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SSS Triangle Congruence Theorem: Dynamic Proof

SSS Triangle Congruence Theorem: Dynamic Proof
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2018-02-27 21:25
TriangleAngleVectorTransformationGeometric Construction
① This example dynamically demonstrates the proof of the SSS (Side-Side-Side) triangle congruence theorem, showing that if three corresponding sides of two triangles are equal, the triangles are congruent. ② Core mathematical concepts include triangle congruence, the SSS criterion, geometric transformations (translation, rotation, reflection), and vector operations. ③ Construction highlights: Using GeoGebra commands such as Circle, Intersect, Segment, Polygon, Point, Line, Angle, Dilate, Vector, Translate, Rotate, Mirror, Ray, Min, and CircleSector, the construction builds a base triangle and then applies transformations to move it onto another triangle, visually verifying congruence; key objects include points A, B, C, sides a, b, c, and auxiliary circles, segments, and polygons. ④ Suitable for middle school geometry (e.g., grades 8-9), this resource helps students grasp the intuitive basis of congruence criteria and serves as a classroom demonstration or self-exploration tool.

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

Create a dynamic diagram illustrating the Side-Side-Side (SSS) triangle congruence theorem. Start with a base triangle, then construct a second triangle with all three corresponding sides equal. Use transformations such as translation, rotation, and reflection to move the second triangle onto the first, visually demonstrating congruence. Include auxiliary circles, segments, and angle markers to highlight the equal sides and angles. The diagram should clearly show that when three sides match, the triangles are congruent.
The prompt above was compiled from the original drawing commands and is for reference only.