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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,372

Yesterday, our prof. asked me to name the criterion for triangle congruences which I did. Then he asked me to provide the proof for them which I also did (except SAS of coufrse). He then scolded me and replied that there is a proof for SAS. Until then I was pretty sure that it was an axiom which we use to prove the other criterias.

*Last edited by Shivamcoder3013 (2013-03-25 01:59:05)*

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**ShivamS****Member**- Registered: 2011-02-07
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So I was just wondering, is there a proof to SAS?

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**bob bundy****Moderator**- Registered: 2010-06-20
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hi Shivamcoder3013

What are your starting points for a proof? Perhaps you could post one that you have done.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**ShivamS****Member**- Registered: 2011-02-07
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Well isn't SAS just an axiom? How do you prove that: If 2 sides and angles in a triangle are congruent to two sides and angles of another triangle, the triangles are congruent. Like take 2 triangles: ABC, DEF. If AB=DE, AC=DF and angle A = angle D, the triangles ABC and DEF are congruent.

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**bob bundy****Moderator**- Registered: 2010-06-20
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My difficulty is I have never seen any proved so I'm not sure where to start. That's why I need to see one that you have already done. Then I can see what you are after.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
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I have found this: http://www.proofwiki.org/wiki/Triangle_ … ity#Part_1.

Seeing how it has been two years since I have done axiomatic geometry, I do not remember what can and what cannot be used in the proof...

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bob bundy****Moderator**- Registered: 2010-06-20
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My difficulty is I have never seen any proved so I'm not sure where to start. That's why I need to see one that you have already done. Then I can see what you are after.

Let me be specific. The diagram below has two triangles.

Corresponding sides are equal as shown. So what can you prove from this?

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**ShivamS****Member**- Registered: 2011-02-07
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For example, to prove Side-Side-Side, using Side-Angle-Side, we just make two triangles such that they make a kite (being diagonal). That proves that the triangles are congruent.

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**bob bundy****Moderator**- Registered: 2010-06-20
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thanks Stefy

Followed the link to

http://www.proofwiki.org/wiki/Triangle_ … e_Equality

Is that enough?

Bob

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**ShivamS****Member**- Registered: 2011-02-07
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Oh okay then. So because the sides and angles coincide (ABC and DEF) making them congruent, that would be a valid proof for Side-Angle-Side? Anyways, thank-you both.

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**bob bundy****Moderator**- Registered: 2010-06-20
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Shivamcoder3013 wrote:

.... they make a kite................. That proves that the triangles are congruent.

Why does it prove that?

The kite has the right properties but are you sure that those properties don't depend on congruency (circular argument).

Bob

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**ShivamS****Member**- Registered: 2011-02-07
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Well if we have a perpendicular bisector as the diagonal, then by SAS we can see that they are congruent.

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**bob bundy****Moderator**- Registered: 2010-06-20
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Prove 'perpendicular'.

Sorry, I'm not being awkward just to be nasty. I really have never seen any proofs for this sort of thing. I think congruency appears very early in Euclid (trying to check this now) and properties of kites much later. So I'm worried that this proof leads to a circular argument. I'll try to find out.

Bob

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**ShivamS****Member**- Registered: 2011-02-07
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Check this out.

http://www.math.washington.edu/~king/coursedir/m444a03/notes/congruence%20html/tri-congruence-summ.html

That includes proof for the congruence criterias.

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**bob bundy****Moderator**- Registered: 2010-06-20
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Thanks. And that uses SAS (axiomatic) to prove the others. Hhhhmm!

Will this page do?

http://www.proofwiki.org/wiki/Triangle_ … e_Equality

Bob

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**ShivamS****Member**- Registered: 2011-02-07
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Yes, so it is congruent because it coincides. Thank-you both.

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**bob bundy****Moderator**- Registered: 2010-06-20
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You're welcome. I have learnt something too.

Bob

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**ShivamS****Member**- Registered: 2011-02-07
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Glad to see that we both were clarified.

*Last edited by Shivamcoder3013 (2013-03-25 03:43:28)*

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