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 disputes about definitions. 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. You can read a discussion about this in the geometry pre-college 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. Could you please help me? 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/
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