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Prove triangles congruent with SSS, SAS, and ASA

I can explain how the SSS, SAS, and ASA criteria for triangle congruence follow from the definition of congruence in terms of rigid motions, and use them to prove two triangles congruent.

Domain
Geometry
Level
advanced
Status
draft · v0.2.0
Also called
SSS, SAS, ASA, Triangle congruence theorems, Triangle congruence postulates
License
CC BY-SA 4.0

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Scope

The three criteria that let a proof stop early: SSS, SAS and ASA. Two triangles, some parts known equal, and a conclusion that the whole triangles are congruent — without checking the other three pairs.

In scope: recognising which criterion the given parts supply, writing the proof, and explaining why the criterion is entitled to its conclusion. Also in scope: the two arrangements that look like criteria and are not. AAS is a restatement of ASA once the angle sum is known, and is treated here as a consequence rather than a fourth criterion.

Why the criteria follow from rigid motions

A criterion is a shortcut, and the shortcut has to be earned. Congruence was defined as the existence of a sequence of rigid motions carrying one figure exactly onto another, so each criterion is the claim that the named parts already determine that sequence.

Take SAS. Translate the first triangle so that the vertex between the two known sides lands on its partner. Rotate so that one known side lies along its partner; because those sides are equal, the endpoints coincide. The known angle then forces the second side onto its partner, and because that side is equal too, its endpoint coincides as well. Reflect if the third vertex landed on the wrong side of the line. Every vertex now sits on its partner, so the triangles are congruent.

SSS and ASA are the same argument with different parts doing the forcing. A learner who can narrate one of the three can usually build the other two, and that narration is the outcome, not the memorised acronym.

Why SSA and AAA are not criteria

  • SSA. Two sides and a non-included angle can often be assembled two ways: the third vertex has two possible positions, giving one triangle with an acute angle and another with an obtuse one. Drawing both from the same given parts is the counterexample.
  • AAA. Three equal angles force the same shape but not the same size. AAA is the criterion for similarity, and calling it congruence is the most common error in this topic.

Common misconceptions

  • “The letters are an order to memorise.” They are a description of which parts are known and how they sit relative to each other. “Included” is the whole content of SAS.
  • “Any three equal parts are enough.” SSA and AAA are three equal parts each, and neither is a criterion.
  • “A criterion needs a picture that looks right.” A diagram drawn to look congruent proves nothing; the given parts do the work, and a proof should survive a badly drawn figure.
  • “Congruent means same shape.” That is similarity. Congruent means same shape and size, which is what rigid motions preserve.

Notes for teachers

Ask for the rigid-motion narration before the two-column proof. A learner who can say where each vertex goes and what forces it has understood the criterion; one who writes “SAS” has recalled a label.

The SSA counterexample repays construction rather than description: with a compass, or with GeoGebra, fix two sides and a non-included angle and let learners find the second triangle for themselves.

Resources

ResourceKindCostProvider
Khan Academy, CongruencevideofreeKhan Academy
GeoGebra GeometrytoolfreeGeoGebra
Illustrative Mathematics: tasks for HSG-CO.B.8practicefreeIllustrative Mathematics

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Evidence of mastery

  • Given two triangles and some equal parts, names the criterion that applies and writes a proof that they are congruent.
  • Explains why the criterion works by describing the sequence of rigid motions that carries one triangle onto the other.
  • Shows with a counterexample why SSA and AAA are not congruence criteria, and says what each does establish.
  • Uses a congruence to justify a further claim about corresponding sides or angles inside a longer proof.

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