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Triangles – Congruence Theorems

One counter-example ends a claim

A claim passes a hundred tries. Is that enough to trust it?

Many tries pass. One that fails ends the claim.

The rule

No: a case nobody tried can still break the claim.

A counter-example is one case where a claim fails. A claim that says always is false if one case fails.

Take the claim "two sides and an angle always make one triangle." Take sides 8 and 5, with 30° next to the 8. Those measurements build two triangles. That one case ends the claim.

The other direction matters too. A search that finds no failure is not a proof. It only shows the claim held in the cases someone tried. A proof covers every case.

Try cases on each claim. Does any case break it?

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This track

  1. 0 Start here
  2. 1 SAS: two sides and the angle between
  3. 2 SSA can give two triangles
  4. 3 RHS: a right angle, the hypotenuse, a leg
  5. 4 AAS: two angles and any side
  6. 5 Equal sides give equal base angles
  7. 6 Checkpoint: SAS, SSA, RHS and AAS
  8. 7 Equal angles give equal sides
  9. 8 One counter-example ends a claim
  10. 9 Matching parts decide congruence
  11. 10 The apex bisector does three jobs
  12. 11 Checkpoint: converse, counter-example and bisector
  13. 12 The big picture
  14. 13 Track quiz

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