Drawtwo polygonsthat arenotsimilar, but whichdohaveall correspondinganglescongruent.Rectanglessuchas the onesbelowmakegoodexamples.Note:All rectangleshavecongruent(right)angles.However, we saw in an earlierlessonthat rectangles
can havedifferentshapes—longand narrowvs. stubbyand square-ish.In formalterms,theserectangles
havecongruentangles,but their side lengthsare obviouslynot proportional.The rectanglesare not similar.
Congruentanglesare not enoughto ensuresimilarityfor rectangles.
The AA Rulefor SimilarTriangles
Someartistsand designersapplythe principlethat “lessis more.”This idea has a placein geometryas well.
Somegeometryscholarsfeel that it is moresatisfyingto provesomethingwith the least possibleinformation.
Similartrianglesare a goodexampleof this principle.
The AAA rule was developedfor similartrianglesearlier. Let’s take anotherlook at this rule, and see if we
can reduceit to “less”ratherthan “more.”
Supposethat triangles and havetwo pairsof congruentangles,say
and.But we knowthat if triangleshavetwopairsof congruentangles,then thethirdpair of anglesare also con-
gruent(by the TriangleSum Theorem).
Summary:Lessis more.The AAA rule for similartrianglesreducesto the AA trianglesimilaritypostulate.
The AA TriangleSimilarityPostulate:If two pairsof correspondinganglesin two trianglesare congruent,
then the trianglesare similar.Example 2
Lookat the diagrambelow.A.Are the trianglessimilar?Explainyour answer.Yes. Theyboth havecongruentright angles,and they both havea angle.The trianglesare similarby
AA.