Exercise 54 Page 476

A regular pentagon is a pentagon where all its sides and angles are congruent.

As we can see, in the two regular pentagons above all of their corresponding angles are congruent and their corresponding sides are proportional. In consequence, the two regular pentagons are similar. However, will any two regular polygons with the same number of sides be similar? Let's consider. The formula to find the sum of the measures of the interior angles of a $n\text{-}$sided polygon is the one written below. Since all the angles of a regular polygon have the same measure, to find it we divide the formula above by the number of sides $n.$ From the formula above, we conclude that the corresponding angles of two regular $n$-sided polygons are congruent. If ${\ell }_1$ is the length of each side of the first polygon and ${\ell }_2$ is the length of each side of the second polygon, we can write the following proportions. We can conclude that the corresponding sides between two regular $n\text{-}$sided polygons are proportional. Thus, they are similar. This implies that any two polygons with the same number of sides are similar.