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James King
Posts:
64
From:
University of Washington
Registered:
12/3/04
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triangles and labels
Posted:
Oct 14, 2009 11:52 AM
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I would like to add my two cents to this discussion. [I read all the
messages but have lopped off all quotes but the most recent from this
email. Since I get this as a digest, the number of messages quoting
previous messages in the thread that all arrived in one digest message
was an interesting illustration of the quadratic nature of the partial
sums of an arithmetic series. It is really challenging to read
recursive quoting.]
I interpret the statement "triangle ABC is congruent to triangle DEF"
differently from Ted Stanford. I agree that triangle ABC and
triangle BCA are the same triangle as sets or polygons. But when one
states that two triangles are congruent, the congruence relation
imposes a correspondence between vertex angles and sides. Of course
it would be possible to say "triangle ABC is congruent to triangle
FED, with angle A congruent to angle E and angle B congruent to angle
D and angle C congruent to angle F" but geometers find it more
convenient to build this into the notation by saying "triangle ABC is
congruent to triangle EDF". So the order required by congruence is
expressed compactly by the notation (viz ABC -> EDF). Of course one
can also say "triangle BAC is congruent to triangle DEF" with the same
meaning. But if one states that "triangle ABC is congruent to
triangle BAC", this asserts that the triangle has a symmetry (and is
therefore isosceles). The same convention holds for similarity.
But this exchange made me curious about usage in geometry books by
authors of the highest qualification, so I looked into Coxeter's
Introduction to Geometry. He often avoids using this compact notation
by using words instead (not a bad idea), but he does use it. In fact
this theorem about transformations does not make sense without it:
"Any two similar triangles ABC, A'B'C' are related by a unique
similarity ABC -> A'B'C' ..."
Of course there are, as is often the case in elementary geometry, some
subtle points than can lead to confusion in a classroom or in a
textbook (the definition of "angle" is very tricky also). There
really is no practice that avoids all possibility of confusion. But I
think that the convention of using labeling to indicate the
corresponding vertices in a congruence is valuable and sensible.
However, there is nothing to prevent one from using more words and
less compact notation to get ideas across in textbooks and in the
classroom.
Secondly, I was seriously intrigued by the subtlety of Ted's comment
about isosceles triangles not being triangles at all if the base is
built into the definition. So an isosceles triangle is just a
triangle with an additional property: it is known that two sides are
congruent and we can label them AB and AC if we want. But the base BC
is actually not quite determined uniquely; for if the triangle is
equilateral, then we can label another pair of sides as AB and AC. An
algebraic analogy is given by definition of a cyclic group. The group
is known to have a generator, but there may be more than one choice of
generator -- but we usually choose one and give it a label if we want
to prove something.
Of course in practice, if we prove that a triangle is isosceles, we
normally know exactly which sides are congruent, but this does not
obviate Ted's point, as I understand it. In some cases there may be
no ambiguity at all (e.g. a right triangle has only one right angle --
and yes it is a triangle).
Jim King
> ----------------------------------------------------------------------
>
> Date: Tue, 13 Oct 2009 04:14:57 EDT
> From: Jonathan Groves <JGroves@KAPLAN.EDU>
> Subject: Re: Why do we do proofs?
>
> Ted Stanford wrote:
>
>
>> This reminds me of a similar issue. Suppose I have
>> triangle ABC
>> and triangle DEF. Suppose I know that segments AB
>> and DE are congruent,
>> segments BC and EF are congruent, and that segments
>> AC and DF are
>> congruent. Then everyone would agree that triangle
>> ABC is congruent to triangle
>> DEF. But what about the question, "Is triangle BCA
>> congruent to triangle DEF?"
>> High school geometry texts that I have looked at
>> would say no. In stating
>> congruence, you have to match up the letters
>> correctly. (I heard that someone
>> once sued the company that runs the SAT over this
>> issue, but I haven't been
>> able to verify that.)
>>
>> But if we insist that triangle ABC is not the same
>> thing as triangle BCA, then
>> we aren't really considering the set of triangles.
>> We are instead considering the set
>> of labeled triangles. There is an obvious map from
>> the set of labeled triangles
>> to the set of triangles, where we just forget the
>> labeling, but the sets aren't the
>> same.
>>
>> Coming back to the discussion of isosceles triangles.
>> If we say that the base is inherent in the definition
>> of isosceles triangle, then
>> an isosceles triangle really isn't a triangle
>> anymore. It's an element of a set
>> consisting of pairs (T,B), where T is a triangle with
>> at least two sides equal,
>> and B is a side of T such that the other two sides of
>> T are equal.
>>
>> I agree that these issues aren't just "pure
>> mathematical pedantry". It's
>> important for high school students to learn to appeal
>> effectively to definitions in their
>> reasoning. I would prefer that the books took the
>> point of view that ABC
>> is congruent to BCA, since this puts the focus on the
>> definitions of "triangle"
>> and "congruent" rather than on some canned procedure
>> derived from the
>> SSS theorem or its relations. And I would prefer
>> that isosceles triangles
>> were actually triangles, not triangles with extra
>> structure.
>
>
> Ted, I agree with you here. When one insists that triangle ABC and
> triangle BCA are not the same, then a triangle is no longer just a
> three-sided polygon but a three-sided polygon labeled in a certain
> way.
> Likewise, insisting that an equilateral triangle becomes a different
> triangle
> when a different side is chosen as a base, then again a triangle is
> not just
> a three-sided polygon. Names of mathematical objects are important,
> and
> names and labels should be chosen carefully so that confusion is
> minimized
> as much as possible. But a mathematical object is rarely dependent
> on the
> choice of labels for it. For example, whether I call the set
> {1,2,3,4,5} A or B
> or capital sigma, I am still talking about the same set.
>
> I'm not against geometry textbooks insisting that triangles that are
> congruent
> be labeled as precisely as you had described it above. But going so
> far as to
> insist that triangle ABC and triangle BCA are different triangles is
> going too
> far, especially since the order of the vertices A,B,C doesn't tell
> us anything
> different about the triangle in terms of side lengths and angle
> measures.
> However, such rules for labeling triangles, especially in the cases
> of discussing
> congruent or similar triangles, may cause more confusion or may
> cause students to
> think that triangles ABC and BCA are not the same triangle. If so,
> then it may be
> better not to insist on being that picky. I don't know what such
> pickiness does
> to students, so I can't say for sure if it causes more harm than
> good or not.
>
> On the other hand, ray AB and ray BA are different rays (if the
> convention that the
> first letter refers to the vertex of the ray and the second letter
> indicates a point
> on the ray that is not the vertex) because the rays have different
> endpoints.
>
>
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