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What is Congruence (SSS)? – Side-Side-Side Theorem Demo

What is Congruence (SSS)? – Side-Side-Side Theorem Demo
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2019-04-18 19:42
TrianglePolygonAngleTransformationGeometric Construction
① This GeoGebra example dynamically demonstrates the SSS (Side-Side-Side) congruence theorem: two triangles are congruent if their three corresponding sides are equal in length. ② Core mathematical concepts include triangle congruence, corresponding sides and angles, rigid transformations (rotation, translation, reflection), and geometric construction using circles and intersections. ③ Construction highlights: using commands such as Circle, Point, Segment, Rotate, Angle, Ray, Intersect, Polygon, Mirror, Line, Dilate, Vector, and Translate, the original triangle is copied and transformed while ensuring equal side lengths via circle intersections; polygons and angle labels visually confirm congruence. ④ Suitable for middle school geometry lessons, self-study, or interactive teacher demonstrations to verify the SSS criterion dynamically.

GeoGebra Commands & Parameters

  1. Circle(A, a) → c
  2. Point(c) → B
  3. Segment(A, B) → f
  4. Circle(A, b) → d
  5. Rotate(B, α, A) → B'
  6. Angle(B, A, B') → β
  7. Ray(A, B') → g
  8. Intersect(d, g, 1) → C
  9. Segment(B, C) → h
  10. Segment(A, C) → i
  11. Min(f, i) → e
  12. Circle(A, (0.25 * e)) → k
  13. Intersect(k, f) → D,
  14. Intersect(k, i) → E,
  15. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  16. CircleSector(A, D, E) → p
  17. Mirror(A, h) → A'
  18. Mirror(B, h) → B'_1
  19. Mirror(C, h) → C'
  20. Polygon(A', B'_1, C') → poly1', c', a', b'_1
  21. Mirror(p, h) → p'
  22. Mirror(A, h) → A'_1
  23. Mirror(B, h) → B'_2
  24. Segment(A'_1, B'_2) → f'
  25. Mirror(A, h) → A'_2
  26. Mirror(C, h) → C'_1
  27. Segment(A'_2, C'_1) → i'
  28. Line(C, B) → j
  29. Point(j) → G
  30. Point(j) → H
  31. Ray(B, H) → l
  32. Ray(C, G) → m
  33. Circle(F, a) → q
  34. Point(q) → I
  35. Segment(F, I) → n
  36. Rotate(I, α, F) → I'
  37. Angle(I, F, I') → γ
  38. Circle(F, b) → r
  39. Ray(F, I') → s
  40. Intersect(r, s, 1) → J
  41. Segment(I, J) → t
  42. Segment(F, J) → f_1
  43. Dilate(B, If[0 < u ≤ 1, 1 - u, 0], I) → B'_3
  44. Vector(B, B'_3) → v
  45. Translate(A, v) → A'_3
  46. Translate(B, v) → B'_4
  47. Segment(A'_3, B'_4) → f'_1
  48. Line(B, n) → g_1
  49. Point(g_1) → K
  50. Angle(K, B, A) → δ
  51. Rotate(A'_3, If[1 < u ≤ 2, ((-(u - 1)) * δ), (-δ)], I) → L
  52. Rotate(B'_4, If[1 < u ≤ 2, ((-(u - 1)) * δ), (-δ)], I) → M
  53. Segment(L, M) → h_1
  54. Dilate(A, If[2 < u ≤ 3, 3 - u, 0], F) → A'_4
  55. Vector(A, A'_4) → w
  56. Translate(A, w) → A'_5
  57. Translate(C, w) → C'_2
  58. Segment(A'_5, C'_2) → i'_1
  59. Line(A, f_1) → i_1
  60. Point(i_1) → N
  61. Angle(N, A, C) → ε
  62. Rotate(A'_5, If[3 < u ≤ 4, ((-(u - 3)) * ε), (-ε)], F) → O
  63. Rotate(C'_2, If[3 < u ≤ 4, ((-(u - 3)) * ε), (-ε)], F) → P
  64. Segment(O, P) → j_1
  65. Translate(p, w) → p'_1
  66. Rotate(p'_1, If[3 < u ≤ 4, ((-(u - 3)) * ε), (-ε)], F) → d_1
  67. Dilate(B, If[4 < u ≤ 5, 5 - u, 0], I) → B'_5
  68. Vector(B, B'_5) → u_1
  69. Translate(A, u_1) → A'_6
  70. Translate(B, u_1) → B'_6
  71. Translate(C, u_1) → C'_3
  72. Polygon(A'_6, B'_6, C'_3) → poly1'_1, c'_3, a'_6, b'_6
  73. Line(B, t) → k_1
  74. Point(k_1) → Q
  75. Angle(Q, B, C) → ζ
  76. Rotate(A'_6, If[5 < u ≤ 6, ((-(u - 5)) * ε), (-ε)], I) → R
  77. Rotate(B'_6, If[5 < u ≤ 6, ((-(u - 5)) * ε), (-ε)], I) → S
  78. Rotate(C'_3, If[5 < u ≤ 6, ((-(u - 5)) * ε), (-ε)], I) → T
  79. Polygon(R, S, T) → poly2, t_1, r_1, s_1
  80. Segment(B, A) → l_1
  81. Segment(A, C) → m_1
  82. Segment(C, B) → n_1
  83. CircleSector(A, D, E) → e_1

AI Drawing Prompt Reference

Draw a triangle ABC, then copy and rotate it to form a congruent triangle A'B'C' using the Side-Side-Side theorem. Start by drawing a circle centered at A with radius AB, find point B on the circle, and connect AB. Then draw a circle centered at A with radius AC, rotate B to B' to find angle BAC, and draw a circle centered at B' with radius BC to find point C'. Connect BC' and AC' to form triangle A'B'C'.
The prompt above was compiled from the original drawing commands and is for reference only.