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### SSA Theorem: Valid or Invalid?

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Date: 12/19/2001 at 20:48:14
From: Chip Collins

You said that the SSA Theorem cannot be used to prove congruence. My
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```
Date: 12/19/2001 at 22:57:37
From: Doctor Peterson

Hi, Chip.

The problem is that what most people would expect an SSA theorem to
say (that if two sides and a non-included angle in one triangle are
congruent to the corresponding parts of another triangle, then the
triangles are congruent) is false.

Our archives, which you have apparently seen, give counterexamples to
show that these conditions can be true for non-congruent triangles.
Here are two:

Angle-Side-Side Does Not Work
http://mathforum.org/library/drmath/view/54663.html

SSA and Non-congruent Triangles
http://mathforum.org/library/drmath/view/55036.html

But some of our discussions of this issue suggest ways to "salvage"
the SSA idea and make a theorem out of it by adding extra conditions:

AAA, ASS, SSA Theorems
http://mathforum.org/library/drmath/view/54659.html

SSA Proof
http://mathforum.org/library/drmath/view/55278.html

The last of these states a valid SSA theorem explicitly, and also
points out that the Hypotenuse-Leg theorem is a special case of this.
If your text has stated and proved such a modified SSA theorem, or if
you have done so yourself, then you can use that theorem. But if all
you have is a naive version that has not been proved, then it is not a
theorem and you can't use it.

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

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