Transformations and SymmetrySimilar Polygons

The two quadrilaterals on the right are similar. Our first important observation is that in similar polygons, all the matching pairs of angles are congruent. This means that

∡ABC ≅ ∡A'B'C' ∡BCD ≅ ∡B'C'D' ∡CDE ≅ ∡C'D'E' ∡DEA ≅ ∡D'E'A'

The second important fact is that in similar polygons, all sides are scaled proportionally by the scale factor of the corresponding dilation. If the scale factor is ${k}, then

AB× ${k} =A’B’BC× ${k} =B’C’ CD× ${k} =C’D’DE× ${k} =D’E’

We can instead rearrange these equations and eliminate the scale factor entirely:

ABA’B’=BCB’C’=ABA’B’=ABA’B’

We can use this to solve real life problems that involve similar polygons – for example finding the length of missing sides, if we know some of the other sides. In the following section you will see a few examples.