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AA Similarity Theorem Proof

AA Similarity Theorem Proof
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2020-04-23 13:31
TriangleAngleGeometric Construction
In collections Geometry - Mini Similarity Book
This example demonstrates the proof of the Angle-Angle (AA) Triangle Similarity Theorem using transformational geometry tools. It involves core concepts like angles, congruence, similar triangles, triangle distributions, general quadrilaterals, upper and lower sums or Riemann sums, expected value, and scalene triangles. The construction utilizes commands/objects such as Segment, Circle, Point, Angle, Rotate, Ray, Intersect, Line, Dilate, Vector, Polygon, Translate, and If from GeoGebra. Suitable for high school mathematics education to understand and apply the AA similarity theorem.

GeoGebra Commands & Parameters

  1. Segment(A, B) → a
  2. Segment(B, C) → b
  3. Segment(C, A) → c
  4. Circle(D, (k * a)) → d
  5. Point(d) → E
  6. Segment(D, E) → e
  7. Circle(E, (k * b)) → f
  8. Angle(A, B, C) → α
  9. Rotate(D, α, E) → D'
  10. Angle(D, E, D') → β
  11. Ray(E, D') → g
  12. Intersect(f, g, 1) → F
  13. Angle(C, A, B) → γ
  14. Rotate(E, (-γ), D) → E'
  15. Angle(E', D, E) → δ
  16. Ray(D, E') → h
  17. Segment(D, E') → i
  18. Segment(E', E) → j
  19. Line(E, D') → l
  20. Line(E, D') → n
  21. Circle(D, (k * c)) → p
  22. Ray(D, E') → m
  23. Intersect(p, m, 1) → G
  24. Segment(E, G) → q
  25. Segment(D, G) → r
  26. Dilate(D, If[0 < s ≤ 1, 1 - s], A) → D'_1
  27. Vector(D, D'_1) → u
  28. Polygon(A, C, B) → poly1, b_1, a_1, c_1
  29. Polygon(D, E, G) → poly2, g_1, d_1, e_1
  30. Translate(D, u) → D'_2
  31. Translate(E, u) → E'_1
  32. Translate(G, u) → G'
  33. Polygon(D'_2, E'_1, G') → poly2', g', d'_2, e'_1
  34. Vector(D, A) → v
  35. Translate(D, v) → D'_3
  36. Translate(E, v) → E'_2
  37. Translate(G, v) → G'_1
  38. Polygon(D'_3, E'_2, G'_1) → poly2'_1, g'_1, d'_3, e'_2
  39. Segment(D'_1, G') → t
  40. Segment(E'_1, D'_1) → h_1
  41. Segment(A, G'_1) → i_1
  42. Segment(G'_1, E'_2) → j_1
  43. Segment(E'_2, A) → k_1
  44. Angle(C, A, E'_2) → ε
  45. If(ε > γ, ε - γ, γ - ε) → ζ
  46. Rotate(D'_3, If[1 < s ≤ 2, ((-(s - 1)) * ζ), (-ζ)], A) → H
  47. Rotate(E'_2, If[1 < s ≤ 2, ((-(s - 1)) * ζ), (-ζ)], A) → I
  48. Rotate(G'_1, If[1 < s ≤ 2, ((-(s - 1)) * ζ), (-ζ)], A) → J
  49. Polygon(H, I, J) → poly3, j_2, h_2, i_2
  50. Segment(A, J) → l_1
  51. Segment(A, I) → m_1
  52. Segment(I, J) → n_1
  53. Dilate(J, If[2 < s ≤ 3, 3 - s, 0], C) → J'
  54. Dilate(I, If[2 < s ≤ 3, 3 - s, 0], B) → I'
  55. Segment(I', J') → p_1
  56. Angle(A, I', J') → η
  57. Angle(A, I, J) → θ
  58. Angle(D'_2, E'_1, G') → ι
  59. Angle(J, A, I) → λ
  60. Angle(G', D'_1, E'_1) → μ
  61. Polygon(A, J', I') → poly4, i', a_2, j'
  62. Angle(A, I, J) → ν
  63. Segment(G'_1, E'_2) → q_1
  64. Segment(G', E'_1) → f_1
  65. Angle(G'_1, A, E'_2) → κ
  66. Angle(A, E'_2, G'_1) → ξ
  67. Rotate(D'_3, ((s - 1) * ζ), A) → K
  68. Rotate(D'_3, If[1 < s ≤ 2, ((s - 1) * ζ), ζ], A) → L
  69. Rotate(E'_2, If[1 < s ≤ 2, ((s - 1) * ζ), ζ], A) → M
  70. Rotate(G'_1, If[1 < s ≤ 2, ((s - 1) * ζ), ζ], A) → N
  71. Polygon(L, M, N) → poly5, n_2, l_2, m_2
  72. Dilate(M, If[2 < s ≤ 3, 3 - s, 0], B) → M'
  73. Dilate(N, If[2 < s ≤ 3, 3 - s, 0], C) → N'
  74. Segment(M', N') → r_1
  75. Polygon(A, M', N') → poly6, n', a_3, m'
  76. Angle(A, M', N') → ο
  77. Angle(A, M, N) → ρ
  78. Segment(A, M) → t_1
  79. Segment(M, N) → b_2
  80. Segment(A, N) → c_2
  81. Angle(B, A, C) → σ
  82. Angle(G, E, D) → τ
  83. Angle(N, M, A) → υ
  84. Angle(N', M', A) → ϕ

AI Drawing Prompt Reference

Draw triangle ABC. Create a circle centered at A and mark a point E on it. Then draw a circle centered at E and mark a point F on it. Connect D to F and rotate triangle DEF so that angle DEF equals angle ABC. Next, draw another circle centered at E and mark a point G on it. Connect D to G and rotate triangle DEG so that angle DEG equals angle ACB. Finally, connect the corresponding vertices to illustrate the proof of the AA similarity theorem.
The prompt above was compiled from the original drawing commands and is for reference only.