SSA and Non-congruent Triangles

Date: 08/13/98 at 13:23:24
From: Carolyn White
Subject: Congruence of triangles

I'm trying to stay ahead of my teenage daughter, but it's been 35 some 
years since high school geometry. I'm using Barron's "Geometry The Easy 
Way," and things are coming back into focus.

But I can't visualize why you cannot necessarily conclude that two 
triangles are congruent when "two sides and an angle that is not 
included of one triangle are congruent to the corresponding parts of 
the other triangle" (page 97). I know SAS proves congruence, but it 
seems to me that if one set of angles is congruent and two sets of 
sides are congruent, you have enough to establish congruency of the two 
triangles by SSA.

What am I overlooking?  Could you provide an example to illustrate 
non-congruent triangles that meet the criteria?

I'd appreciate some guidance. Thanks!

Date: 08/13/98 at 13:56:20
From: Doctor Barrus
Subject: Re: Congruence of triangles

Hi, Carolyn!

In some SPECIAL cases (which depend on the lengths of the sides and the 
size of the angle involved), SSA is enough to show congruence. However, 
it's not always enough. Consider the following triangles:
        
       . A
      .   .
     .       .
    .............
   B             C

       D
       .
        . .
         .   .
          .......
      E      F

Here:

   side AB is congruent to side DE (S)
   side AC is congruent to side DF (S)
   angle C is congruent to angle F (A)

But the triangles are not congruent, as you can see.

What happens is this: if you draw a vertical line through point A in 
the first triangle, you can sort of "flip" side AB around this line to 
get the second triangle. If we were to lay one triangle on top of the 
other and draw the vertical line, this is how it'd look.

       A,D
       |
      /|\ .
     / | \   .
    /..|..\......
   B   | E      C,F

You can see that side DE is just side AB flipped around the line. So 
we haven't changed the length of the side, and the other side AC (or 
DF) is unchanged, as is angle C (or F). So these two triangles that 
have the same SSA information, but they're not congruent.

I hope that makes sense. If not, feel free to write back.

Good luck!

- Doctor Barrus, The Math Forum
Check out our web site! http://mathforum.org/dr.math/