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Algebra Proof of Vertical Angles Theorem

Algebra Proof of Vertical Angles Theorem
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2016-05-03 19:03
AngleCoordinate GeometryGeometric Construction
This example demonstrates how to prove the Vertical Angles Theorem algebraically, showing that vertical angles formed by intersecting lines are equal. Key math concepts include angles, intersecting lines, and algebraic proofs. GeoGebra commands and objects used in the construction are Circle, Point, Mirror, Semicircle, Line, Intersect, CircleSector, Midpoint, Segment, and Dilate. Suitable for high school students learning geometric theorem proofs and math competition preparation.

GeoGebra Commands & Parameters

  1. Circle(A_1, 2) → c
  2. Circle(A_1, 4) → d
  3. Point(d) → A
  4. Mirror(A, A_1) → B'
  5. Semicircle(A, B') → e
  6. Point(e) → B
  7. Line(A_1, B) → a
  8. Line(A_1, A) → b
  9. Intersect(c, a, 2) → D
  10. Intersect(c, b, 2) → E
  11. Intersect(c, a, 1) → F
  12. Intersect(c, b, 1) → G
  13. CircleSector(A_1, D, E) → f
  14. CircleSector(A_1, G, D) → g
  15. CircleSector(A_1, F, G) → h
  16. Midpoint(E, D) → H
  17. Midpoint(D, G) → I
  18. Midpoint(G, F) → J
  19. Point(xAxis) → K
  20. Segment(H, K) → j
  21. Dilate(H, If[0 ≤ i < 1, 1 - i, 0], K) → H'
  22. Point(xAxis) → L
  23. Segment(J, L) → k
  24. Point(xAxis) → M
  25. Segment(I, M) → l
  26. Dilate(I, If[0 ≤ i < 1, 1 - i, 0], M) → I'
  27. Dilate(J, If[0 ≤ i < 1, 1 - i, 0], L) → J'
  28. Point(xAxis) → N
  29. Segment(I, N) → m
  30. Dilate(I, If[0 ≤ i < 1, 1 - i, 0], N) → I'_1
  31. Dilate(N, If[2 ≤ i < 3, 3 - i, 0], O) → N'
  32. Dilate(K, If[2 ≤ i < 3, 3 - i, 0], (0, 0)) → K'