Proof of ASA Theorem, instead of it being a postulate
Guy Brandenburg
theorem, instead of it being a postulate. I think he is correct. It's
something I've been trying to prove for some time, without success.
To do it, he assumes 2 things: The SSS postulate (i.e., if 2 triangles
have 3 pairs of corresponding, congruent sides, then the triangles are
congruent) and the AAA postulate (i.e., if 2 triangles have 3 pairs of
corresponding, congruent angles, then the triangles are similar).
His proof went something like this.
Given: triangle ABC and triangle DEF, with angle A congruent to angle D,
angle B congruent to angle E, and side AB congruent to side DE.
Prove: triangle ABC is congruent to triangle DEF.
Since angles A and D are congruent, and angle B is congruent to angle E,
then we already know that angles C and F must be congruent as a
consequence of the triangle-angle-sum theorem. Thus, the 2 triangles are
congruent by the AAA postulate.
Since the triangles are similar, that means there is some factor k by
which all of the sides in one triangle could be multiplied to get the
lengths in the other triangle. Thus, AB = k * DC, BC = k * EF, and AC =
k * DF. But we already know, from the given, that sides AB and DE are
congruent, hence have equal lengths, thus k must be equal to 1 (one).
Thus, AB = DC, BC = EF, and AC = DF, thus all of the pairs of sides are
congruent, thus the two triangles are congruent by SSS.
Not bad, huh?
Guy Brandenburg
(PS - I really don't like Euclid's supposed 'proof' of SAS and of SSS
and of ASA in the first few propositions of Book 1. I tried doing proofs
by reductio ad absurdum, myself, and got nowhere. Most geometry books in
the US assume ASA, SAS, SSS, and ASA, and try --more or less-- to prove
things from them. My goal is to prove more of them and assume fewer things.)
Guy Brandenburg
Guy Brandenburg wrote:
> One of my students - an 8th grader - today proposed a proof of the ASA
> theorem, instead of it being a postulate. I think he is correct. It's
> something I've been trying to prove for some time, without success.
>
> To do it, he assumes 2 things: The SSS postulate (i.e., if 2 triangles
> have 3 pairs of corresponding, congruent sides, then the triangles are
> congruent) and the AAA postulate (i.e., if 2 triangles have 3 pairs of
> corresponding, congruent angles, then the triangles are similar).
>
> His proof went something like this.
> Given: triangle ABC and triangle DEF, with angle A congruent to angle D,
> angle B congruent to angle E, and side AB congruent to side DE.
> Prove: triangle ABC is congruent to triangle DEF.
>
> Since angles A and D are congruent, and angle B is congruent to angle E,
> then we already know that angles C and F must be congruent as a
> consequence of the triangle-angle-sum theorem. Thus, the 2 triangles are
> congruent by the AAA postulate.
^^^^^^^^^^
SIMILAR!!!!
>
> Since the triangles are similar, that means there is some factor k by
> which all of the sides in one triangle could be multiplied to get the
> lengths in the other triangle. Thus, AB = k * DC, BC = k * EF, and AC =
> k * DF. etc ....
BTW, the student's name is Daniel Poore. You may find him joining the
ranks of real professional mathematicians one day. He already, this
year, qualified to take the USAMO, and also placed #13 at the national
Mathcounts competitions.
Guy Brandenburg
Guy Brandenburg
Thanks for the book recommendations.
Which of those 3 books would you recommend for a student, and which one
for a teacher?
Guy (not Gary)
Karl M. Bunday wrote:
> Gary,
>
> Has that bright young man you are teaching mathematics read much of
> Hilbert's axiomatization of Euclidean geometry? I have recently been
> reading, for recreational reading and "teacher training" as a
> homeschooling dad,
>
> Artmann, Euclid: The Creation of Mathematics
>
> Hartshorne, Geometry: Euclid and Beyond
>
> and
>
> Cederberg, A Course in Modern Geometries 2nd ed.
>
> All of those titles go over the issue of how much should be an axiom and
> how much can be turned into theorems in Euclidean geometry. The latter two
> titles, especially, discuss the usual approach to axioms for Euclidean
> geometry taken in United States school mathematics, which was largely
> advocated by the elder Birkhoff.
>
> It's always interesting to get a young mind thinking about axioms of
> geometry. I continually remind my son that such "facts" as that the sum of
> the angles of a triangle will always be 180 degrees assume the parallel
> postulate and that there are forms of geometry in which there are no such
> "facts."
>
> Karl M. Bunday "Christ has set us free." Galatians 5:1
> Learn in Freedom (TM) http://learninfreedom.org/
> kmbunday AT earthlink DOT net (preferred email address)
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------- End of Forwarded Message
Walter Whiteley
{I can say, in case there is a doubt, that I am not a fan of the
elder Birkhoff, or of the axiomatic approach to plane geometry).
Congruence is, at its heart, about transformations.
ASA is a claim that one figure can be transformed into
another, using isometries (translations, reflections, rotations).
In fact, it holds quite nicely in absolute geometry (Euclidean,
spherical or elliptic, hyperbolic).
So the fact that two points are congruent, that two sides
of equal measure are congruent, that two angels of equal
measure are congruent, are all pieces which are carefully
combined to find the larger transformation of the entire figures
(and therefore the equality of other measures). In that understanding,
SSS is less obvious than ASA. AAA is something that does take
different forms for Eucliean (Similarity) and Spherical or hyperbolic
(congruence - the polar of SSS) geometries.
Does the reasoning below give any sense of WHICH transformation
is available to show the congruence?
Walter Whiteley
Wayne Bishop
>One of my students - an 8th grader - today proposed a proof of the ASA
>theorem, instead of it being a postulate. I think he is correct. It's
>something I've been trying to prove for some time, without success.
>
>To do it, he assumes 2 things: The SSS postulate (i.e., if 2 triangles
>have 3 pairs of corresponding, congruent sides, then the triangles are
>congruent) and the AAA postulate (i.e., if 2 triangles have 3 pairs of
>corresponding, congruent angles, then the triangles are similar).
Assuming the rest of neutral (absolute is another name for the first 28 of
Euclid's propositions and other results that do not require some equivalent
of "Euclid's 5th", the parallel postulate), all of the usual triangle
congruence implications are equivalent. That is, assuming any one of them
as an axiom implies the others as theorems. It is no more appropriate to
call them all "postulates" than to speak of the Pythagorean Axiom, which
most of us would be reluctant to do except, perhaps, as an educational
exercise to be "cleaned up" at some later time. Invoking AAA similarity
forces Euclidean geometry (i.e., some form of the 5th) unnecessarily, ASA
has been being proved in neutral geometry for a couple thousand years, in
fact, for a couple thousand years prior to the invention of neutral
geometry, it is half of Proposition 26 of Book 1 to be precise. It is
easier to start with SAS than SSS, however.
>(PS - I really don't like Euclid's supposed 'proof' of SAS
Euclid's proof of SAS really is suspect (superposition is a better word)
but the rest are fine.
>and of SSS
It's fine along with the others.
>My goal is to prove more of them and assume fewer things.)
That's a good goal. Fundamental to the subject, in fact.
Wayne.
- - ------- End of Forwarded Message
- ------- End of Forwarded Message
Wayne Bishop
>It's always interesting to get a young mind thinking about axioms of
>geometry. I continually remind my son that such "facts" as that the sum of
>the angles of a triangle will always be 180 degrees assume the parallel
>postulate and that there are forms of geometry in which there are no such
>"facts."
Such as in our "real" world; i.e., life on a sphere (where AAA is a
*congruency* theorem, there is no "similarity" of triangles).
Larry Deack
>That is, assuming any one of them
> as an axiom implies the others as theorems.
I have a problem with the proofs of parallel lines in "Basic Geometry" by
Brown. We are supposed to tell the difference between a postulate and a
theorem but they look the same to me.
Postulate 7 - If two parallel lines are cut by a transversal, then
corresponding angles are equal.
Postulate 8 - If two lines and a transversal form equal corresponding
angles, then the lines are parallel.
To me these are flip flops of each other so they are not really 2 separate
things but 2 side to the same coin.
Then it goes on the 'prove' the theorems about the other angles like this
one:
Theorem 6 - If two lines and a transversal form equal ALTERNATE INTERIOR
angles, then the lines are parallel.
Why couldn't we flip theorem 6 (and its flip side) with postulate 7/8 ?
Then there is the naming of the angles which is totally stupid since they
create 3 names
corresponding
alternate interior
supplementary same-side interior
They mention some books also use 'alternate exterior' but this whole thing
seems like a lot of confusion to introduce for almost no gain.
The angles form many rather obvious pairings all of them either the same
angle or supplementary. When I think about the relations I tend to see it as
2 of the same thing offset from each other. I can see all the patterns but I
spend all my time trying to learn which term applies to which hand of the
mirror images.
Why _corresponding_? Each of the 4 angles at an intersection 'corresponds'
to another angle at the other intersection. Why create such messy words to
describe a simple symmetry? Even more confusing are the instructions on how
to find the pattern using the F, Z or U shape to find them... What? Is this
for real?
Am I nuts or is this way more confusing than it needs to be? I really don't
get why I have to learn all this junk when I can easily see the symmetrical
patterns formed by these lines.
Thanks for reading this. I'm an older returning student who has to take
some remedial math but it sure is looking a lot different after this many
years. Hope somebody who knows this stuff can help me understand why it
seems like I'm learning a lot of vocabulary that is more confusing than
edifying.
Guy Brandenburg
powerful. Not all-powerful of course, but very powerful.
The young man was not sure whether his approach would work, because he
was explicitly unsure as to whether he could assume AAA similarity in
plane geometry. He is aware of spherical geometry, but has not looked at
hyperbolic geometry -- yet. He was specifically avoiding a
transformational approach, though he has studied reflections, rotations,
translations, glide relfections and dilations. He has on his own proved
that the angles of a triangle (in Euclidean geometry) add up to 180
degrees, if he is allowed to assume a single postulate about parallel lines.
Guy Brandenburg