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### SSS, ASA, SAS Proofs

```
Date: 01/21/2002 at 18:08:09
From: Sarah
Subject: Proofs

Dear Dr. Math,

Hello! I know you've answered a lot of questions about proofs, but
I'm still having trouble with them. Can you please explain them in
fuller detail? My problem is that I understand the ideas, but I'm
not sure when and where to use them. I especially need help with SSS,
ASA, SAS, etc. I get confused with what they mean and how to tell
which one is which.

Thank you,
Sarah
```

```
Date: 01/21/2002 at 23:37:04
From: Doctor Peterson
Subject: Re: Proofs

Hi, Sarah.

I assume you have looked through the Dr. Math FAQ on Proofs:

http://mathforum.org/dr.math/faq/faq.proof.html

and have read problems like these:

Congruent Triangles in a Rectangle
http://mathforum.org/dr.math/problems/lacey.11.11.99.html

Writing a Proof
http://mathforum.org/dr.math/problems/tony.05.16.01.html

Triangle Proof
http://mathforum.org/dr.math/problems/ashley.11.19.01.html

Congruent Parts Congruent Triangles Congruent (CPCTC)
http://mathforum.org/dr.math/problems/brittany.11.28.01.html

Here is a quick summary of what each method of proof means:

SSS                SAS                ASA

+                  +                  +
/  \               /  \               /  \
c/     \a          c/     \            /     \
/        \         /A       \         /A     C \
+-----------+      +-----------+      +-----------+
b                  b                  b

+                  +                  +
/  \               /  \               /  \
c/     \a          c/     \            /     \
/        \         /A       \         /A     C \
+-----------+      +-----------+      +-----------+
b                  b                  b

All three sides    Two sides and      Two angles and
of one triangle    the included       the included
agree with all     angle of one       side of one
three sides of     triangle agree     triangle agree
the other.         with the other.    with the other.

One way to think of it is to say that, if you were given one of these
three sets of three numbers, it would be enough to be able to draw
that triangle correctly. For example, If I told you angle A and sides
b and c, you could draw line b, make a ray A degrees from it, and
measure c units along that ray. Then you would have all three vertices
of the triangle, and would know that you had exactly the shape I had
in mind. If I just gave you, say, the three angles, you would have to
choose a length for the sides, so you would not be sure. That's why
there's no AAA congruence theorem.

The answers I recommended show how to use these in practice. If you
find that you have the appropriate set of congruent pairs of parts
between two triangles, then you can use that theorem to say they must
be congruent.

If you need more direct help, please write back and show me a sample
problem that gave you trouble, so I can see where you get confused and
suggest specific ideas. You can also find more examples at this
ThinkQuest site:

Geometry: Congruent Triangles - Math for Morons Like Us
http://library.thinkquest.org/20991/geo/ctri.html

- Doctors Peterson and Sarah, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Triangles and Other Polygons

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