Congruent and Similar Triangle Theorems

Date: 05/14/98 at 13:07:12
From: Amanda Crane
Subject: Request 11 grade levels

Dear Dr. Math,

I am currently in a statistics class, but my true math love will 
always be geometry. So this is the subject I will ask a question 
about. When there are so many theorems about triangle congruency, like 
Angle-Side-Angle (ASA) and Side-Side-Side (SSS), why doesn't Angle-
Angle-Angle (AAA) work?

I will earnestly be awaiting your reply.

Amanda Crane

Date: 05/16/98 at 09:25:15
From: Doctor Bob
Subject: Re: Request 11 grade levels

Hello Amanda,

Geometry is fascinating! I hope that you will keep studying it. There 
are some amazing things to be learned.

Now, about your question. There _is_ an Angle-Angle-Angle theorem. It 
just has a different conclusion than those other theorems. Remember 
that congruent triangles are ones in which all pairs of corresponding 
sides and angles have the same measurements. That means you can place 
one triangle on top of any congruent triangle so that all the parts 
coincide.

The Angle-Side-Angle and Side-Side-Side theorems conclude with: 
"...then the two triangles are congruent." The Angle-Angle-Angle 
theorem I am talking about concludes with "... then the two triangles 
are similar." Triangles are similar if they have the same shape, but 
not necessarily the same size. That is, they might have congruent 
angles (as paired up), but the paired sides might not be congruent.

To see this, take a small square cake and cut it diagonally from one 
corner to the opposite corner. Then throw away one half and look at 
the triangle remaining. (Don't do this with real cakes, your mother 
won't like it!) Now take a much larger square cake and cut it the same 
way, discard half, and look at that triangle. Those two remaining 
triangle cakes have the same shape but not the same size. They are 
"similar" but not congruent because the pairs of sides are not 
congruent.

It is interesting that there is a Side-Angle-Side (SAS) theorem that 
concludes with "...then the two triangles are congruent" but there is 
not such a simple Side-Side-Angle (SSA) theorem, the case when the 
matching angles are not between the matching sides. See if you can 
draw a picture to show why.

There are other kinds of geometry than the one you are studying. If 
you draw your figures on the surface of a sphere, such as a 
basketball, then there is an Angle-Angle-Angle theorem which gives 
congruent triangles. You may have to think about that for a while! 
Enjoy!

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