Associated Topics || Dr. Math Home || Search Dr. Math

### Congruent and Similar Triangle Theorems

```
Date: 05/14/98 at 13:07:12
From: Amanda Crane

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
Angle-Side-Angle (ASA) and Side-Side-Side (SSS), why doesn't Angle-
Angle-Angle (AAA) work?

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.

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/
```
Associated Topics:
High School Geometry
High School Triangles and Other Polygons

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search