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Proof that Complements of Congruent Angles Are Congruent – GeoGebra Demonstratio

Proof that Complements of Congruent Angles Are Congruent – GeoGebra Demonstratio
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2016-10-20 05:47
AngleTransformationGeometric Construction
This GeoGebra example demonstrates that the complements of congruent (or equal) angles are themselves congruent. It illustrates the geometric theorem concerning angle complements, involving concepts such as angle measurement, supplementary angles, and congruence relations. The construction uses commands like Ray, Rotate, Angle, Circle, Line, Intersect, LaTeX, CircleSector, Dilate, Translate, Point, CircleArc, OrthogonalLine, and objects such as A, B, α, B′, β, etc., to build the figure and verify the equality of the complementary angles. The example is suitable for high‑school geometry classes and proof‑based mathematics courses.

GeoGebra Commands & Parameters

  1. Ray(A, B) → f
  2. Rotate(B, α, A) → B'
  3. Angle(B, A, B') → β
  4. Ray(A, B') → g
  5. Circle(A, 1) → c
  6. Line(B, A) → h
  7. Intersect(c, h, 2) → C
  8. Ray(A, C) → i
  9. Ray(D, E) → j
  10. Rotate(E, α, D) → E'
  11. Angle(E, D, E') → γ
  12. Ray(D, E') → k
  13. Line(E, D) → l
  14. Circle(D, 1) → d
  15. Intersect(d, l, 2) → F
  16. Ray(D, F) → m
  17. LaTeX(B', true, true) → text1
  18. Circle(D, 2) → e
  19. Intersect(e, j, 1) → G
  20. Intersect(e, k, 1) → H
  21. Line(D, i) → n
  22. CircleSector(D, G, H) → p
  23. Dilate(D, If[0 < a ≤ 1, 1 - a, 0], A) → D'
  24. Vector(D, D') → u
  25. Translate(p, u) → p'
  26. Point(n) → I
  27. Angle(I, D, E') → δ
  28. Line(D, g) → q
  29. Intersect(e, q, 2) → J
  30. Angle(H, D, J) → ε
  31. Rotate(p', If[0 < a ≤ 1, (a * ε), ε], D') → p''
  32. CircleArc(D, G, H) → r
  33. Intersect(e, m) → K,
  34. Dilate(D, If[2.75 < a ≤ 3.75, 3.75 - a, 0], A) → D'_1
  35. Vector(D, D'_1) → v
  36. CircleArc(D, H, K) → t
  37. OrthogonalLine(A, f) → b
  38. OrthogonalLine(D, j) → f_1
  39. Intersect(e, f_1) → L, M
  40. Angle(G, D, M) → ζ
  41. Ray(D, M) → s
  42. Line(D, b) → g_1
  43. Intersect(d, g_1, 2) → N
  44. Angle(M, D, N) → η
  45. CircleSector(D, H, M) → c_1
  46. Dilate(D, If[2.25 < a ≤ 3.25, 3.25 - a, 0], A) → D'_2
  47. Vector(D, D'_2) → w
  48. Translate(c_1, w) → d_1
  49. Rotate(d_1, If[2.25 < a ≤ 3.25, ((a - 2.25) * η), η], D'_2) → e_1
  50. Point(b) → O
  51. Angle(B, A, O) → θ
  52. CircleArc(D, H, M) → h_1
  53. Circle(A, 5) → k_1
  54. Intersect(k_1, b, 2) → Q
  55. Ray(A, Q) → i_1