Triangle Congruence

Date: 05/13/2003 at 20:29:48
From: Candice
Subject: Congruent Triangles

I don't understand how to tell if two triangles are congruent. My 
teacher told me to use SSS (Side Side Side) SAS (Side Angle Side) and 
ASA (Angle Side Angle) to figure out if they are congruent, but I 
don't understand.

I think it has something to do with seeing if all the sides and angles 
match up.

Date: 05/13/2003 at 23:04:52
From: Doctor Peterson
Subject: Re: Congruent Triangles

Hi, Candice.

Suppose you wanted to tell if a twenty-dollar bill is genuine. You 
would compare it with a real one, and see that everything is the same 
(except for serial numbers and so on; ignore that). Is the number in 
the upper left printed the same way on both? Is the right ear in the 
portrait the same in both? And so on.

To see if two triangles are identical (congruent), you could do the 
same thing: is the lower left corner the same angle in both? How about 
the length of the bottom edge? And so on. Once you had compared every 
single part, you would know they were the same.

But we have several theorems (or postulates) that tell us we don't 
have to check EVERY part; it's enough to check just three parts, if 
we choose the right three. Specifically,

      +              +          SSS tells us that if each side
     /  \           /  \        of one matches the corresponding
    s     s        s     s      side of the other, then the
   /        \     /        \    triangles are congruent.
  +-----s-----+  +-----s-----+

      +              +          ASA tells us that if two angles of
     /  \           /  \        one match the corresponding angles
    /     \        /     \      of the other, and the sides between
   /        \     /        \    those angles match, then the
  A-----s-----A  A-----s-----A  triangles are congruent.


      +              +          SAS tells us that if two sides of
     /  \           /  \        one triangle match the corresponding
    s     \        s     \      sides of the other, and the angles
   /        \     /        \    between them match, then the
  A-----s-----+  A-----s-----+  triangles are congruent.

Now, it is not enough to say that a pair of angles LOOK the same; you 
have to KNOW they are. Typically in geometry proof, you are told (or 
know for other reasons) that certain parts are congruent. For example, 
you might be told in this figure that

      Q-----------R    PQ is the same length as RS, and
     /A \        /     angle PQS has the same measure as RSQ.
   x/     \     /x
   /       A\  /       You know, without having to be told, that any 
  P-----------S        segment is congruent to itself, so QS = QS.

So if we think about triangles PQS and RSQ (with the letters given in 
the order we want to compare them), we can flip the latter around in 
our minds so that they line up with the letters in corresponding spots 
like this:

      Q                  S  
     /A \               /A \
   x/     \y          x/     \y
   /        \         /        \
  P-----------S      R-----------Q

Now we have three pairs of matching parts:

   PQ = RS
   PQS = RSQ
   QS = SQ

And these are in the order SAS, with the angle between the sides. So 
the SAS theorem tells us that the two triangles are congruent. We 
don't have to know anything about the other side and angles; it's 
like saying that if the left ears match, the right ears will match 
too, so we don't need to check them.

Here are some pages where this process is demonstrated:

   Congruence and Triangles
   http://mathforum.org/library/drmath/view/54673.html 

   Congruent Triangles in a Rectangle
   http://mathforum.org/library/drmath/view/54674.html 

   SSS, ASA, SAS Proofs
   http://mathforum.org/library/drmath/view/55423.html 

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/