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Topic: Proving the base angles theorem w/out constructing anything
Replies: 1   Last Post: Nov 27, 2005 7:32 AM

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Walter Whiteley

Posts: 418
Registered: 12/3/04
Re: Proving the base angles theorem w/out constructing anything
Posted: Nov 27, 2005 7:32 AM
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As someone who teaches geometry and practices as an applied geometry, I
find it very disturbing to consider introducing Congruence as some form
of abstract 'equivalence' not related to transformations.

There is so much to be learned from using transformations, so many
connections to be made to 'measurement' and to the skills which are
embedded in the k-8 curriculum. The transformational insights are
also key to later use of geometry. I recommend looking at Henderson,
Experiencing Geometry and Plane and Sphere. The unifying theme is the
use of appropriate transformations in both contexts, to prove shared
properties (and to explore the differences). My university students
(often future teachers) find it quite possible to prove the theorem
about base angles in both contexts, several ways. It is not hard
after they have explored why SAS and ASA do guarantee there is a
transformation of one to the other (ie. are sufficient for congruence).

I recall a conversation with someone who developed a Tangram software
to help elementary students explore plane transformations. The only
permitted moves were reflections and translations. The students in
elementary school did not find it too hard to work with that. There
were different levels of interfaces, some of which offered more
scaffolding (e.g. indicating the ghost where the object would be, as
GSP does), In fact, what seems to work well was gradually decreasing
the scaffolding so the student developed internal, visual reasoning

One nice story was a grade 3 student (age 9) who played for an hour and
then observed: I can always get an object from one place to the other
in at most 3 moves (reflections). Wonderful geometric thinking. That
student would be ready to explore congruence of the type we are talking
about, without the concrete experience of congruence as transformation
(which it is).

I am reminded of the article 'Why do we teach so little math in
elementary school'!. In fact, this type of experience is perhaps
harder for a 15 year old who has been cut off from what they learned as
younger children, and is struggling with geometry as this strange game
played by weird people.

Walter Whiteley

On 26-Nov-05, at 8:38 PM, wrote:

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I agree with you that geometry students can grasp the concept of
corresponding parts being congruent. And if you define congruence of
triangles in that
way (corresponding angles and sides congruent), I think they
understand it
fine. It's the transformational definition that bothers me as the
definition (rather than something they take up after alot of
experience with

I agree with you also that proving that an angle bisector of a triangle
intersects the 3rd side is very difficult. If memory serves, Moise
offers a
proof of this proposition, which he calls the "Crossbar Theorem"
(Actually the
crossbar theorem is more general, but the angle bisector situation
would be a
special case in which the theorem applies). This is a theorem far
beyond the
scope of a high school course in geometry. This is one reason, among
several, why the "traditional" proof (introducing the bisector of the
vertex angle)
is deficient.(This is the reason given by Moise, in his famous
textbook, for
rejecting this proof.) In one or both of her geometry books (Modern
Geometry: Structure & Method, and Modern School Mathematics:
Geometry), Prof.
Dolciani actually confronts this deficiency, and tells students that
she is
postulating that the angle bisector will intersect the 3rd side. To
me, this is a
superior approach than rejecting this proof entirely in favor of the
where triangle ABC is congruent to triangle CBA, which I find just too
ethereal. Sorry, but I just do.

OK!!! Have dug out the reference...Modern Geometry, Structure & Method
(Houghton Mifflin 1965), Jurgensen, Donnelly, & Dolciani, page 208.
"In the proof
of Theorem 31 (base angles of isosceles triangle congruent), and again
the proof of Theorem 32 (the converse), the statement that the
bisector of
angle C intersects segment AB could be supported by this postulate:
The bisector
of an angle of a triangle intersects the opposite side. The postulate
used in this section without further mention."

Now, right on point!! In her Teacher's Manual, Prof. Dolciani states:
question as to whether or not the bisector of an angle of a triangle
intersects the opposite side (segment) is not interesting or meaningful
to most
high-school students. The problem can be avoided by using the method
shown on p.
226 [ie, triangle ABC congruent to triangle CBA] . . . That method
elusive to many students, however. The authors prefer to present
proofs that
are meaningful to typical students, mentioning -- but not emphasizing
-- the
assumption that is being made." I think the authors are exactly right
but, of course, you are free to disagree.

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