Chapter 7.3, Problem 1WE

To determine

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

Answer to Problem 1WE

Two equilateral triangles are always similar.

Explanation of Solution

Given information: Two polygons are equilateral triangles

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 to equilateral triangles ABC and PQR.

As ABC and PQR are equilateral triangles and sides of equilateral triangles are equal,

  $\begin{array}{l}AB=BC=AC\\ PQ=QR=PR\end{array}$

So,

  $\frac{AB}{PQ}=\frac{BC}{QR}=\frac{AC}{PR}=1$

That is $\frac{AB}{PQ}=\frac{BC}{QR}=\frac{AC}{PR}$

Condition 1 is satisfied.

Now check condition 2.

As each angle of the equilateral triangle is equal to $60°$ ,

  $\begin{array}{l}\angle A=\angle P=60°\\ \angle B=\angle Q=60°\\ \angle C=\angle R=60°\end{array}$

So, condition 2 is also satisfied.

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