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SSA Theorem: Valid or Invalid?

Date: 12/19/2001 at 20:48:14
From: Chip Collins
Subject: An earlier answer

You said that the SSA Theorem cannot be used to prove congruence. My 
teacher said you CAN use it. Please help straighten this problem out.

Date: 12/19/2001 at 22:57:37
From: Doctor Peterson
Subject: Re: An earlier answer

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   

   SSA and Non-congruent Triangles   

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   

   SSA Proof   

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   
Associated Topics:
High School Geometry
High School Triangles and Other Polygons
Middle School Geometry
Middle School Triangles and Other Polygons

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