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Similar Triangles (AAA) Construction and Demonstration

Similar Triangles (AAA) Construction and Demonstration
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2018-01-29 09:33
TrianglePolygonAngleTransformationGeometric Construction
This GeoGebra example dynamically demonstrates the Angle-Angle-Angle (AAA) criterion for triangle similarity. The core mathematical concept is that if three corresponding angles of two triangles are equal, the triangles are similar, meaning their corresponding sides are proportional. The construction uses commands such as Segment, Ray, Line to create triangle sides, Angle to mark angles, and Rotate, Dilate, Translate to generate a second triangle with identical angles but different size. Polygon commands build the triangular regions, while Point and Intersect determine key vertices. The object list (Q, P, a, R, b, c, etc.) shows a step-by-step geometric construction from basic elements to a complete figure. This resource is suitable for middle-to-high school students learning triangle similarity theorems, and serves as an interactive tool for teachers to illustrate the AAA criterion or for students to explore and verify the property independently.

GeoGebra Commands & Parameters

  1. Segment(Q, P) → a
  2. Segment(P, R) → b
  3. Segment(R, Q) → c
  4. Circle(M, (k * a)) → d
  5. Point(d) → L
  6. Segment(M, L) → e
  7. Circle(L, (k * b)) → f
  8. Angle(Q, P, R) → α
  9. Rotate(M, α, L) → D'
  10. Angle(M, L, D') → β
  11. Ray(L, D') → g
  12. Intersect(f, g, 1) → F
  13. Angle(R, Q, P) → γ
  14. Rotate(L, (-γ), M) → E'
  15. Angle(E', M, L) → δ
  16. Ray(M, E') → h
  17. Segment(M, E') → i
  18. Segment(E', L) → j
  19. Line(L, D') → l
  20. Line(L, D') → n
  21. Circle(M, (k * c)) → p
  22. Ray(M, E') → m
  23. Intersect(p, m, 1) → G
  24. Segment(L, G) → q
  25. Segment(M, G) → r
  26. Dilate(M, If[0 < s ≤ 1, 1 - s], Q) → D'_1
  27. Vector(M, D'_1) → u
  28. Polygon(Q, R, P) → poly1, b_1, a_1, c_1
  29. Polygon(M, L, G) → poly2, g_1, d_1, e_1
  30. Translate(M, u) → D'_2
  31. Translate(L, u) → E'_1
  32. Translate(G, u) → G'
  33. Polygon(D'_2, E'_1, G') → poly2', g', d'_2, e'_1
  34. Vector(M, Q) → v
  35. Translate(M, v) → D'_3
  36. Translate(L, 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(Q, G'_1) → i_1
  42. Segment(G'_1, E'_2) → j_1
  43. Segment(E'_2, Q) → k_1
  44. Angle(R, Q, E'_2) → ε
  45. If(ε > γ, ε - γ, γ - ε) → ζ
  46. Rotate(D'_3, If[1 < s ≤ 2, ((-(s - 1)) * ζ), (-ζ)], Q) → H
  47. Rotate(E'_2, If[1 < s ≤ 2, ((-(s - 1)) * ζ), (-ζ)], Q) → I
  48. Rotate(G'_1, If[1 < s ≤ 2, ((-(s - 1)) * ζ), (-ζ)], Q) → J
  49. Polygon(H, I, J) → poly3, j_2, h_2, i_2
  50. Segment(Q, J) → l_1
  51. Segment(Q, I) → m_1
  52. Segment(I, J) → n_1
  53. Dilate(J, If[2 < s ≤ 3, 3 - s, 0], R) → J'
  54. Dilate(I, If[2 < s ≤ 3, 3 - s, 0], P) → I'
  55. Segment(I', J') → p_1
  56. Angle(Q, I', J') → η
  57. Angle(Q, I, J) → θ
  58. Angle(D'_2, E'_1, G') → ι
  59. Angle(J, Q, I) → λ
  60. Angle(G', D'_1, E'_1) → μ
  61. Polygon(Q, J', I') → poly4, i', a_2, j'
  62. Angle(Q, I, J) → ν
  63. Segment(G'_1, E'_2) → q_1
  64. Segment(G', E'_1) → f_1
  65. Angle(G'_1, Q, E'_2) → κ
  66. Angle(Q, E'_2, G'_1) → ξ
  67. Rotate(D'_3, ((s - 1) * ζ), Q) → K
  68. Rotate(D'_3, If[1 < s ≤ 2, ((s - 1) * ζ), ζ], Q) → L_1
  69. Rotate(E'_2, If[1 < s ≤ 2, ((s - 1) * ζ), ζ], Q) → M_1
  70. Rotate(G'_1, If[1 < s ≤ 2, ((s - 1) * ζ), ζ], Q) → N_1
  71. Polygon(L_1, M_1, N_1) → poly5, n_2, l_2, m_2
  72. Dilate(M_1, If[2 < s ≤ 3, 3 - s, 0], P) → M'
  73. Dilate(N_1, If[2 < s ≤ 3, 3 - s, 0], R) → N'
  74. Segment(M', N') → r_1
  75. Polygon(Q, M', N') → poly6, n', a_3, m'
  76. Angle(Q, M', N') → ο
  77. Angle(Q, M_1, N_1) → ρ
  78. Segment(Q, M_1) → t_1
  79. Segment(M_1, N_1) → b_2
  80. Segment(Q, N_1) → c_2
  81. Angle(P, Q, R) → σ
  82. Angle(G, L, M) → τ
  83. Angle(N_1, M_1, Q) → υ
  84. Angle(N', M', Q) → ϕ
  85. Angle(Q, R, P) → χ
  86. Angle(M, G, L) → ω
  87. Intersect(q, r) → N