Geometry Section 3-2 1112

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Geometry Section 3-2 1112

  1. 1. SECTION 3-2 ANGLES AND PARALLEL LINESWednesday, December 14, 2011
  2. 2. ESSENTIAL QUESTIONS HOW DO YOU USE THEOREMS TO DETERMINE THE RELATIONSHIPS BETWEEN SPECIFIC PAIRS OF ANGLES? HOW DO YOU USE ALGEBRA TO FIND ANGLE MEASUREMENTS?Wednesday, December 14, 2011
  3. 3. POSTULATES AND THEOREMS CORRESPONDING ANGLES POSTULATEWednesday, December 14, 2011
  4. 4. POSTULATES AND THEOREMS CORRESPONDING ANGLES POSTULATE IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF CORRESPONDING ANGLES IS CONGRUENT.Wednesday, December 14, 2011
  5. 5. POSTULATES AND THEOREMS CORRESPONDING ANGLES POSTULATE IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF CORRESPONDING ANGLES IS CONGRUENT. ∠1 ≅ ∠5Wednesday, December 14, 2011
  6. 6. POSTULATES AND THEOREMS ALTERNATE INTERIOR ANGLES THEOREMWednesday, December 14, 2011
  7. 7. POSTULATES AND THEOREMS ALTERNATE INTERIOR ANGLES THEOREM IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF ALTERNATE INTERIOR ANGLES IS CONGRUENT.Wednesday, December 14, 2011
  8. 8. POSTULATES AND THEOREMS ALTERNATE INTERIOR ANGLES THEOREM IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF ALTERNATE INTERIOR ANGLES IS CONGRUENT. ∠4 ≅ ∠6Wednesday, December 14, 2011
  9. 9. POSTULATES AND THEOREMS CONSECUTIVE INTERIOR ANGLES THEOREMWednesday, December 14, 2011
  10. 10. POSTULATES AND THEOREMS CONSECUTIVE INTERIOR ANGLES THEOREM IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF CONSECUTIVE INTERIOR ANGLES IS SUPPLEMENTARY.Wednesday, December 14, 2011
  11. 11. POSTULATES AND THEOREMS CONSECUTIVE INTERIOR ANGLES THEOREM IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF CONSECUTIVE INTERIOR ANGLES IS SUPPLEMENTARY. ∠4 AND ∠5 ARE SUPPLEMENTARYWednesday, December 14, 2011
  12. 12. POSTULATES AND THEOREMS ALTERNATE EXTERIOR ANGLES THEOREMWednesday, December 14, 2011
  13. 13. POSTULATES AND THEOREMS ALTERNATE EXTERIOR ANGLES THEOREM IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF ALTERNATE EXTERIOR ANGLES IS CONGRUENT.Wednesday, December 14, 2011
  14. 14. POSTULATES AND THEOREMS ALTERNATE EXTERIOR ANGLES THEOREM IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF ALTERNATE EXTERIOR ANGLES IS CONGRUENT. ∠2 ≅ ∠7Wednesday, December 14, 2011
  15. 15. EXAMPLE 1 IN THE FIGURE, m∠4 = 51°. FIND THE MEASURE OF EACH ANGLE. GIVE A JUSTIFICATION TO YOUR ANSWER. a. m∠2 b. m∠3 c. m∠6Wednesday, December 14, 2011
  16. 16. EXAMPLE 1 IN THE FIGURE, m∠4 = 51°. FIND THE MEASURE OF EACH ANGLE. GIVE A JUSTIFICATION TO YOUR ANSWER. a. m∠2 51°; VERTICAL ANGLES WITH ∠4 b. m∠3 c. m∠6Wednesday, December 14, 2011
  17. 17. EXAMPLE 1 IN THE FIGURE, m∠4 = 51°. FIND THE MEASURE OF EACH ANGLE. GIVE A JUSTIFICATION TO YOUR ANSWER. a. m∠2 51°; VERTICAL ANGLES WITH ∠4 b. m∠3 129°; SUPPLEMENTARY ANGLES WITH ∠4 c. m∠6Wednesday, December 14, 2011
  18. 18. EXAMPLE 1 IN THE FIGURE, m∠4 = 51°. FIND THE MEASURE OF EACH ANGLE. GIVE A JUSTIFICATION TO YOUR ANSWER. a. m∠2 51°; VERTICAL ANGLES WITH ∠4 b. m∠3 129°; SUPPLEMENTARY ANGLES WITH ∠4 c. m∠6 51°; ALTERNATE INTERIOR ANGLES WITH ∠4Wednesday, December 14, 2011
  19. 19. EXAMPLE 2 USE THE FIGURE, IN WHICH a||b, m∠2 = 125°, AND c||d||e, TO FIND THE MEASURE OF EACH NUMBERED ANGLE. PROVIDE A REASON FOR THE ANSWER FOR EACH MEASURE.Wednesday, December 14, 2011
  20. 20. EXAMPLE 2 USE THE FIGURE, IN WHICH a||b, m∠2 = 125°, AND c||d||e, TO FIND THE MEASURE OF EACH NUMBERED ANGLE. PROVIDE A REASON FOR THE ANSWER FOR EACH MEASURE. m∠1 = 125°; VERTICAL WITH ∠2Wednesday, December 14, 2011
  21. 21. EXAMPLE 2 USE THE FIGURE, IN WHICH a||b, m∠2 = 125°, AND c||d||e, TO FIND THE MEASURE OF EACH NUMBERED ANGLE. PROVIDE A REASON FOR THE ANSWER FOR EACH MEASURE. m∠1 = 125°; VERTICAL WITH ∠2 m∠3 = 55°; CONSECUTIVE INTERIOR WITH ∠2Wednesday, December 14, 2011
  22. 22. EXAMPLE 2 USE THE FIGURE, IN WHICH a||b, m∠2 = 125°, AND c||d||e, TO FIND THE MEASURE OF EACH NUMBERED ANGLE. PROVIDE A REASON FOR THE ANSWER FOR EACH MEASURE. m∠1 = 125°; VERTICAL WITH ∠2 m∠3 = 55°; CONSECUTIVE INTERIOR WITH ∠2 m∠4 = 125°; CONSECUTIVE INTERIOR WITH ∠3Wednesday, December 14, 2011
  23. 23. EXAMPLE 2 USE THE FIGURE, IN WHICH a||b, m∠2 = 125°, AND c||d||e, TO FIND THE MEASURE OF EACH NUMBERED ANGLE. PROVIDE A REASON FOR THE ANSWER FOR EACH MEASURE. m∠1 = 125°; VERTICAL WITH ∠2 m∠3 = 55°; CONSECUTIVE INTERIOR WITH ∠2 m∠4 = 125°; CONSECUTIVE INTERIOR WITH ∠3 m∠5 = 55°; SUPPLEMENTARY WITH ∠4Wednesday, December 14, 2011
  24. 24. EXAMPLE 3 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. a. IF m∠2 = 2x − 10 AND m∠6 = x + 15, FIND x.Wednesday, December 14, 2011
  25. 25. EXAMPLE 3 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. a. IF m∠2 = 2x − 10 AND m∠6 = x + 15, FIND x. 2x − 10 = x + 15Wednesday, December 14, 2011
  26. 26. EXAMPLE 3 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. a. IF m∠2 = 2x − 10 AND m∠6 = x + 15, FIND x. 2x − 10 = x + 15 −x −xWednesday, December 14, 2011
  27. 27. EXAMPLE 3 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. a. IF m∠2 = 2x − 10 AND m∠6 = x + 15, FIND x. 2x − 10 = x + 15 −x +10 −x +10Wednesday, December 14, 2011
  28. 28. EXAMPLE 3 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. a. IF m∠2 = 2x − 10 AND m∠6 = x + 15, FIND x. 2x − 10 = x + 15 −x +10 −x +10 x = 25Wednesday, December 14, 2011
  29. 29. EXAMPLE 3 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. a. IF m∠2 = 2x − 10 AND m∠6 = x + 15, FIND x. 2x − 10 = x + 15 −x +10 −x +10 x = 25 SINCE THE ANGLES ARE CORRESPONDING, THEY ARE CONGRUENT, MEANING THEY ARE EQUAL.Wednesday, December 14, 2011
  30. 30. EXAMPLE 3 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. b. IF m∠7 = 4(y − 25) AND m∠1 = 4y, FIND y.Wednesday, December 14, 2011
  31. 31. EXAMPLE 3 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. b. IF m∠7 = 4(y − 25) AND m∠1 = 4y, FIND y. 4(y − 25) + 4y = 180Wednesday, December 14, 2011
  32. 32. EXAMPLE 3 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. b. IF m∠7 = 4(y − 25) AND m∠1 = 4y, FIND y. 4(y − 25) + 4y = 180 4y − 100 + 4y = 180Wednesday, December 14, 2011
  33. 33. EXAMPLE 3 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. b. IF m∠7 = 4(y − 25) AND m∠1 = 4y, FIND y. 4(y − 25) + 4y = 180 4y − 100 + 4y = 180 8y − 100 = 180Wednesday, December 14, 2011
  34. 34. EXAMPLE 3 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. b. IF m∠7 = 4(y − 25) AND m∠1 = 4y, FIND y. 4(y − 25) + 4y = 180 4y − 100 + 4y = 180 8y − 100 = 180 8y = 280Wednesday, December 14, 2011
  35. 35. EXAMPLE 3 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. b. IF m∠7 = 4(y − 25) AND m∠1 = 4y, FIND y. 4(y − 25) + 4y = 180 4y − 100 + 4y = 180 8y − 100 = 180 8y = 280 y = 35Wednesday, December 14, 2011
  36. 36. CHECK YOUR UNDERSTANDING CHECK OUT PROBLEMS 1-10 ON PAGE 181.Wednesday, December 14, 2011
  37. 37. PROBLEM SETWednesday, December 14, 2011
  38. 38. PROBLEM SET P. 181 #11-35 ODD “YOU CAN FOOL TOO MANY OF THE PEOPLE TOO MUCH OF THE TIME.” - JAMES THURBERWednesday, December 14, 2011

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