# Similarities among polygons

Given two polygons $P$ and $Q$ like those in the following diagram

we say that $P$ and $Q$ are similar if the homologous angles and the sides are equal. Or, if the following equalities are satisfied: $$\widehat{A}=\widehat{A'}, \ \widehat{B}=\widehat{B'}, \ \widehat{C}=\widehat{C'}, \ \widehat{D}=\widehat{D'}, \ \widehat{E}=\widehat{E'}, \ \widehat{F}=\widehat{F'}$$ $$\dfrac{a}{a'}=\dfrac{b}{b'}=\dfrac{c}{c'}=\dfrac{d}{d'}=\dfrac{e}{e'}=\dfrac{f}{f'}$$

where $\widehat{A}$ is the angle that is on apex $A$ and $a=\overline{AB}$ is the length of the edge $AB$.

Consider the rectangle $ABCD$ and a square $A'B'C'D'$. Are both similar polygons?

The answer is NO, because even if the angles of each apex are equal, the reasons of the sides are different since the square has all the equal sides while the rectangle has the sides equal two by two.

This leads us to say that a square will always be similar to another square. More generally, a regular polygon will always be similar to the same regular polygon since the angles will be equal and the sides will be proportional.