Finally, since is an angle,we are lookingforExample 3
is a rectanglewith length and width.is a rectanglewith length and width.A.Are correspondinganglesin the rectanglescongruent?Yes. Sinceboth are rectangles,all the anglesin both are congruentright angles.B.Are the lengthsof the sidesof the rectanglesproportional?No. The ratio of the lengthsis. The ratio of the widthsis. Therefore,
the lengthsof the sidesare not proportional.
C. Are the rectanglessimilar?No. Correspondinganglesare congruent,but lengthsof correspondingsidesare not proportional.Example 4
Provethat all squaresare similar.Our proofis a “paragraph”proofin bulletform,ratherthan a two-columnproof:Giventwo squares.- All the anglesof both squaresare right angles,so all anglesof both squaresare congruent—andthis
includescorrespondingangles. - Let the lengthof eachside of one squarebe , and the lengthof eachside of the othersquarebe
. Thenthe ratio of the lengthof any side of the first squareto the lengthof any side of the secondsquare
is. So the lengthsof the sidesare proportional. - The squaressatisfythe definitionof similarpolygons:congruentanglesand proportionalside lengths-
so they are similar
ScaleFactors
If two polygonsare similar, we knowthat the lengthsof correspondingsidesare proportional.If is the
lengthof a side in one polygon,and is the lengthof the correspondingside in the otherpolygon,then
the ratio is calledthescalefactorrelatingthe first polygonto the second.Anotherway to say this is: