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SSS Triangle Congruence: Dynamic Proof and Visualization

SSS Triangle Congruence: Dynamic Proof and Visualization
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2016-07-30 15:15
TrianglePolygonAngleTransformationGeometric Construction
In collections Congruence (CO) Cluster Congruence (Vol 1) Congruence (Vol 1) Over 80 Resources Geometry (Congruence) Volume 1 CCSS High School: Geometry (Congruence) Volume 1 S N ATTAR CCSS High School: Geometry (Congruence) Volume 1 CCSS High School: Geometry (Congruence) Volume 1 CCSS High School: Geometry (Congruence) Volume 1 CCSS High School: Geometry (Congruence) Volume 1 CCSS High School: Geometry (Congruence) Volume 1 CCSS High School: Geometry (Congruence) Volume 1 CCSS High School: Geometry (Congruence) Volume 1 CCSS High School: Geometry (Congruence) Volume 1 CCSS High School: Geometry (Congruence) Volume 1 CCSS High School: Geometry (Congruence) Volume 1 CCSS High School: Geometry (Congruence) Volume 1 Triangle Theorems CCSS High School: Geometry (Congruence) Volume 1 Unit 5: Triangle Congruence Triangle Congruence Theorems 1 Triangle Congruence Book 1 Proving Triangles Congruent Triangle Congruence Triangle Congruence
① This example dynamically illustrates the SSS (Side-Side-Side) triangle congruence theorem: if three corresponding sides of two triangles are equal, the triangles are congruent. ② The core mathematical concept is the SSS congruence criterion, involving geometric transformations such as translation, rotation, and reflection, and the invariance of side lengths under these transformations. ③ Construction highlights: using GeoGebra commands including Circle, Intersect, Segment, Polygon, Point, Line, Angle, Dilate, Vector, Translate, Rotate, Mirror, and Ray, the applet constructs a triangle and its dynamic copy, allowing users to drag vertices and observe that the triangles remain congruent. ④ Suitable for middle school geometry (grades 8-9) to teach congruence theorems, and useful for classroom demonstrations or student self-exploration to intuitively grasp the sufficiency of the SSS condition.

GeoGebra Commands & Parameters

  1. Circle(A, a) → d
  2. Intersect(d, 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. Polygon(D, C'', E) → poly2, e_1, d_1, c''
  52. Dilate(A, If[(6 * 0.5) < o ≤ 7.5, 7.5 - o, 0], D) → A'_4
  53. Vector(A, A'_4) → u_1
  54. Translate(A, u_1) → A'_5
  55. Translate(C, u_1) → C'_3
  56. Translate(F, u_1) → F'_2
  57. Polygon(A'_5, C'_3, F'_2) → poly1', f'_2, a'_5, c'_3
  58. Rotate(A'_5, If[7.5 < o ≤ 8.5, ((-(o - 7.5)) * γ), (-γ)], D) → P
  59. Rotate(C'_3, If[7.5 < o ≤ 8.5, ((-(o - 7.5)) * γ), (-γ)], D) → Q
  60. Rotate(F'_2, If[7.5 < o ≤ 8.5, ((-(o - 7.5)) * γ), (-γ)], D) → R
  61. Polygon(P, Q, R) → poly1'', r_1, p_1, q_1
  62. Line(F, C) → j_1
  63. Mirror(A, j_1) → A'_6
  64. Mirror(C, j_1) → C'_4
  65. Mirror(F, j_1) → F'_3
  66. Polygon(A'_6, C'_4, F'_3) → poly1'_1, f'_3, a'_6, c'_4
  67. Point(j_1) → S
  68. Point(j_1) → T
  69. Ray(C'_4, T) → k_1
  70. Ray(F'_3, S) → l_1
  71. Segment(A'_6, F'_3) → m_1
  72. Segment(A'_6, C'_4) → n_1

AI Drawing Prompt Reference

Draw a triangle ABC, then create a dynamic triangle A'B'C' by translating, rotating, and reflecting triangle ABC around point A, while keeping the two triangles congruent. First, draw triangle ABC, then generate triangle A'B'C' through translation and rotation, and observe that they remain congruent.
The prompt above was compiled from the original drawing commands and is for reference only.