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# Geometry Section 6-2 1112

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Parallelograms

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### Geometry Section 6-2 1112

1. 1. SECTION 6-2 Parallelograms Friday, May 2, 14
2. 2. Essential Questions How do you recognize and apply properties of the sides and angles of parallelograms? How do you recognize and apply properties of the diagonals of parallelograms? Friday, May 2, 14
3. 3. Vocabulary 1. Parallelogram: Friday, May 2, 14
4. 4. Vocabulary 1. Parallelogram: A quadrilateral that has two pairs of parallel sides Friday, May 2, 14
5. 5. Properties and Theorems 6.3 - Opposite Sides: 6.4 - Opposite Angles: 6.5 - Consecutive Angles: 6.6 - Right Angles: Friday, May 2, 14
6. 6. Properties and Theorems 6.3 - Opposite Sides: If a quadrilateral is a parallelogram, then its opposite sides are congruent 6.4 - Opposite Angles: 6.5 - Consecutive Angles: 6.6 - Right Angles: Friday, May 2, 14
7. 7. Properties and Theorems 6.3 - Opposite Sides: If a quadrilateral is a parallelogram, then its opposite sides are congruent 6.4 - Opposite Angles: If a quadrilateral is a parallelogram, then its opposite angles are congruent 6.5 - Consecutive Angles: 6.6 - Right Angles: Friday, May 2, 14
8. 8. Properties and Theorems 6.3 - Opposite Sides: If a quadrilateral is a parallelogram, then its opposite sides are congruent 6.4 - Opposite Angles: If a quadrilateral is a parallelogram, then its opposite angles are congruent 6.5 - Consecutive Angles: If a quadrilateral is a parallelogram, then its consecutive angles are supplementary 6.6 - Right Angles: Friday, May 2, 14
9. 9. Properties and Theorems 6.3 - Opposite Sides: If a quadrilateral is a parallelogram, then its opposite sides are congruent 6.4 - Opposite Angles: If a quadrilateral is a parallelogram, then its opposite angles are congruent 6.5 - Consecutive Angles: If a quadrilateral is a parallelogram, then its consecutive angles are supplementary 6.6 - Right Angles: If a quadrilateral has one right angle, then it has four right angles Friday, May 2, 14
10. 10. Properties and Theorems 6.7 - Bisecting Diagonals: 6.8 - Triangles from Diagonals: Friday, May 2, 14
11. 11. Properties and Theorems 6.7 - Bisecting Diagonals: If a quadrilateral is a parallelogram, then its diagonals bisect each other 6.8 - Triangles from Diagonals: Friday, May 2, 14
12. 12. Properties and Theorems 6.7 - Bisecting Diagonals: If a quadrilateral is a parallelogram, then its diagonals bisect each other 6.8 - Triangles from Diagonals: If a quadrilateral is a parallelogram, then each diagonal separates the parallelogram into two congruent triangles Friday, May 2, 14
13. 13. Example 1 In ABCD, suppose m∠B = 32°, CD = 80 inches, and BC = 15 inches. Find each measure. a. AD b. m∠C c. m∠D Friday, May 2, 14
14. 14. Example 1 In ABCD, suppose m∠B = 32°, CD = 80 inches, and BC = 15 inches. Find each measure. a. AD 15 inches b. m∠C c. m∠D Friday, May 2, 14
15. 15. Example 1 In ABCD, suppose m∠B = 32°, CD = 80 inches, and BC = 15 inches. Find each measure. a. AD 15 inches b. m∠C = 180 − 32 c. m∠D Friday, May 2, 14
16. 16. Example 1 In ABCD, suppose m∠B = 32°, CD = 80 inches, and BC = 15 inches. Find each measure. a. AD 15 inches b. m∠C = 180 − 32 = 148° c. m∠D Friday, May 2, 14
17. 17. Example 1 In ABCD, suppose m∠B = 32°, CD = 80 inches, and BC = 15 inches. Find each measure. a. AD 15 inches b. m∠C = 180 − 32 = 148° c. m∠D = 32° Friday, May 2, 14
18. 18. Example 2 If WXYZ is a parallelogram, ﬁnd the value of the indicated variable. m∠VWX = (2t)°, m∠VYX = 40°, m∠VYZ =18°, WX = 4r, ZV = 8s, VX = 7s + 3, ZY =18 a. r b. s c. t Friday, May 2, 14
19. 19. Example 2 If WXYZ is a parallelogram, ﬁnd the value of the indicated variable. m∠VWX = (2t)°, m∠VYX = 40°, m∠VYZ =18°, WX = 4r, ZV = 8s, VX = 7s + 3, ZY =18 a. r WX = ZY b. s c. t Friday, May 2, 14
20. 20. Example 2 If WXYZ is a parallelogram, ﬁnd the value of the indicated variable. m∠VWX = (2t)°, m∠VYX = 40°, m∠VYZ =18°, WX = 4r, ZV = 8s, VX = 7s + 3, ZY =18 a. r WX = ZY 4r = 18 b. s c. t Friday, May 2, 14
21. 21. Example 2 If WXYZ is a parallelogram, ﬁnd the value of the indicated variable. m∠VWX = (2t)°, m∠VYX = 40°, m∠VYZ =18°, WX = 4r, ZV = 8s, VX = 7s + 3, ZY =18 a. r WX = ZY 4r = 18 r = 4.5 b. s c. t Friday, May 2, 14
22. 22. Example 2 If WXYZ is a parallelogram, ﬁnd the value of the indicated variable. m∠VWX = (2t)°, m∠VYX = 40°, m∠VYZ =18°, WX = 4r, ZV = 8s, VX = 7s + 3, ZY =18 a. r WX = ZY 4r = 18 r = 4.5 b. s ZV = VX c. t Friday, May 2, 14
23. 23. Example 2 If WXYZ is a parallelogram, ﬁnd the value of the indicated variable. m∠VWX = (2t)°, m∠VYX = 40°, m∠VYZ =18°, WX = 4r, ZV = 8s, VX = 7s + 3, ZY =18 a. r WX = ZY 4r = 18 r = 4.5 b. s ZV = VX 8s = 7s + 3 c. t Friday, May 2, 14
24. 24. Example 2 If WXYZ is a parallelogram, ﬁnd the value of the indicated variable. m∠VWX = (2t)°, m∠VYX = 40°, m∠VYZ =18°, WX = 4r, ZV = 8s, VX = 7s + 3, ZY =18 a. r WX = ZY 4r = 18 r = 4.5 b. s ZV = VX 8s = 7s + 3 s = 3 c. t Friday, May 2, 14
25. 25. Example 2 If WXYZ is a parallelogram, ﬁnd the value of the indicated variable. m∠VWX = (2t)°, m∠VYX = 40°, m∠VYZ =18°, WX = 4r, ZV = 8s, VX = 7s + 3, ZY =18 a. r WX = ZY 4r = 18 r = 4.5 b. s ZV = VX 8s = 7s + 3 s = 3 c. t m∠VWX = m∠VYZ Friday, May 2, 14
26. 26. Example 2 If WXYZ is a parallelogram, ﬁnd the value of the indicated variable. m∠VWX = (2t)°, m∠VYX = 40°, m∠VYZ =18°, WX = 4r, ZV = 8s, VX = 7s + 3, ZY =18 a. r WX = ZY 4r = 18 r = 4.5 b. s ZV = VX 8s = 7s + 3 s = 3 c. t m∠VWX = m∠VYZ 2t = 18 Friday, May 2, 14
27. 27. Example 2 If WXYZ is a parallelogram, ﬁnd the value of the indicated variable. m∠VWX = (2t)°, m∠VYX = 40°, m∠VYZ =18°, WX = 4r, ZV = 8s, VX = 7s + 3, ZY =18 a. r WX = ZY 4r = 18 r = 4.5 b. s ZV = VX 8s = 7s + 3 s = 3 c. t m∠VWX = m∠VYZ 2t = 18 t = 9 Friday, May 2, 14
28. 28. Example 3 What are the coordinates of the intersection of the diagonals of parallelogram MNPR with vertices M(−3, 0), N(−1, 3), P(5, 4) , and R(3, 1)? Friday, May 2, 14
29. 29. Example 3 What are the coordinates of the intersection of the diagonals of parallelogram MNPR with vertices M(−3, 0), N(−1, 3), P(5, 4) , and R(3, 1)? M = x1 + x2 2 , y1 + y2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Friday, May 2, 14
30. 30. Example 3 What are the coordinates of the intersection of the diagonals of parallelogram MNPR with vertices M(−3, 0), N(−1, 3), P(5, 4) , and R(3, 1)? M = x1 + x2 2 , y1 + y2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ M(MP) = −3+ 5 2 , 0 + 4 2 ⎛ ⎝⎜ ⎞ ⎠⎟ Friday, May 2, 14
31. 31. Example 3 What are the coordinates of the intersection of the diagonals of parallelogram MNPR with vertices M(−3, 0), N(−1, 3), P(5, 4) , and R(3, 1)? M = x1 + x2 2 , y1 + y2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ M(MP) = −3+ 5 2 , 0 + 4 2 ⎛ ⎝⎜ ⎞ ⎠⎟ = 2 2 , 4 2 ⎛ ⎝⎜ ⎞ ⎠⎟ Friday, May 2, 14
32. 32. Example 3 What are the coordinates of the intersection of the diagonals of parallelogram MNPR with vertices M(−3, 0), N(−1, 3), P(5, 4) , and R(3, 1)? M = x1 + x2 2 , y1 + y2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ M(MP) = −3+ 5 2 , 0 + 4 2 ⎛ ⎝⎜ ⎞ ⎠⎟ = 2 2 , 4 2 ⎛ ⎝⎜ ⎞ ⎠⎟ = 1,2( ) Friday, May 2, 14
33. 33. Example 3 What are the coordinates of the intersection of the diagonals of parallelogram MNPR with vertices M(−3, 0), N(−1, 3), P(5, 4) , and R(3, 1)? M = x1 + x2 2 , y1 + y2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ M(MP) = −3+ 5 2 , 0 + 4 2 ⎛ ⎝⎜ ⎞ ⎠⎟ = 2 2 , 4 2 ⎛ ⎝⎜ ⎞ ⎠⎟ = 1,2( ) M(NR) = −1+ 3 2 , 3+1 2 ⎛ ⎝⎜ ⎞ ⎠⎟ Friday, May 2, 14
34. 34. Example 3 What are the coordinates of the intersection of the diagonals of parallelogram MNPR with vertices M(−3, 0), N(−1, 3), P(5, 4) , and R(3, 1)? M = x1 + x2 2 , y1 + y2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ M(MP) = −3+ 5 2 , 0 + 4 2 ⎛ ⎝⎜ ⎞ ⎠⎟ = 2 2 , 4 2 ⎛ ⎝⎜ ⎞ ⎠⎟ = 1,2( ) M(NR) = −1+ 3 2 , 3+1 2 ⎛ ⎝⎜ ⎞ ⎠⎟ = 2 2 , 4 2 ⎛ ⎝⎜ ⎞ ⎠⎟ Friday, May 2, 14
35. 35. Example 3 What are the coordinates of the intersection of the diagonals of parallelogram MNPR with vertices M(−3, 0), N(−1, 3), P(5, 4) , and R(3, 1)? M = x1 + x2 2 , y1 + y2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ M(MP) = −3+ 5 2 , 0 + 4 2 ⎛ ⎝⎜ ⎞ ⎠⎟ = 2 2 , 4 2 ⎛ ⎝⎜ ⎞ ⎠⎟ = 1,2( ) M(NR) = −1+ 3 2 , 3+1 2 ⎛ ⎝⎜ ⎞ ⎠⎟ = 2 2 , 4 2 ⎛ ⎝⎜ ⎞ ⎠⎟ = 1,2( ) Friday, May 2, 14
36. 36. Example 4 Given ABCD, AC and BD are diagonals, P is the intersection of AC and BD, prove that AC and BD bisect each other. Friday, May 2, 14
37. 37. Example 4 Given ABCD, AC and BD are diagonals, P is the intersection of AC and BD, prove that AC and BD bisect each other. 1. ABCD is a parallelogram, AC and BD are diagonals, P is the intersection of AC and BD Friday, May 2, 14
38. 38. Example 4 Given ABCD, AC and BD are diagonals, P is the intersection of AC and BD, prove that AC and BD bisect each other. 1. ABCD is a parallelogram, AC and BD are diagonals, P is the intersection of AC and BD 1. Given Friday, May 2, 14
39. 39. Example 4 Given ABCD, AC and BD are diagonals, P is the intersection of AC and BD, prove that AC and BD bisect each other. 1. ABCD is a parallelogram, AC and BD are diagonals, P is the intersection of AC and BD 1. Given 2. AB is parallel to CD Friday, May 2, 14
40. 40. Example 4 Given ABCD, AC and BD are diagonals, P is the intersection of AC and BD, prove that AC and BD bisect each other. 1. ABCD is a parallelogram, AC and BD are diagonals, P is the intersection of AC and BD 1. Given 2. Deﬁnition of parallelogram2. AB is parallel to CD Friday, May 2, 14
41. 41. Example 4 Given ABCD, AC and BD are diagonals, P is the intersection of AC and BD, prove that AC and BD bisect each other. 1. ABCD is a parallelogram, AC and BD are diagonals, P is the intersection of AC and BD 1. Given 2. Deﬁnition of parallelogram 3. ∠BAC ≅ ∠DCA, ∠BDC ≅ ∠DBA 2. AB is parallel to CD Friday, May 2, 14
42. 42. Example 4 Given ABCD, AC and BD are diagonals, P is the intersection of AC and BD, prove that AC and BD bisect each other. 1. ABCD is a parallelogram, AC and BD are diagonals, P is the intersection of AC and BD 1. Given 2. Deﬁnition of parallelogram 3. ∠BAC ≅ ∠DCA, ∠BDC ≅ ∠DBA 2. AB is parallel to CD 3.Alt. int. ∠’s of parallel lines ≅ Friday, May 2, 14
43. 43. Example 4 Given ABCD, AC and BD are diagonals, P is the intersection of AC and BD, prove that AC and BD bisect each other. 1. ABCD is a parallelogram, AC and BD are diagonals, P is the intersection of AC and BD 1. Given 2. Deﬁnition of parallelogram 3. ∠BAC ≅ ∠DCA, ∠BDC ≅ ∠DBA 2. AB is parallel to CD 3.Alt. int. ∠’s of parallel lines ≅ 4. AB ≅ CD Friday, May 2, 14
44. 44. Example 4 Given ABCD, AC and BD are diagonals, P is the intersection of AC and BD, prove that AC and BD bisect each other. 1. ABCD is a parallelogram, AC and BD are diagonals, P is the intersection of AC and BD 1. Given 2. Deﬁnition of parallelogram 3. ∠BAC ≅ ∠DCA, ∠BDC ≅ ∠DBA 2. AB is parallel to CD 3.Alt. int. ∠’s of parallel lines ≅ 4. AB ≅ CD 4. Opposite sides of parallelograms ≅ Friday, May 2, 14
45. 45. Example 4 Friday, May 2, 14
46. 46. Example 4 5. ∆APB ≅ ∆CPD Friday, May 2, 14
47. 47. Example 4 5. ∆APB ≅ ∆CPD 5.ASA Friday, May 2, 14
48. 48. Example 4 5. ∆APB ≅ ∆CPD 5.ASA 6. AP ≅ CP, DP ≅ BP Friday, May 2, 14
49. 49. Example 4 5. ∆APB ≅ ∆CPD 5.ASA 6. AP ≅ CP, DP ≅ BP 6. CPCTC Friday, May 2, 14
50. 50. Example 4 5. ∆APB ≅ ∆CPD 5.ASA 6. AP ≅ CP, DP ≅ BP 6. CPCTC 7. AC and BD bisect each other Friday, May 2, 14
51. 51. Example 4 5. ∆APB ≅ ∆CPD 5.ASA 6. AP ≅ CP, DP ≅ BP 6. CPCTC 7. AC and BD bisect each other 7. Def. of bisect Friday, May 2, 14
52. 52. Problem Set Friday, May 2, 14
53. 53. Problem Set p. 403 #1-23 odd, 27-35 odd, 43, 51, 57 “The best way to predict the future is to invent it.” - Alan Kay Friday, May 2, 14