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ShivamS

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Registered: 2011-02-07

Posts: 3,648

Side-Angle-Side

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)

ShivamS

Member

Registered: 2011-02-07

Posts: 3,648

Re: Side-Angle-Side

So I was just wondering, is there a proof to SAS?

Bob

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Registered: 2010-06-20

Posts: 10,646

Re: Side-Angle-Side

hi Shivamcoder3013

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

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

ShivamS

Member

Registered: 2011-02-07

Posts: 3,648

Re: Side-Angle-Side

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.

Bob

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Registered: 2010-06-20

Posts: 10,646

Re: Side-Angle-Side

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

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Side-Angle-Side

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...

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“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
The knowledge of some things as a function of age is a delta function.

Bob

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Registered: 2010-06-20

Posts: 10,646

Re: Side-Angle-Side

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

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

ShivamS

Member

Registered: 2011-02-07

Posts: 3,648

Re: Side-Angle-Side

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.

Bob

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Registered: 2010-06-20

Posts: 10,646

Re: Side-Angle-Side

thanks Stefy

Followed the link to

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

Is that enough?

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

ShivamS

Member

Registered: 2011-02-07

Posts: 3,648

Re: Side-Angle-Side

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.

Bob

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Registered: 2010-06-20

Posts: 10,646

Re: Side-Angle-Side

.... 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

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

ShivamS

Member

Registered: 2011-02-07

Posts: 3,648

Re: Side-Angle-Side

Well if we have a perpendicular bisector as the diagonal, then by SAS we can see that they are congruent.

Bob

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Registered: 2010-06-20

Posts: 10,646

Re: Side-Angle-Side

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

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

ShivamS

Member

Registered: 2011-02-07

Posts: 3,648

Re: Side-Angle-Side

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.

Bob

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Registered: 2010-06-20

Posts: 10,646

Re: Side-Angle-Side

Thanks.  And that uses SAS (axiomatic) to prove the others.  Hhhhmm!

Will this page do?

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

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

ShivamS

Member

Registered: 2011-02-07

Posts: 3,648

Re: Side-Angle-Side

Yes, so it is congruent because it coincides. Thank-you both.

Bob

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Registered: 2010-06-20

Posts: 10,646

Re: Side-Angle-Side

You're welcome.  I have learnt something too. 

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

ShivamS

Member

Registered: 2011-02-07

Posts: 3,648

Re: Side-Angle-Side

  Glad to see that we both were clarified.

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