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Exploring Alternate Exterior Angles

Exploring Alternate Exterior Angles
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2019-06-24 22:44
AngleTransformationGeometric Construction
This example demonstrates the geometric concept that alternate exterior angles are congruent when a transversal intersects parallel lines. It involves core math concepts like parallel lines, transversal, alternate exterior angles, and geometric proofs. The construction uses GeoGebra commands such as Point, Rotate, Angle, Line, Circle, Intersect, CircleSector, Dilate, Vector, Midpoint, Segment, Ray, and CircleArc. Suitable for high school students learning geometric proofs.

GeoGebra Commands & Parameters

  1. Point(a) → B
  2. Rotate(B, α, A) → B'
  3. Angle(B, A, B') → β
  4. Line(A, B') → b
  5. Circle(A, 1) → e
  6. Intersect(b, d) → C
  7. Circle(C, 1) → f
  8. Intersect(e, a) → D, E
  9. Intersect(e, b) → F, G
  10. Intersect(f, b) → H, I
  11. Intersect(f, d) → J, K
  12. CircleSector(C, J, H) → k
  13. CircleSector(C, H, K) → p
  14. CircleSector(A, F, E) → t
  15. Dilate(A, If[0 < i < 1, 1 - i], C) → A'
  16. Vector(A, A') → u
  17. Vector(A, C) → v
  18. Midpoint(C, A) → L
  19. Rotate(A, (((-i) * 180))°, L) → A'_1
  20. Segment(L, A'_1) → g
  21. Segment(C, L) → n
  22. Segment(L, A) → s
  23. Point(b) → M
  24. Point(b) → N
  25. Ray(C, M) → a_1
  26. Ray(A, N) → b_1
  27. Rotate(E, (((-i) * 180))°, L) → E'
  28. Ray(A'_1, E') → d_1
  29. Circle(A'_1, Distance[L, A'_1]) → e_1
  30. Intersect(e_1, d_1, 1) → O
  31. Rotate(A, ((i * 180))°, L) → A'_2
  32. Rotate(D, ((i * 180))°, L) → D'
  33. Circle(A'_2, L) → k_1
  34. Ray(A'_2, D') → i_1
  35. Intersect(k_1, i_1) → P,
  36. Segment(A'_2, P) → j_1
  37. Intersect(e, b, 1) → Q
  38. Intersect(f, n, 1) → R
  39. CircleSector(A, Q, E) → c_1
  40. Rotate(c_1, (((-i) * 180))°, L) → q
  41. Rotate(Q, (((-i) * 180))°, L) → Q'
  42. Circle(A, g) → g_1
  43. Intersect(g_1, b_1, 1) → S
  44. Intersect(g_1, a, 2) → T
  45. Segment(A, S) → h_1
  46. Rotate(A, (((-i) * 180))°, L) → A'_3
  47. Rotate(S, (((-i) * 180))°, L) → S'
  48. Segment(A'_3, S') → l_1
  49. Segment(A, T) → m_1
  50. Rotate(A, (((-i) * 180))°, L) → A'_4
  51. Rotate(T, (((-i) * 180))°, L) → T'
  52. Segment(A'_4, T') → n_1
  53. CircleArc(A, Q, E) → p_1
  54. Intersect(e_1, a, 1) → U
  55. Segment(A, U) → q_1
  56. Intersect(a, q_1) → V
  57. Intersect(a, q_1) → W
  58. Intersect(e, a, 1) → Z
  59. CircleSector(A, Z, Q) → r
  60. Rotate(A, ((i * 180))°, L) → A'_5
  61. Rotate(r, ((i * 180))°, L) → r'
  62. Rotate(A, ((i * 180))°, L) → A'_6
  63. Rotate(S, ((i * 180))°, L) → S'_1
  64. Segment(A'_6, S'_1) → h
  65. Segment(L, A'_5) → r_1
  66. CircleArc(A, Z, Q) → f_1