Similarityby AA
LearningObjectives
- Determinewhethertrianglesare similar.
- UnderstandAAA and AA rulesfor similartriangles.
- Solveproblemsaboutsimilartriangles.
Introduction
You havean understandingof whatsimilarpolygonsare and how to recognizethem.Becausetrianglesare
the mostbasicbuildingblockon whichotherpolygonscan be based,we now focusspecificallyon similar
triangles.We’ll find that there’s a surprisinglysimplerule for trianglesto be similar.
Anglesin SimilarTriangles
Tech Note- GeometrySoftware
Use your geometrysoftwareto experimentwith triangles.Try this:- Set up two triangles, and.
- Measurethe anglesof both triangles.
- Movethe verticesuntil the measuresof the correspondinganglesare the samein both triangles.
- Computethe ratiosof the lengthsof the sides
.
Repeatsteps1-4 with differenttriangles.Observewhathappensin step 4 eachtime.Recordyour obser-
vations.Whatdid you see duringyour experiment?You mighthavenoticedthis: Whenyou adjusttrianglesto make
their anglescongruent,you automaticallymakethe sidesproportional(the ratiosin step 4 are the same).
Oncewe havetriangleswith congruentanglesand sideswith proportionallengths,we knowthat the triangles
are similar.
Conclusion:If the anglesof a triangleare congruentto the correspondinganglesof anothertriangle,then
the trianglesare similar. This is a handyrule for similartriangles—arule basedon just the anglesof the tri-
angles.We call this the AAA rule.
Caution:The AAA rule is a rule fortrianglesonly. We alreadyknowthat otherpairsof polygonscan have
all correspondinganglescongruenteventhoughthe polygonsarenotsimilar.
Example 1
The followingis false statement:If the correspondinganglesof two polygonsare congruent,then the polygons
are similar.
Whatis a counterexampleto thefalsestatementabove?