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Today we were in our Maths Class. The teacher asked us what is congruence. Most of the people of our class could not answer but I was able to give the definition of it. Now he started asking us what is the total number of congruence criteria in triangles.

This is what I said:

1. Side-Side-Side

2. Side-Angle-Side

3. Angle-Angle-Side

4. Right Angle- Hypotenuse-Side

Then he said:

You are missing a basic point

I said:

Is it Angle-Side-Angle

He said:

Yes

I said:

But that's only a special case of Angle-Angle-Side

He said:

The basic criteria is Angle-Side-Angle.

However, its application is Angle-Angle-Side

*I found it difficult to continue arguing*

My question is whether he is correct.

If he is what does he mean?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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

I'm just considering this. Give me 5 mins

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

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,015

In school we learned it without the Right Angle- Hypotenuse-Side and with Angle-Side-Angle.

EDIT: Btw,SAS and SSA are two different criteria.I don't think either is a special case of the other one.

*Last edited by anonimnystefy (2012-06-20 21:59:00)*

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.

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

**angle-side-angle** and **angle-angle-side** amount to the same thing as you can always work out the third angle anyway.

I'm assuming when you say **side-angle-side** you are putting the angle between the two known sides.

That distinquishes it from **side-side-angle** where the angle isn't between the known sides.

That case needs to be considered separately because SAS has a unique construction but SSA may not, depending on the measurements.

I'll work on a diagram to illustrate 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

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Look,

I am talking about AAS and ASA

I want to know is he correct when he says:

AAS is the application of ASA

*Last edited by Agnishom (2012-06-20 22:08:03)*

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,015

AAS and ASA are the same thing. But you are missing SSA in your list. SSA is the generalization of RHS.

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.

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Does SSA exist?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,015

Of course it does! As I said,it is a generalization of the Right Angle-Hypotenuse-Side rule.

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.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,579

You need to look at post #11 as well to make sense of this.

Say AB = 8.31

AC = 3.81

ABC = 23.3 degrees

There are two possible places for C (second labelled E)

Therefore having SS and the non included angle may lead to a congruence or it may not.

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

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Thanks

Anyway, I want to know why did he tell me that I am missing out "A-S-A"

I think he shouldn't have told it bcause I had already mentioned A-A-S

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,579

Yes, I agree with you there. When you know two angles, you also know the third, so AAS and ASA are the same thing.

SSA case: Now I've got this clear in my head. Hope I can make it clear for you.

SSA definitely exists. Diagram below shows an example.

Because AB is shorter than AC the point C is uniquely constructed ( draw AB, then the angle, then the curve to pinpoint C).

But the earlier example in post #9 shows that you don't always get a pair of congruent triangles with the same SSA, so you have to be careful.

Do you know the SINE RULE because it all becomes clear once you use that.

Bob

ps. What has construction got to do with congruence?

Just this: If I give two people some measurements for a triangle and ask them to draw it, will they land up with congruent triangles? A test for congruency cases is "When you try to construct a triangle with these measurements, is the solution unique?" This assumes that rotations and reflections are considered to be the same and that it doesn't matter what you use as letter labels for the triangle.

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

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Do you mean this?:

For three sides a, b, c and angles alpha,beta and gama

sine(a)/alpha = sine(b)/beta = sine(c)/gama

I have only seen it but I don' have any Idea how to use it

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,579

Yes, that's it.

I have edited post #11 and 9 by the way.

If you know the right information about a triangle you can substitute in the values into the sine rule and so work out all the missing sides and angles.

http://www.mathsisfun.com/algebra/trig-sine-law.html

There is a direct link between some of the congruency cases and the sine rule.

eg. If you have AAS you can use the sine rule to work out the unknown sides. Thus a pair of triangles with the same AAS will also be SSS once you've worked out the missing sides.

But when you try the same trick with SSA you will have to do an inv sine to get a second angle. Because there is an obtuse angle that has the same sine as every acute angle {sin (180 - x) = sinx} you end up with two triangles with the same SSA . This is what happens in post #9

The cases that aren't covered by the sine rule are covered by the cosine rule

So rather neatly, congruence, constructability and advanced trig. all lead to the same results.

Bob

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

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Ya, but what actually is happening is that I can't tell my Maths Teacher that because we don't have yet learnt that law in school

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,579

hi Agnishom

Well you'll have to decide what to do. If it were me, I wouldn't mind that one of my students had done some independent research.

I'd probably get him in front of the class to explain it.

Bob

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

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Argument against SSA

See The Image

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,015

Hi Agnishom

I have just realized that the SSA rule is invalid, but not for the reason provided in your post. In your post one triangle has the angle opposing the 5cm side and in the other it is opposing the 4cm side.

But there are in fact two triangles where the angles stay in he same places with respect to the sides.

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.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,579

I thought I covered this in posts #9 and #11.

Provided you know which cases are valid then SSA works for those cases.

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: Harlan's World
- Registered: 2011-05-23
- Posts: 16,015

I have just remembered how the SSA rule truly goes."Two triangles are congruent if they have two pairs of equal sides and equal angles opposite one of them and all other angles are of the same type (meaning either acute, right or obtuse).

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.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,579

That sounds right to me.

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: Harlan's World
- Registered: 2011-05-23
- Posts: 16,015

Yes, I remembered it now. That is how we learned it.

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.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,579

Arrhhh............ the ideal student who remembers what has been taught.

Me, I always have to work everything out from first principles as I'm hopeless at remembering anything. I cannot even remember a phone number in the time from seeing it written down to the time when my fingers are on the buttons!

That's why it takes me longer to sort out someone's "help me" problem.

Bob

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

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Hi

Do you derive phone numbers from the first principal?

*Last edited by Agnishom (2012-07-10 01:00:53)*

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,015

bob bundy wrote:

Arrhhh............ the ideal student who remembers what has been taught.

Me, I always have to work everything out from first principles as I'm hopeless at remembering anything. I cannot even remember a phone number in the time from seeing it written down to the time when my fingers are on the buttons!

That's why it takes me longer to sort out someone's "help me" problem.

Bob

I don't agree with the first part. I mostly remember things I learned in mthematics, but since I self-taught and bobbym-taught myself most of the stuff, it really isn't hard for me to remember things from class.

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.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,579

Do you derive phone numbers from the first principal?

You may well ask! If I could, then I would not have such trouble punching them in. For this reason the only one I can manage easily is

01 1235 81321

The person who has this number is getting fed up with my calls.

Bob

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

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