a Rhombus 6.15: If a parallelogram is a rhombus, then its diagonals are perpendicular 6.16: Tuesday, April 29, 14
  10. 10. Theorems Diagonals of a Rhombus 6.15: If a parallelogram is a rhombus, then its diagonals are perpendicular 6.16: If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles Tuesday, April 29, 14
  11. 11. Theorems Conditions for Rhombi and Squares 6.17: 6.18: 6.19: Tuesday, April 29, 14
  12. 12. Theorems Conditions for Rhombi and Squares 6.17: If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus 6.18: 6.19: Tuesday, April 29, 14
  13. 13. Theorems Conditions for Rhombi and Squares 6.17: If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus 6.18: If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus 6.19: Tuesday, April 29, 14
  14. 14. Theorems Conditions for Rhombi and Squares 6.17: If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus 6.18: If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus 6.19: If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus Tuesday, April 29, 14
  15. 15. Theorems Conditions for Rhombi and Squares 6.20: Tuesday, April 29, 14
  16. 16. Theorems Conditions for Rhombi and Squares 6.20: If a quadrilateral is both a rectangle and a rhombus, then it is a square Tuesday, April 29, 14
  17. 17. Example 1 The diagonals of a rhombus WXYZ intersect at V. Use the given information to find each measure or value. a. If m∠WZX = 39.5°, find m∠ZYX. Tuesday, April 29, 14
  18. 18. Example 1 The diagonals of a rhombus WXYZ intersect at V. Use the given information to find each measure or value. a. If m∠WZX = 39.5°, find m∠ZYX. Since m∠WZX = 39.5°, we know that m∠WZY = 79°. Tuesday, April 29, 14
  19. 19. Example 1 The diagonals of a rhombus WXYZ intersect at V. Use the given information to find each measure or value. a. If m∠WZX = 39.5°, find m∠ZYX. Since m∠WZX = 39.5°, we know that m∠WZY = 79°. 180° − 79° Tuesday, April 29, 14
  20. 20. Example 1 The diagonals of a rhombus WXYZ intersect at V. Use the given information to find each measure or value. a. If m∠WZX = 39.5°, find m∠ZYX. Since m∠WZX = 39.5°, we know that m∠WZY = 79°. 180° − 79° = 101° Tuesday, April 29, 14
  21. 21. Example 1 The diagonals of a rhombus WXYZ intersect at V. Use the given information to find each measure or value. a. If m∠WZX = 39.5°, find m∠ZYX. Since m∠WZX = 39.5°, we know that m∠WZY = 79°. 180° − 79° = 101° m∠ZYX = 101° Tuesday, April 29, 14
  22. 22. Example 1 The diagonals of a rhombus WXYZ intersect at V. Use the given information to find each measure or value. b. If WX = 8x − 5 and WZ = 6x + 3, find x. Tuesday, April 29, 14
  23. 23. Example 1 The diagonals of a rhombus WXYZ intersect at V. Use the given information to find each measure or value. b. If WX = 8x − 5 and WZ = 6x + 3, find x. WX = WZ Tuesday, April 29, 14
  24. 24. Example 1 The diagonals of a rhombus WXYZ intersect at V. Use the given information to find each measure or value. b. If WX = 8x − 5 and WZ = 6x + 3, find x. WX = WZ 8x − 5 = 6x + 3 Tuesday, April 29, 14
  25. 25. Example 1 The diagonals of a rhombus WXYZ intersect at V. Use the given information to find each measure or value. b. If WX = 8x − 5 and WZ = 6x + 3, find x. WX = WZ 8x − 5 = 6x + 3 2x = 8 Tuesday, April 29, 14
  26. 26. Example 1 The diagonals of a rhombus WXYZ intersect at V. Use the given information to find each measure or value. b. If WX = 8x − 5 and WZ = 6x + 3, find x. WX = WZ 8x − 5 = 6x + 3 2x = 8 x = 4 Tuesday, April 29, 14
  27. 27. Example 2 Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6 Prove: LMNP is a rhombus Tuesday, April 29, 14
  28. 28. Example 2 Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6 Prove: LMNP is a rhombus 1. LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6 Tuesday, April 29, 14
  29. 29. Example 2 Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6 Prove: LMNP is a rhombus 1. LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6 1. Given Tuesday, April 29, 14
  30. 30. Example 2 Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6 Prove: LMNP is a rhombus 1. LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6 1. Given 2. LM ∣∣ PN and MN ∣∣ PL Tuesday, April 29, 14
  31. 31. Example 2 Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6 Prove: LMNP is a rhombus 1. LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6 1. Given 2. Def. of parallelogram2. LM ∣∣ PN and MN ∣∣ PL Tuesday, April 29, 14
  32. 32. Example 2 Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6 Prove: LMNP is a rhombus 1. LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6 1. Given 2. Def. of parallelogram 3. ∠5 ≅ ∠1 2. LM ∣∣ PN and MN ∣∣ PL Tuesday, April 29, 14
  33. 33. Example 2 Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6 Prove: LMNP is a rhombus 1. LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6 1. Given 2. Def. of parallelogram 3. ∠5 ≅ ∠1 3. Alt. int. ∠’s of ∣∣ lines ≅ 2. LM ∣∣ PN and MN ∣∣ PL Tuesday, April 29, 14
  34. 34. Example 2 Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6 Prove: LMNP is a rhombus 1. LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6 1. Given 2. Def. of parallelogram 3. ∠5 ≅ ∠1 3. Alt. int. ∠’s of ∣∣ lines ≅ 4. ∠1 ≅ ∠ 6 and ∠6 ≅ ∠5 2. LM ∣∣ PN and MN ∣∣ PL Tuesday, April 29, 14
  35. 35. Example 2 Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6 Prove: LMNP is a rhombus 1. LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6 1. Given 2. Def. of parallelogram 3. ∠5 ≅ ∠1 3. Alt. int. ∠’s of ∣∣ lines ≅ 4. ∠1 ≅ ∠ 6 and ∠6 ≅ ∠5 4. Transitive 2. LM ∣∣ PN and MN ∣∣ PL Tuesday, April 29, 14
  36. 36. Example 2 Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6 Prove: LMNP is a rhombus 1. LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6 1. Given 2. Def. of parallelogram 3. ∠5 ≅ ∠1 3. Alt. int. ∠’s of ∣∣ lines ≅ 4. ∠1 ≅ ∠ 6 and ∠6 ≅ ∠5 4. Transitive 5. LMNP is a rhombus 2. LM ∣∣ PN and MN ∣∣ PL Tuesday, April 29, 14
  37. 37. Example 2 Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6 Prove: LMNP is a rhombus 1. LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6 1. Given 2. Def. of parallelogram 3. ∠5 ≅ ∠1 3. Alt. int. ∠’s of ∣∣ lines ≅ 4. ∠1 ≅ ∠ 6 and ∠6 ≅ ∠5 4. Transitive 5. LMNP is a rhombus 5. One diagonal bisects opposite angles 2. LM ∣∣ PN and MN ∣∣ PL Tuesday, April 29, 14
  38. 38. Example 3 Matt Mitarnowski is measuring the boundary of a new garden. He wants the garden to be square. He has set each of the corner stakes 6 feet apart. What does Matt need to know to make sure that the garden is a square? Tuesday, April 29, 14
  39. 39. Example 3 Matt Mitarnowski is measuring the boundary of a new garden. He wants the garden to be square. He has set each of the corner stakes 6 feet apart. What does Matt need to know to make sure that the garden is a square? Matt knows he at least has a rhombus as all four sides are congruent. To make it a square, it must also be a rectangle, so the diagonals must also be congruent, as anything that is both a rhombus and a rectangle is also a square. Tuesday, April 29, 14
  40. 40. Example 4 Determine whether parallelogram ABCD is a rhombus, rectangle, or square for A(−2, −1), B(−1, 3), C(3, 2), and D(2, −2). List all that apply and explain how you know. Tuesday, April 29, 14
  41. 41. Example 4 Determine whether parallelogram ABCD is a rhombus, rectangle, or square for A(−2, −1), B(−1, 3), C(3, 2), and D(2, −2). List all that apply and explain how you know. AC = (−2 − 3)2 + (−1− 2)2 Tuesday, April 29, 14
  42. 42. Example 4 Determine whether parallelogram ABCD is a rhombus, rectangle, or square for A(−2, −1), B(−1, 3), C(3, 2), and D(2, −2). List all that apply and explain how you know. AC = (−2 − 3)2 + (−1− 2)2 = (−5)2 + (−3)2 Tuesday, April 29, 14
  43. 43. Example 4 Determine whether parallelogram ABCD is a rhombus, rectangle, or square for A(−2, −1), B(−1, 3), C(3, 2), and D(2, −2). List all that apply and explain how you know. AC = (−2 − 3)2 + (−1− 2)2 = (−5)2 + (−3)2 = 25 + 9 Tuesday, April 29, 14
  44. 44. Example 4 Determine whether parallelogram ABCD is a rhombus, rectangle, or square for A(−2, −1), B(−1, 3), C(3, 2), and D(2, −2). List all that apply and explain how you know. AC = (−2 − 3)2 + (−1− 2)2 = (−5)2 + (−3)2 = 25 + 9 = 34 Tuesday, April 29, 14
  45. 45. Example 4 Determine whether parallelogram ABCD is a rhombus, rectangle, or square for A(−2, −1), B(−1, 3), C(3, 2), and D(2, −2). List all that apply and explain how you know. AC = (−2 − 3)2 + (−1− 2)2 = (−5)2 + (−3)2 = 25 + 9 = 34 BD = (−1− 2)2 + (3 + 2)2 Tuesday, April 29, 14
  46. 46. Example 4 Determine whether parallelogram ABCD is a rhombus, rectangle, or square for A(−2, −1), B(−1, 3), C(3, 2), and D(2, −2). List all that apply and explain how you know. AC = (−2 − 3)2 + (−1− 2)2 = (−5)2 + (−3)2 = 25 + 9 = 34 BD = (−1− 2)2 + (3 + 2)2 = (−3)2 + (5)2 Tuesday, April 29, 14
  47. 47. Example 4 Determine whether parallelogram ABCD is a rhombus, rectangle, or square for A(−2, −1), B(−1, 3), C(3, 2), and D(2, −2). List all that apply and explain how you know. AC = (−2 − 3)2 + (−1− 2)2 = (−5)2 + (−3)2 = 25 + 9 = 34 BD = (−1− 2)2 + (3 + 2)2 = (−3)2 + (5)2 = 9 + 25 Tuesday, April 29, 14
  48. 48. Example 4 Determine whether parallelogram ABCD is a rhombus, rectangle, or square for A(−2, −1), B(−1, 3), C(3, 2), and D(2, −2). List all that apply and explain how you know. AC = (−2 − 3)2 + (−1− 2)2 = (−5)2 + (−3)2 = 25 + 9 = 34 BD = (−1− 2)2 + (3 + 2)2 = (−3)2 + (5)2 = 9 + 25 = 34 Tuesday, April 29, 14
  49. 49. Example 4 Determine whether parallelogram ABCD is a rhombus, rectangle, or square for A(−2, −1), B(−1, 3), C(3, 2), and D(2, −2). List all that apply and explain how you know. AC = (−2 − 3)2 + (−1− 2)2 = (−5)2 + (−3)2 = 25 + 9 = 34 BD = (−1− 2)2 + (3 + 2)2 = (−3)2 + (5)2 = 9 + 25 = 34 The diagonals are congruent, so it is a rectangle Tuesday, April 29, 14
  50. 50. Example 4 Determine whether parallelogram ABCD is a rhombus, rectangle, or square for A(−2, −1), B(−1, 3), C(3, 2), and D(2, −2). List all that apply and explain how you know. AC = (−2 − 3)2 + (−1− 2)2 = (−5)2 + (−3)2 = 25 + 9 = 34 BD = (−1− 2)2 + (3 + 2)2 = (−3)2 + (5)2 = 9 + 25 = 34 The diagonals are congruent, so it is a rectangle m( AC) = 2 +1 3 + 2 Tuesday, April 29, 14
  51. 51. Example 4 Determine whether parallelogram ABCD is a rhombus, rectangle, or square for A(−2, −1), B(−1, 3), C(3, 2), and D(2, −2). List all that apply and explain how you know. AC = (−2 − 3)2 + (−1− 2)2 = (−5)2 + (−3)2 = 25 + 9 = 34 BD = (−1− 2)2 + (3 + 2)2 = (−3)2 + (5)2 = 9 + 25 = 34 The diagonals are congruent, so it is a rectangle m( AC) = 2 +1 3 + 2 = 3 5 Tuesday, April 29, 14
  52. 52. Example 4 Determine whether parallelogram ABCD is a rhombus, rectangle, or square for A(−2, −1), B(−1, 3), C(3, 2), and D(2, −2). List all that apply and explain how you know. AC = (−2 − 3)2 + (−1− 2)2 = (−5)2 + (−3)2 = 25 + 9 = 34 BD = (−1− 2)2 + (3 + 2)2 = (−3)2 + (5)2 = 9 + 25 = 34 The diagonals are congruent, so it is a rectangle m( AC) = 2 +1 3 + 2 = 3 5 m(BD) = −2 − 3 2 +1 Tuesday, April 29, 14
  53. 53. Example 4 Determine whether parallelogram ABCD is a rhombus, rectangle, or square for A(−2, −1), B(−1, 3), C(3, 2), and D(2, −2). List all that apply and explain how you know. AC = (−2 − 3)2 + (−1− 2)2 = (−5)2 + (−3)2 = 25 + 9 = 34 BD = (−1− 2)2 + (3 + 2)2 = (−3)2 + (5)2 = 9 + 25 = 34 The diagonals are congruent, so it is a rectangle m( AC) = 2 +1 3 + 2 = 3 5 m(BD) = −2 − 3 2 +1 = −5 3 Tuesday, April 29, 14
  54. 54. Example 4 Determine whether parallelogram ABCD is a rhombus, rectangle, or square for A(−2, −1), B(−1, 3), C(3, 2), and D(2, −2). List all that apply and explain how you know. AC = (−2 − 3)2 + (−1− 2)2 = (−5)2 + (−3)2 = 25 + 9 = 34 BD = (−1− 2)2 + (3 + 2)2 = (−3)2 + (5)2 = 9 + 25 = 34 The diagonals are congruent, so it is a rectangle m( AC) = 2 +1 3 + 2 = 3 5 m(BD) = −2 − 3 2 +1 = −5 3 AC ⊥ BD Tuesday, April 29, 14
  55. 55. Example 4 Determine whether parallelogram ABCD is a rhombus, rectangle, or square for A(−2, −1), B(−1, 3), C(3, 2), and D(2, −2). List all that apply and explain how you know. AC = (−2 − 3)2 + (−1− 2)2 = (−5)2 + (−3)2 = 25 + 9 = 34 BD = (−1− 2)2 + (3 + 2)2 = (−3)2 + (5)2 = 9 + 25 = 34 The diagonals are congruent, so it is a rectangle m( AC) = 2 +1 3 + 2 = 3 5 m(BD) = −2 − 3 2 +1 = −5 3 AC ⊥ BD The diagonals are ⊥, so it is a rhombus, thus also a square. Tuesday, April 29, 14
  56. 56. Problem Set Tuesday, April 29, 14
  57. 57. Problem Set p. 431 #1-33 odd, 46, 55, 57, 59 "Courage is saying, 'Maybe what I'm doing isn't working; maybe I should try something else.'" - Anna Lappe Tuesday, April 29, 14

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