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Why ASS (or SSA) Cannot Determine Triangle Congruence

Why ASS (or SSA) Cannot Determine Triangle Congruence
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2020-07-12 18:38
TriangleAngleTransformationGeometric ConstructionSlider Animation
This GeoGebra example dynamically demonstrates why the Angle-Side-Side (ASS or SSA) condition is not a valid criterion for triangle congruence. It shows that when two sides and a non-included angle are fixed, two distinct triangles can be constructed, violating the uniqueness required for congruence. The core mathematical concepts include triangle congruence theorems (SSS, SAS, ASA, AAS) and counterexample construction. The construction uses GeoGebra commands such as Polygon, Angle, Circle, Point, Segment, Line, Rotate, and Intersect to build two triangles, rotate one for comparison, and use sliders or step controls to explore different configurations. Text labels highlight the key relationships. This resource is ideal for middle school geometry or high school review, helping students grasp the necessity and sufficiency of congruence criteria and avoid common misconceptions.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → pol1, c, a, b
  2. Angle(B, A, C) → α
  3. Circle(D, b) → d
  4. Point(d) → E
  5. Segment(D, E) → f
  6. Circle(E, a) → e
  7. Line(D, E) → g
  8. Rotate(g, (-α), D) → g'
  9. Intersect(e, g', 2) → F
  10. Polygon(F, E, D) → pol2, d_1, f_1, e_1
  11. Angle(F, D, E) → β

AI Drawing Prompt Reference

Draw two triangles, where one triangle has two sides and a non-included angle fixed, and the other triangle is obtained by rotating the first one. Use a slider to adjust the angle and observe different configurations of the two triangles.
The prompt above was compiled from the original drawing commands and is for reference only.