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!
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