Chapter 7.3, Problem 14WE

To determine

To prove: Whether the given two polygons are always, sometimes, or never similar.

Explanation of Solution

Given information: An equilateral triangle and an equiangular triangle.

Proof: In an equilateral triangle, all three sides are equal and each angle is $60°$

Here, $<CAB=60°$ . And $<ABC,<BCA$ are also $60°$

  

And, in an equiangular triangle, all three angles are equal. That means each angle is $60°$

  

Here, $<A\text{'}B\text{'}C\text{'},<B\text{'}C\text{'}A\text{'}$ and $<C\text{'}A\text{'}B\text{'}$ all three angles are $60°$ .

We can see all three angles of equilateral and equiangular triangles are equal.

And, for similar triangles, all three angles must be of the same measurement. So, an equilateral triangle and an equiangular triangle are always similar.