Is an Equilateral Triangle Also Isosceles?

Date: 02/19/2008 at 16:08:48
From: Karen
Subject: the relationship between equilateral and isosceles triangles

Can equilateral triangles also be classified as isosceles?

Our 6th-grade math book defines an isosceles triangle as "a triangle 
with at least two congruent sides."  The use of "at least" implies 
that an equilateral triangle could also be classified as isosceles, 
since it has "at least" two equal sides.  (Perhaps in a similar 
manner to how a square is also a rectangle?)

As a student, I had always learned that the three types of 
triangles, when classifying by side length (equilateral, isosceles, 
and scalene), were distinct and separate.  An equilateral triangle 
has 3 equal sides, an isosceles has 2, and a scalene has different 
lengths for each side.  Our school's text book has made me question 
this assumption, which is why I went to your site!  I don't want to 
teach this incorrectly!

Which is correct?  Can an equilateral triangle also be thought of as
isosceles?

Date: 02/20/2008 at 10:22:11
From: Doctor Rick
Subject: Re: the relationship between equilateral and isosceles triangles

Hi, Karen.

I definitely prefer the inclusive definition, using "at least".  Yes, 
this is the same thing as saying that a square is a type of 
rectangle; an equilateral triangle is a type of isosceles triangle.

I'm glad your textbook is going in this direction; it is the way 
mathematicians think.  By the time the students take a geometry 
class, in which they learn to prove theorems, I would hope they 
would learn the inclusive definitions, because these are in keeping 
with the way theorems work.  A theorem that holds for any rectangle 
will apply to a square, because a square fits the definition of a 
rectangle; there is no need to write a separate theorem for squares. 
Likewise, a theorem that holds for any isosceles triangle also 
applies to equilateral triangles, because they fit the definition of 
an isosceles triangle.

So why not learn this way of thinking in sixth grade?  That sounds 
good to me!

There are a number of discussions of inclusive versus exclusive 
definitions in our Archive, and as far as I am aware, those 
discussions all come down on the side of inclusive definitions.  For 
example:

  Inclusive and Exclusive Definitions
    http://mathforum.org/library/drmath/view/55295.html 

Please feel free to write back if you have further questions on this.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 02/20/2008 at 10:51:33
From: Karen
Subject: Thank you (the relationship between equilateral and isosceles
triangles)

Your reply was very clear and helpful.  Thanks for the clarification!