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Defining Congruent and Equal

Date: 07/07/2005 at 20:44:23
From: Mark
Subject: congruency vs. equal


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 
Associated Topics:
High School Geometry

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