Geometry Section 6-3 1112

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Tests for Paral

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Geometry Section 6-3 1112

  1. 1. Section 6-3 Tests for Parallelograms Tuesday, April 29, 14
  2. 2. Essential Questions How do you recognize the conditions that ensure a quadrilateral is a parallelogram? How do you prove that a set of points forms a parallelogram in the coordinate plane? Tuesday, April 29, 14
  3. 3. Theorems 6.9 - OPPOSITE SIDES: 6.10 - OPPOSITE ANGLES: 6.11 - DIAGONALS: 6.12 - PARALLEL CONGRUENT SET OF SIDES: Tuesday, April 29, 14
  4. 4. Theorems 6.9 - OPPOSITE SIDES: IF BOTH PAIRS OF OPPOSITE SIDES OF A QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A PARALLELOGRAM 6.10 - OPPOSITE ANGLES: 6.11 - DIAGONALS: 6.12 - PARALLEL CONGRUENT SET OF SIDES: Tuesday, April 29, 14
  5. 5. Theorems 6.9 - OPPOSITE SIDES: IF BOTH PAIRS OF OPPOSITE SIDES OF A QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A PARALLELOGRAM 6.10 - OPPOSITE ANGLES: IF BOTH PAIRS OF OPPOSITE ANGLES OF A QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A PARALLELOGRAM 6.11 - DIAGONALS: 6.12 - PARALLEL CONGRUENT SET OF SIDES: Tuesday, April 29, 14
  6. 6. Theorems 6.9 - OPPOSITE SIDES: IF BOTH PAIRS OF OPPOSITE SIDES OF A QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A PARALLELOGRAM 6.10 - OPPOSITE ANGLES: IF BOTH PAIRS OF OPPOSITE ANGLES OF A QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A PARALLELOGRAM 6.11 - DIAGONALS: IF THE DIAGONALS OF A QUADRILATERAL BISECT EACH OTHER, THEN THE QUADRILATERAL IS A PARALLELOGRAM 6.12 - PARALLEL CONGRUENT SET OF SIDES: Tuesday, April 29, 14
  7. 7. Theorems 6.9 - OPPOSITE SIDES: IF BOTH PAIRS OF OPPOSITE SIDES OF A QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A PARALLELOGRAM 6.10 - OPPOSITE ANGLES: IF BOTH PAIRS OF OPPOSITE ANGLES OF A QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A PARALLELOGRAM 6.11 - DIAGONALS: IF THE DIAGONALS OF A QUADRILATERAL BISECT EACH OTHER, THEN THE QUADRILATERAL IS A PARALLELOGRAM 6.12 - PARALLEL CONGRUENT SET OF SIDES: IF ONE PAIR OF OPPOSITES SIDES OF A QUADRILATERAL IS BOTH CONGRUENT AND PARALLEL, THEN THE QUADRILATERAL IS A PARALLELOGRAM Tuesday, April 29, 14
  8. 8. Example 1 DETERMINE WHETHER THE QUADRILATERAL IS A PARALLELOGRAM. JUSTIFY YOUR ANSWER. Tuesday, April 29, 14
  9. 9. Example 1 DETERMINE WHETHER THE QUADRILATERAL IS A PARALLELOGRAM. JUSTIFY YOUR ANSWER. BOTH PAIRS OF OPPOSITE SIDES HAVE THE SAME MEASURE, SO EACH OPPOSITE PAIR IS CONGRUENT, THUS MAKING IT A PARALLELOGRAM. Tuesday, April 29, 14
  10. 10. Example 2 FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM. Tuesday, April 29, 14
  11. 11. Example 2 FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM. 4x − 1= 3(x + 2) Tuesday, April 29, 14
  12. 12. Example 2 FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM. 4x − 1= 3(x + 2) 4x − 1= 3x + 6 Tuesday, April 29, 14
  13. 13. Example 2 FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM. 4x − 1= 3(x + 2) 4x − 1= 3x + 6 x = 7 Tuesday, April 29, 14
  14. 14. Example 2 FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM. 4x − 1= 3(x + 2) 4x − 1= 3x + 6 x = 7 3(y + 1) = 4y − 2 Tuesday, April 29, 14
  15. 15. Example 2 FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM. 4x − 1= 3(x + 2) 4x − 1= 3x + 6 x = 7 3(y + 1) = 4y − 2 3y + 3 = 4y − 2 Tuesday, April 29, 14
  16. 16. Example 2 FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM. 4x − 1= 3(x + 2) 4x − 1= 3x + 6 x = 7 3(y + 1) = 4y − 2 3y + 3 = 4y − 2 5 = y Tuesday, April 29, 14
  17. 17. Example 3 QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS A PARALLELOGRAM. Tuesday, April 29, 14
  18. 18. Example 3 QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS A PARALLELOGRAM. m(TA) = 1− 3 3 − (−1) Tuesday, April 29, 14
  19. 19. Example 3 QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS A PARALLELOGRAM. m(TA) = 1− 3 3 − (−1) = −2 4 Tuesday, April 29, 14
  20. 20. Example 3 QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS A PARALLELOGRAM. m(TA) = 1− 3 3 − (−1) = −2 4 = − 1 2 Tuesday, April 29, 14
  21. 21. Example 3 QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS A PARALLELOGRAM. m(TA) = 1− 3 3 − (−1) = −2 4 = − 1 2 m(CO) = −1− (−3) −2 − 2 Tuesday, April 29, 14
  22. 22. Example 3 QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS A PARALLELOGRAM. m(TA) = 1− 3 3 − (−1) = −2 4 = − 1 2 m(CO) = −1− (−3) −2 − 2 = 2 −4 Tuesday, April 29, 14
  23. 23. Example 3 QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS A PARALLELOGRAM. m(TA) = 1− 3 3 − (−1) = −2 4 = − 1 2 m(CO) = −1− (−3) −2 − 2 = 2 −4 = − 1 2 Tuesday, April 29, 14
  24. 24. Example 3 QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS A PARALLELOGRAM. m(TA) = 1− 3 3 − (−1) = −2 4 = − 1 2 m(CO) = −1− (−3) −2 − 2 = 2 −4 = − 1 2 m(AC) = −3 − 1 2 − 3 Tuesday, April 29, 14
  25. 25. Example 3 QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS A PARALLELOGRAM. m(TA) = 1− 3 3 − (−1) = −2 4 = − 1 2 m(CO) = −1− (−3) −2 − 2 = 2 −4 = − 1 2 m(AC) = −3 − 1 2 − 3 = −4 −1 Tuesday, April 29, 14
  26. 26. Example 3 QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS A PARALLELOGRAM. m(TA) = 1− 3 3 − (−1) = −2 4 = − 1 2 m(CO) = −1− (−3) −2 − 2 = 2 −4 = − 1 2 m(AC) = −3 − 1 2 − 3 = −4 −1 = 4 Tuesday, April 29, 14
  27. 27. Example 3 QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS A PARALLELOGRAM. m(TA) = 1− 3 3 − (−1) = −2 4 = − 1 2 m(CO) = −1− (−3) −2 − 2 = 2 −4 = − 1 2 m(AC) = −3 − 1 2 − 3 = −4 −1 = 4 m(TO) = −1− 3 −2 − (−1) Tuesday, April 29, 14
  28. 28. Example 3 QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS A PARALLELOGRAM. m(TA) = 1− 3 3 − (−1) = −2 4 = − 1 2 m(CO) = −1− (−3) −2 − 2 = 2 −4 = − 1 2 m(AC) = −3 − 1 2 − 3 = −4 −1 = 4 m(TO) = −1− 3 −2 − (−1) = −4 −1 Tuesday, April 29, 14
  29. 29. Example 3 QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS A PARALLELOGRAM. m(TA) = 1− 3 3 − (−1) = −2 4 = − 1 2 m(CO) = −1− (−3) −2 − 2 = 2 −4 = − 1 2 m(AC) = −3 − 1 2 − 3 = −4 −1 = 4 m(TO) = −1− 3 −2 − (−1) = −4 −1 = 4 Tuesday, April 29, 14
  30. 30. Example 3 QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS A PARALLELOGRAM. m(TA) = 1− 3 3 − (−1) = −2 4 = − 1 2 m(CO) = −1− (−3) −2 − 2 = 2 −4 = − 1 2 m(AC) = −3 − 1 2 − 3 = −4 −1 = 4 m(TO) = −1− 3 −2 − (−1) = −4 −1 = 4 SINCE EACH SET OF OPPOSITE SIDES HAVE THE SAME SLOPE, THEY ARE PARALLEL. WITH EACH SET OF OPPOSITE SIDES BEING PARALLEL, TACO IS A PARALLELOGRAM Tuesday, April 29, 14
  31. 31. Example 4 FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM. Tuesday, April 29, 14
  32. 32. Example 4 FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM. 4x − 4 = 72 Tuesday, April 29, 14
  33. 33. Example 4 FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM. 4x − 4 = 72 4x = 76 Tuesday, April 29, 14
  34. 34. Example 4 FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM. 4x − 4 = 72 4x = 76 x = 19 Tuesday, April 29, 14
  35. 35. Example 4 FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM. 4x − 4 = 72 4x = 76 x = 19 180 − 72 Tuesday, April 29, 14
  36. 36. Example 4 FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM. 4x − 4 = 72 4x = 76 x = 19 180 − 72 = 108 Tuesday, April 29, 14
  37. 37. Example 4 FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM. 4x − 4 = 72 4x = 76 x = 19 180 − 72 = 108 8y + 8 = 108 Tuesday, April 29, 14
  38. 38. Example 4 FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM. 4x − 4 = 72 4x = 76 x = 19 180 − 72 = 108 8y + 8 = 108 8y = 100 Tuesday, April 29, 14
  39. 39. Example 4 FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM. 4x − 4 = 72 4x = 76 x = 19 180 − 72 = 108 8y + 8 = 108 8y = 100 y = 12.5 Tuesday, April 29, 14
  40. 40. Problem Set Tuesday, April 29, 14
  41. 41. Problem Set P. 413 #1-23 ODD, 27, 51, 53 “I AM ALWAYS DOING THAT WHICH I CAN NOT DO, IN ORDER THAT I MAY LEARN HOW TO DO IT." – PABLO PICASSO Tuesday, April 29, 14

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