MathPlotAI Logo MathPlotAI

AA Similarity Theorem Proof by Mrs. Cole

AA Similarity Theorem Proof by Mrs. Cole
Loading the math board and drawing, please wait…
2019-02-19 21:01
TriangleAngleTransformationGeometric Construction
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, plane figures, integers, statistics, kites, and correlation. The construction utilizes commands such as Segment, Circle, Point, Angle, Rotate, Ray, Intersect, Line, Dilate, Vector, Polygon, Translate, and If. Suitable for high school mathematics education and geometry learning.

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