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

, 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.