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### Inclusive and Exclusive Definitions

```
Date: 04/05/2001 at 17:46:14
From: hawesrd
Subject: Squares and Rectangles

Are squares rectangles? Are rectangles squares?

Thanks.
```

```
Date: 04/06/2001 at 06:08:25
From: Doctor Floor
Subject: Re: Squares and Rectangles

Hi,

Thanks for writing.

Rectangles are not always squares, because squares need four sides of
equal lengths. The question whether the other way around is true
depends on the type of definition you are using. There are often

In general there are two types of definition for geometric shapes:

INCLUSIVE DEFINITIONS: In the case of rectangles and squares this
means that a square is seen as a special case of a rectangle:

* A rectangle is a quadrilateral with four right angles.

* A square is a quadrilateral with four right angles and four equal
sides.

EXCLUSIVE DEFINITIONS: In the case of rectangles and squares this
means that a square is NOT considered a rectangle:

* A rectangle is a quadrilateral with four right angles but not four
equal sides.

* A square is a quadrilateral with four right angles and four equal
sides.

I prefer inclusive definitions, because they include the basic
mathematical concept of 'generalization': one item (rectangle) is more
general than the other item (square). But others say that exclusive
definitions are very useful for special cases.

newsgroup at:

Trapezoid definition
http://mathforum.org/kb/message.jspa?messageID=1080683

If you have more questions, just write back.

Best regards,
- Doctor Floor, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 04/03/2001 at 20:40:02
From: Michelle
Subject: Isosceles triangles?

I am in the 10th grade and I need to know whether an isosceles
triangle has exactly or only two sides congruent. Is an equilateral
triangle considered isosceles as well? I've looked at over 30 sites
but I never get a full answer on whether there are exactly, only, or
at least 2 sides congruent - or if I do, there is no explanation or
reasoning to prove the statement.

Thanks so much,
Michelle
```

```
Date: 04/04/2001 at 01:39:07
From: Doctor Schwa
Subject: Re: Isosceles triangles?

Hi Michelle,

The question of whether an isosceles triangle has to have at least two
sides congruent versus exactly two sides congruent isn't something you
can prove: it's a question of definition. What does the term
"isosceles triangle" mean?

Definitions are something that we can choose arbitrarily, and books can
have different definitions.

However, some definitions are more useful than others, more
convenient, easier to use... and in this case, one of those two
definition choices is much better than the other. The inclusive
definition is almost always better, as it is in this case. The
inclusive definition, where isosceles triangles include equilateral
triangles, is much more convenient. So isosceles triangles should be
defined as triangles that have *at least* two sides congruent.

Why is the inclusive definition more convenient? Well, consider a
theorem like:

If angle A = angle B in triangle ABC, then the triangle is
isosceles.

If you had the non-inclusive definition, you'd always have to be
saying things like:

If angle A = angle B but is not equal to angle C, then ...

or

If ... then the triangle is isosceles or equilateral.

For the same reason, the definition of rectangle should include
squares, the definition of parallelogram should include rectangles,
and so on. The inclusive definition is almost always better.

- Doctor Schwa, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Definitions
High School Geometry
High School Triangles and Other Polygons
Middle School Definitions
Middle School Geometry
Middle School Triangles and Other Polygons

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