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Triangle Congruence Theorems Summary

Triangle Congruence Theorems Summary
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2020-11-29 22:25
TrianglePolygonAngleTransformationGeometric Construction
In collections Exploring the Conditions for Congruence of 2 Triangles
①This interactive GeoGebra example systematically explores the conditions for triangle congruence (SSS, SAS, ASA, AAS, etc.), showing which combinations of sides and angles are sufficient to prove two triangles congruent and which are not. ②Core mathematical concepts include triangle congruence theorems, corresponding sides and angles, counterexamples, and logical reasoning. ③Construction highlights: using GeoGebra commands such as Point, Polygon, Segment, Intersect, Angle, and LaTeX, the applet builds multiple triangles (e.g., tri_SSS, tri_SA, tri_SSA) with labeled sides and angles, allowing users to compare different congruence criteria dynamically. ④Suitable for middle school geometry (grade 8 and above), this resource supports classroom inquiry, student self-exploration of congruence theorems, or teacher demonstrations to clarify the necessity and sufficiency of each condition.

GeoGebra Commands & Parameters

  1. Point(Circle[A_{SS}, AB]) → B_{SS}
  2. Point(Circle[B_{SS}, BC]) → C_{SS}
  3. Polygon(A_{SS}, B_{SS}, C_{SS}) → tri_{SS}, c_{ss}, a_{ss}, b_{ss}
  4. Point(Circle[A_{SA}, AB]) → B_{SA}
  5. Point(Ray[Rotate[B_{SA}, (-angB), B_{SA}], Rotate[A_{SA}, angB, B_{SA}]]) → C_{SA}
  6. Polygon(A_{SA}, B_{SA}, C_{SA}) → tri_{SA}, c_{sa}, a_{sa}, b_{sa}
  7. Point(Circle[A_{SSS}, AB]) → B_{SSS}
  8. Point(Circle[B_{SSS}, BC]) → C_{SSS}
  9. Point(Circle[C_{SSS}, CA]) → AC_{SSS}
  10. Segment(A_{SSS}, B_{SSS}) → a
  11. Segment(B_{SSS}, C_{SSS}) → b
  12. Segment(C_{SSS}, AC_{SSS}) → c
  13. Polygon(A_{SSS}, B_{SSS}, C_{SSS}) → tri_{SSS}, c_{sss}, a_{sss}, b_{sss}
  14. Point(Circle[A_{SSA}, AB]) → B_{SSA}
  15. Point(Circle[B_{SSA}, BC]) → C_{SSA}
  16. Point(Ray[Rotate[C_{SSA}, (-angC), C_{SSA}], Rotate[B_{SSA}, angC, C_{SSA}]]) → AC_{SSA}
  17. Polygon(A_{SSA}, B_{SSA}, C_{SSA}) → tri_{SSA}, c_{ssa}, a_{ssa}, b_{ssa}
  18. Point(Circle[A_{AAS}, CA]) → C_{AAS}
  19. Point(Ray[Rotate[A_{AAS}, angA, A_{AAS}], Rotate[C_{AAS}, angA, A_{AAS}]]) → BA_{AAS}
  20. Intersect(Ray[Rotate[A_{AAA}, (-angA), A_{AAA}], Rotate[B_{AAA}, (-angA), A_{AAA}]], Ray[Rotate[B_{AAA}, angB, B_{AAA}], Rotate[A_{AAA}, angB, B_{AAA}]]) → C_{AAA}
  21. Polygon(A_{AAA}, B_{AAA}, C_{AAA}) → tri_{AAA}, c_{aaa}, a_{aaa}, b_{aaa}
  22. Angle(A_{SA}, B_{SA}, C_{SA}) → α
  23. Segment(A_{SSA}, B_{SSA}) → d
  24. Segment(B_{SSA}, C_{SSA}) → e
  25. Segment(C_{SSA}, AC_{SSA}) → f
  26. Angle(B_{SSA}, C_{SSA}, AC_{SSA}) → β
  27. Segment(A_{AAS}, C_{AAS}) → g
  28. Segment(A_{AAS}, BA_{AAS}) → h
  29. Angle(C_{AAS}, A_{AAS}, BA_{AAS}) → γ
  30. Point(Circle[A_{ASA}, CA]) → C_{ASA}
  31. Segment(A_{ASA}, C_{ASA}) → m
  32. Point(Ray[Rotate[A_{ASA}, angA, A_{ASA}], Rotate[C_{ASA}, angA, A_{ASA}]]) → BA_{ASA}
  33. Segment(A_{ASA}, BA_{ASA}) → n
  34. Point(Ray[Rotate[C_{ASA}, (-angC), C_{ASA}], Rotate[A_{ASA}, (-angC), C_{ASA}]]) → BC_{ASA}
  35. Segment(C_{ASA}, BC_{ASA}) → p
  36. Intersect(Ray[A_{ASA}, BA_{ASA}], Ray[C_{ASA}, BC_{ASA}]) → B_{ASA}
  37. Polygon(A_{ASA}, B_{ASA}, C_{ASA}) → tri_{ASA}, c_{asa}, a_{asa}, b_{asa}
  38. Angle(BA_{ASA}, A_{ASA}, C_{ASA}) → ζ
  39. Angle(BC_{ASA}, C_{ASA}, A_{ASA}) → η
  40. Angle(C_{AAA}, B_{AAA}, A_{AAA}) → θ
  41. Angle(B_{AAA}, A_{AAA}, C_{AAA}) → ι
  42. Angle(A_{AAA}, C_{AAA}, B_{AAA}) → κ
  43. Point(Ray[Rotate[BA_{AAS}, angB, BA_{AAS}], Rotate[A_{AAS}, angB, BA_{AAS}]]) → CB_{AAS}
  44. Segment(BA_{AAS}, CB_{AAS}) → i
  45. Angle(CB_{AAS}, BA_{AAS}, A_{AAS}) → δ
  46. Intersect(Ray[Rotate[C_{AAS}, (-angC), C_{AAS}], Rotate[A_{AAS}, (-angC), C_{AAS}]], Ray[Rotate[A_{AAS}, angA, A_{AAS}], Rotate[C_{AAS}, angA, A_{AAS}]]) → B_{AAS}
  47. Polygon(A_{AAS}, B_{AAS}, C_{AAS}) → tri_{AAS}, c_{aas}, a_{aas}, b_{aas}
  48. Point(Circle[A_{SAS}, AB]) → B_{SAS}
  49. Intersect(Circle[A_{SAS}, CA], Ray[Rotate[A_{SAS}, (-angA), A_{SAS}], Rotate[B_{SAS}, (-angA), A_{SAS}]]) → C_{SAS},
  50. Polygon(A_{SAS}, B_{SAS}, C_{SAS}) → tri_{SAS}, c_{sas}, a_{sas}, b_{sas}
  51. Angle(C_{SAS}, A_{SAS}, B_{SAS}) → ε
  52. Point(Circle[A_1, AB]) → B_1
  53. Point(Ray[Rotate[B_1, (-angB), B_1], Rotate[A_1, angB, B_1]]) → C_1
  54. Polygon(A_1, B_1, C_1) → tri_1, c_{sa1}, a_{sa1}, b_{sa1}
  55. Angle(A_1, B_1, C_1) → α_1
  56. LaTeX(c_{sa}, true, true) → text1
  57. Point(Circle[A_2, AB]) → B_2
  58. Point(Circle[B_2, BC]) → C_2
  59. Segment(A_2, B_2) → a_2
  60. Segment(B_2, C_2) → b_2
  61. Polygon(A_2, B_2, C_2) → tri_2, c_1, a_1, b_1
  62. Point(Circle[A_3, CA]) → C_3
  63. Segment(A_3, C_3) → g_1
  64. Intersect(Ray[Rotate[C_3, (-angC_1), C_3], Rotate[A_3, (-angC_1), C_3]], Ray[Rotate[A_3, angA, A_3], Rotate[C_3, angA, A_3]]) → B_3
  65. Polygon(A_3, B_3, C_3) → tri_3, c_2, a_3, b_3
  66. Point(Circle[A_4, AB]) → B_4
  67. Intersect(Circle[A_4, CA], Ray[Rotate[A_4, (-angA), A_4], Rotate[B_4, (-angA), A_4]]) → C_4,
  68. Polygon(A_4, B_4, C_4) → tri_4, c_3, a_4, b_4
  69. Angle(C_4, A_4, B_4) → ε_1
  70. Intersect(Ray[Rotate[A_5, (-angA), A_5], Rotate[B_5, (-angA), A_5]], Ray[Rotate[B_5, angB, B_5], Rotate[A_5, angB, B_5]]) → C_5
  71. Polygon(A_5, B_5, C_5) → tri_5, c_4, a_5, b_5
  72. Angle(C_5, B_5, A_5) → θ_1
  73. Angle(B_5, A_5, C_5) → ι_1
  74. Angle(A_5, C_5, B_5) → κ_1
  75. Point(Circle[A_6, AB]) → B_6
  76. Point(Circle[B_6, BC]) → C_6
  77. Polygon(A_6, B_6, C_6) → tri_6, c_5, a_6, b_6
  78. Point(Circle[A_7, AB]) → B_7
  79. Point(Circle[B_7, BC]) → C_7
  80. Polygon(A_7, B_7, C_7) → tri_7, c_6, a_7, b_7
  81. Segment(A_7, B_7) → d_1
  82. Segment(B_7, C_7) → e_1
  83. Point(Circle[A_8, CA]) → C_8
  84. Segment(A_8, C_8) → m_1
  85. Point(Ray[Rotate[A_8, angA, A_8], Rotate[C_8, angA, A_8]]) → BA_1
  86. Segment(A_8, BA_1) → n_1
  87. Point(Ray[Rotate[C_8, (-angC_4), C_8], Rotate[A_8, (-angC_4), C_8]]) → BC_1
  88. Segment(C_8, BC_1) → p_1
  89. Intersect(Ray[A_8, BA_1], Ray[C_8, BC_1]) → B_8
  90. Polygon(A_8, B_8, C_8) → tri_8, c_7, a_8, b_8
  91. Angle(BA_1, A_8, C_8) → ζ_1
  92. Angle(BC_1, C_8, A_8) → η_1
  93. Point(Circle[A_9, CA_1]) → C_9
  94. Segment(A_9, C_9) → m_2
  95. Point(Ray[Rotate[A_9, angA_1, A_9], Rotate[C_9, angA_1, A_9]]) → BA_2
  96. Segment(A_9, BA_2) → n_2
  97. Point(Ray[Rotate[C_9, (-angC_8), C_9], Rotate[A_9, (-angC_8), C_9]]) → BC_2
  98. Segment(C_9, BC_2) → p_2
  99. Intersect(Ray[A_9, BA_2], Ray[C_9, BC_2]) → B_9
  100. Polygon(A_9, B_9, C_9) → tri_9, c_8, a_9, b_9
  101. Angle(BA_2, A_9, C_9) → ζ_2
  102. Angle(BC_2, C_9, A_9) → η_2

AI Drawing Prompt Reference

Create a single scene containing several triangle-congruence comparison configurations. Construct multiple triangles one after another, where each triangle is formed by defining points on circles using given side lengths, then using rotations and ray intersections to place the third vertex so the required angle/side relationships are satisfied. For each constructed triangle, draw the three sides to form a closed polygon and include labels for the side lengths (like a, b, c in each configuration) and for the key angles at relevant vertices (such as α, β, γ, ζ, η, etc.). Arrange the different triangles so their corresponding sides and corresponding angles can be visually compared across the cases.
The prompt above was compiled from the original drawing commands and is for reference only.