Defining Congruent and EqualDate: 07/07/2005 at 20:44:23 From: Mark Subject: congruency vs. equal Hi, What is the difference between being congruent and being equal? I can't seem to get a definitive answer. Date: 07/07/2005 at 22:35:48 From: Doctor Peterson Subject: Re: congruency vs. equal Hi, Mark. Here is an answer I gave to a similar question earlier this year: ===================================================================== >How do congruence and equality relate to each other. How are they >different, yet the same? > >I'm not really sure why there is a big difference according to my >teacher. So could you please help define it more. I think the main problem is that "equal" is a little vague, so we want a more precise word to say that two figures are identical (except for orientation and location). You might say, instead, that "equal" is used for numbers, and when used of geometrical figures it is not clearly defined. Euclid used the word "equal" of two figures that had the same AREA. (He had no word for "congruent"!) We no longer use "equal" as he did, but we do reserve it generally for equal numbers (such as areas or lengths), and use the "new" word "congruent" to specifically say that the figures themselves are "the same" in a carefully defined sense. So it's not really that "equal" is wrong, as that we have defined a precise word for what you want to say, and you should use that word. In math it's essential that we define words carefully and then use those words, so that it can be clear what we mean. When we talk about a line segment, it is easier to use "equal" more or less as Euclid would, to mean that their lengths are equal; so there "equal" and "congruent" are sometimes taken to be equivalent. But properly, you should say that if all the sides of a triangle are CONGRUENT to those of another, then the whole triangles are congruent. The word "equal" never has to come up here, until you say "two segments are congruent when their lengths are equal". ===================================================================== Here is a fuller answer I gave several years ago: ===================================================================== >Why must we say that sides or angles of triangles are congruent >rather than just equal? > >My teacher doesn't really know. I've wondered why such a big point is made of this, myself. There's really no ambiguity in just saying two segments are equal if they have the same length, and it's a perfectly normal and understandable shorthand to say that the sides of the triangles are equal rather than "of equal length". So why force people to say "congruent"? I think in part this is just a pedagogical issue: teachers want to get students used to using more formal terms, so the word "congruent" is introduced as soon as it can be used (for segments), even though we could get along very well with "equal". On the other hand, mathematicians do like to be consistent in the use of words. "Equal" would not be as suitable for any figure other than a segment or angle, because it can be taken different ways; it is too broad a term. Euclid uses "same" to mean that there is actually only one figure, not two that are alike; "equal" to mean that two figures have the same size (length for segments, area for plane figures); but does not, as far as I know, use any word for "congruent" as we know it, which is yet a third concept. For example, here are three of his propositions from Book I: Proposition 4. If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides. [If he could say "congruent" he would have!] Proposition 35. Parallelograms which are on the same base and in the same parallels equal one another. [There is only one base segment, shared by the parallelograms.] Proposition 36. Parallelograms which are on equal bases and in the same parallels equal one another. [Here the parallelograms do not share one base, but have congruent bases -- but the bases are segments of the same pair of parallel lines!] Since "equal" when applied to geometrical figures could be taken in any of these three senses (same, equal size, congruent), it is not clear enough for general use, and we define "congruent" to mean precisely that two shapes can be superimposed on one another (or more informally, are "the same shape and the same size"). Having defined that, we want to use it consistently, even though we could very well go back to Euclid's usage and call congruent segments "equal". We no longer use "equal" in that sense, but have replaced it with more precise statements that the length, or "measure", of two entities are equal. We don't use the word "equal" of shapes, but only of numbers. Therefore, when we are using technical terms, we must say that two segments either are congruent, or have equal lengths, not that they are equal. I personally think some texts get too pedantic about this, using very awkward phrases that can only make students dislike geometry. I would be happy to allow informal shorthand everywhere but in formal proofs. That's why I suggest that this is largely a matter of trying to develop a habit of precise language in students, so they will see the distinctions clearly; apart from that we could probably relax the rules where segments are concerned. ===================================================================== If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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