Chapter 7.3, Problem 5WE

To determine

To check: Whether two squares are always, sometimes or never similar.

Answer to Problem 5WE

Two squares are always similar.

Explanation of Solution

Given information: Two polygons are squares

Two polygons are said to be similar if the following conditions are satisfied:

1. Corresponding sides of the polygons are proportional.
2. Corresponding angles of the polygons are equal.

Consider squares ABCD and PQRS.

As ABCD and PQRS are squares and sides of squares are equal,

  $\begin{array}{l}AB=BC=CD=AD\\ PQ=QR=RS=PS\end{array}$

So,

  $\frac{AB}{PQ}=\frac{BC}{QR}=\frac{CD}{RS}=\frac{AD}{PS}=1$

That is $\frac{AB}{PQ}=\frac{BC}{QR}=\frac{CD}{RS}=\frac{AD}{PS}$

Condition 1 is satisfied.

Now check condition 2.

As each angle of the square is equal to $90°$ ,

  $\begin{array}{l}\angle A=\angle P=90°\\ \angle B=\angle Q=90°\\ \angle C=\angle R=90°\\ \angle D=\angle S=90°\end{array}$

So, condition 2 is also satisfied.

Hence, it can be concluded that two squares are always similar.