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SSS Triangle Congruence Theorem: Dynamic Proof by Aaron Dankman

SSS Triangle Congruence Theorem: Dynamic Proof by Aaron Dankman
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2017-02-01 07:47
TriangleAngleTransformationGeometric ConstructionSlider Animation
In collections Triangle Congruence Book 1
①This GeoGebra example dynamically demonstrates the proof of the SSS (Side-Side-Side) triangle congruence theorem, showing that if three sides of one triangle are equal to the corresponding sides of another, the triangles are congruent. ②Core mathematical concept: the SSS congruence criterion in Euclidean geometry, emphasizing the logical sufficiency of side lengths to determine triangle shape. ③Construction highlights: uses Circle and Intersect to create points, Segment and Polygon to form triangles, applies transformations such as Dilate, Translate, Rotate, and Mirror, and employs Angle, Vector, Ray, and dynamic sliders to visualize correspondence and enable interactive manipulation. ④Suitable for middle school geometry (grades 8-9) to teach triangle congruence, ideal for classroom demonstrations or student-led exploration of geometric proofs.

GeoGebra Commands & Parameters

  1. Circle(A, a) → d_1
  2. Intersect(d_1, e) → B, C
  3. Circle(C, b) → f
  4. Segment(A, C) → g
  5. Circle(A, c) → k
  6. Intersect(k, f) → F, G, ,
  7. Segment(C, F) → h
  8. Segment(F, A) → i
  9. Polygon(A, C, F) → poly1, f_1, a_1, c_1
  10. Circle(D, i) → p
  11. Point(p) → E
  12. Segment(D, E) → n
  13. Circle(D, g) → q
  14. Circle(E, h) → r
  15. Intersect(q, r, 2) → H
  16. Segment(D, H) → s
  17. Segment(H, E) → t
  18. Line(A, n) → g_1
  19. Point(g_1) → I
  20. Angle(I, A, F) → α
  21. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], D) → A'
  22. Vector(A, A') → u
  23. Translate(F, u) → F'
  24. Translate(A, u) → A'_1
  25. Segment(F', A'_1) → i'
  26. Rotate(F', If[1 < o ≤ 2, ((-(o - 1)) * α), (-α)], D) → F''
  27. Rotate(A'_1, If[1 < o ≤ 2, ((-(o - 1)) * α), (-α)], D) → J
  28. Segment(F'', J) → i''
  29. Dilate(A, If[2 < o ≤ 3, 3 - o, 0], D) → A'_2
  30. Vector(A, A'_2) → v
  31. Translate(A, v) → A'_3
  32. Translate(C, v) → C'
  33. Segment(A'_3, C') → g'
  34. Line(D, g) → h_1
  35. Point(h_1) → K
  36. Angle(H, D, K) → β
  37. Rotate(A'_3, If[3 < o ≤ 4, ((-(o - 3)) * β), (-β)], D) → L
  38. Rotate(C', If[3 < o ≤ 4, ((-(o - 3)) * β), (-β)], D) → C''
  39. Segment(L, C'') → g''
  40. Line(C, t) → i_1
  41. Point(i_1) → M
  42. Angle(M, C, F) → γ
  43. Dilate(C, If[4 < o ≤ 5, 5 - o, 0], C'') → C'_1
  44. Vector(C, C'_1) → w
  45. Translate(C, w) → C'_2
  46. Translate(F, w) → F'_1
  47. Segment(C'_2, F'_1) → h'
  48. Rotate(C'_2, If[5 < o ≤ 6, ((-(o - 5)) * γ), (-γ)], C'') → N
  49. Rotate(F'_1, If[5 < o ≤ 6, ((-(o - 5)) * γ), (-γ)], C'') → O
  50. Segment(N, O) → h''
  51. Line(F, C) → j_1
  52. Mirror(A, j_1) → A'_6
  53. Mirror(C, j_1) → C'_4
  54. Mirror(F, j_1) → F'_3
  55. Polygon(A'_6, C'_4, F'_3) → poly1'_1, f'_3, a'_6, c'_4
  56. Point(j_1) → S
  57. Point(j_1) → T
  58. Ray(C'_4, T) → k_1
  59. Ray(F'_3, S) → l_1
  60. Segment(A'_6, F'_3) → m_1
  61. Segment(A'_6, C'_4) → n_1
  62. Dilate(F, If[0 < d ≤ 1, 1 - d, 0], E) → F'_2
  63. Vector(F, F'_2) → u_1
  64. Translate(A'_6, u_1) → P
  65. Translate(C'_4, u_1) → Q
  66. Translate(F'_3, u_1) → R
  67. Polygon(P, Q, R) → poly2, r_1, p_1, q_1
  68. Translate(A'_6, u_1) → U
  69. Translate(C'_4, u_1) → V
  70. Segment(U, V) → s_1
  71. Translate(A'_6, u_1) → W
  72. Translate(F'_3, u_1) → Z
  73. Segment(W, Z) → t_1
  74. Translate(C, u_1) → C'_3
  75. Translate(F, u_1) → F'_4
  76. Segment(C'_3, F'_4) → h'_1
  77. Translate(A'_6, u_1) → A_1
  78. Translate(F'_3, u_1) → B_1
  79. Segment(A_1, B_1) → e_1
  80. Line(E, k_1) → f_2
  81. Point(f_2) → C_1
  82. Angle(C'', E, C_1) → δ
  83. Rotate(P, If[1 < d ≤ 2, ((-(d - 1)) * δ), (-δ)], E) → P'
  84. Rotate(Q, If[1 < d ≤ 2, ((-(d - 1)) * δ), (-δ)], E) → Q'
  85. Rotate(R, If[1 < d ≤ 2, ((-(d - 1)) * δ), (-δ)], E) → R'
  86. Polygon(P', Q', R') → poly2', r', p', q'
  87. Rotate(A_1, If[1 < d ≤ 2, ((-(d - 1)) * δ), (-δ)], E) → D_1
  88. Rotate(B_1, If[1 < d ≤ 2, ((-(d - 1)) * δ), (-δ)], E) → E_1
  89. Segment(D_1, E_1) → g_2
  90. Rotate(U, If[1 < d ≤ 2, ((-(d - 1)) * δ), (-δ)], E) → U'
  91. Rotate(V, If[1 < d ≤ 2, ((-(d - 1)) * δ), (-δ)], E) → V'
  92. Segment(U', V') → h_2
  93. Rotate(C'_3, If[1 < d ≤ 2, ((-(d - 1)) * δ), (-δ)], E) → F_1
  94. Rotate(F'_4, If[1 < d ≤ 2, ((-(d - 1)) * δ), (-δ)], E) → G_1
  95. Segment(F_1, G_1) → i_2
  96. Polygon(D, C'', E) → poly3, e_2, d_2, c''
  97. Dilate(D, If[2.5 < d ≤ 3.5, 3.5 - d, 0], D_1) → D'
  98. Segment(D, D') → j_2
  99. Segment(E, C'') → k_2
  100. Min(c, k_2 / 2) → o_1
  101. Circle(D, (0.3 * o_1)) → c_2
  102. Intersect(c_2, i'') → H_1,
  103. Intersect(c_2, j_2) → I_1,
  104. CircleSector(D, I_1, H_1) → p_2
  105. Intersect(j_2, k_2) → J_1
  106. Polygon(D, D_1, E) → poly4, e_3, d_3, d_4
  107. Dilate(D, If[3.5 < d ≤ 4.5, 4.5 - d, 0], D_1) → D'_1
  108. Vector(D, D'_1) → v_1
  109. Translate(p_2, v_1) → q_2
  110. Angle(E, D_1, D) → ε
  111. Rotate(q_2, If[4.5 < d ≤ 5.5, ((d - 4.5) * (π - ε)), π - ε], D_1) → r_2
  112. Intersect(c_2, g'') → K_1, L_1
  113. CircleSector(D, K_1, I_1) → s_2
  114. Polygon(D, C'', D_1) → poly5, d_5, d_6, c''_1
  115. Dilate(D, If[6 < d ≤ 7, 7 - d, 0], D_1) → D'_2
  116. Vector(D, D'_2) → w_1
  117. Translate(s_2, w_1) → t_2
  118. Angle(D, D_1, C'') → ζ
  119. Rotate(t_2, If[7 < d ≤ 8, ((-(d - 7)) * (π - ζ)), (-(π - ζ))], D_1) → c_3
  120. Dilate(E, If[8.5 < d ≤ 9.5, 9.5 - d, 0], C'') → E'
  121. Segment(E, E') → l_2
  122. Segment(E, C'') → m_2
  123. Polygon(A, C, F) → poly6, f_3, a_2, c_4
  124. Polygon(D, C'', E) → poly7, e_4, d_7, c''_2

AI Drawing Prompt Reference

Draw a dynamic illustration of the SSS triangle congruence theorem, showing that two triangles with equal corresponding sides are congruent. Start by creating triangle vertices using circles and intersection points, then connect them with line segments to form two triangles. Apply transformations like dilation, translation, rotation, and reflection to superimpose one triangle onto the other, and use angles and vectors to visualize the correspondences. Incorporate sliders to control transformation parameters for interactive manipulation.
The prompt above was compiled from the original drawing commands and is for reference only.