We began the study of similar triangles with the
following questions: "*What are the conditions that tell us two
triangles are similar? Why do these conditions work and not some
other combinations of sides, angles, equality and proportion? How do
the conditions that determine similar triangles differ from the
conditions that tell us triangles are congruent?"*

The students set aside their textbooks, as I requested, and worked together in small groups of 3 or 4 students.

The first thing that I did was to ask them to draw
a triangle on a piece of paper. Then I gave each group a small
magnifying glass, and asked them to look at their triangle through
the magnifying glass and tell me what they saw. They said, of course
"it looks bigger". So I then asked for more information - what,
specifically, did they mean by bigger? Did they mean bigger like a
giraffe is bigger than a snake? Well, maybe a baby giraffe bigger
than a python, if the giraffe is standing and the python is lying on
the ground. But the python certainly has greater horizontal length
when stretched out than the baby giraffe has when standing. Did they
mean taller, or wider, or thicker? Of course they thought that was a
silly question, but I reminded them that the language of mathematics
is a language with great precision, and that "big" has many meanings;
we needed to be more precise. After some discussion, we agreed that
it would be more precise to say that in our investigation of
triangles, we could only say that one was bigger than the other, in
the general sense of the word, if the two triangles were
**similar** triangles. (We also discussed the idea that two
triangles **could** have the same area without having the same
shape, and that two triangles could **not** have the same
dimensions without having the same shape.) This was an interesting
conversation, bordering on argument at some points, but much more
alive than our usual math class.

But, I digress. Here is a comment that one of the students wrote, in her reflections on this topic:

*"What made similar triangles so easy to
understand was when the teacher showed us a magnifying glass and we
looked at a drawing of a triangle through the class - the triangle
was, of course, the same shape but looked bigger through the
magnifying glass, This is what similar means." Sara and
Megan.*

The students constructed triangles using the
Geometer's Sketchpad software, and experimented to determine what
combination of sides and angles would form similar triangles. They
discussed their findings, compared notes, and came up with the
following conclusions: *"The first thing we discovered is that two
triangles are similar if 2 angles of one triangle are congruent to 2
angles of another. This was surprising at first because it seemed
like too little information, just 2 pieces, but that is because once
you know 2 angles you automatically know the third, so it's kind of
like 3 pieces of information. (We had expected we'd need 3 pieces of
information because that's what we needed for proving triangles
congruent.) Then we experimented some more and decided that 2
triangles would have to be similar if all 3 sides were proportional.
What we mean by this is, for example, the sides of one triangle might
be 3, 4 and 5 and the sides of the other triangle might be 6", 8",
and 10". So the triangle with sides 6", 8" and 10" would have each
side exactly twice as long as the 3", 4"and 5" triangle. What made
similar triangles so easy to understand was when the teacher showed
us a magnifying glass and we looked at a drawing of a triangle
through the class - the triangle was, of course, the same shape but
looked bigger through the magnifying glass, This is what similar
means."* Sara and Megan.

Eventually, after experimentation and discussion, the students agreed on the following: Two triangles are similar if:

1) *... all three sides of one triangle are
proportional to all three sides of the other triangle. (This means
the ratios of the sides are equal). This is like SSS for congruent
triangles but we will call it
**SSS**~**
**for similarity.*

2) *...*** ***two angles of one triangle
are congruent to any two angles of the other. We'll call that
**AA**~**
**for similarity.*

3) *...* a pair of sides are proportional and
the included angles are congruent. *We'll call that
**SAS**~**
**for similarity.*

The next question was* "Why do these conditions
work and not other combinations of sides and angles?"*

Anya wrote the following essay, in answer to that
question: *"Only those conditions work because if you try to construct
the triangles using SSS, AA, or SAS you will find that there is only
one way to put those three measurements together. With SSS, imagine
you have 3 straws and you run a piece of string through all 3. When
you tie the string tightly, you will have a triangle and you can't
change that shape. That's why they use triangles in the construction
of bridges; the triangle is a strong shape."*

The students did a very interesting project on using similar triangles and "indirect measurement" to find the height of a tree on campus. As we often did with these projects, the students worked in groups of 3 or 4. Sometimes I let them choose their groups, but more often they drew names from a small box containing small cards; each card having a student's name on it. The problem was stated in a "worksheet" as follows:

"*Your assignment is to find the height of tree
on campus.*" (or the teacher could ask the students to find the
height of a building) "Y*ou will calculate the height using two
different methods, and then compare the answers that you get using
the two methods.*

*Method 1) Shadow Reckoning: Go to your tree and
measure the shadow cast by the tree. Then measure the length of the
shadow of a student from your group, who stands by the tree (but not
in it's shadow). Measure the student's height. Then draw a sketch of
the tree and the student, label the sketch with the measurements, and
calculate the height of the assigned object.*

*Draw the tree, the student, and both shadows on
a piece of poster board. Explain every step of your work, and write
these explanations clearly and completely, including all of equations
and any theorems that you use. This explanation will be part of your
"poster".*

*Method 2) Mirror Method: Use the small hand
mirror that has been provided. Place the mirror on the ground about
10 or 15 feet from the base of the tree. Then, looking in the mirror,
walk forwards or backwards until you can see the top of the tree
centered in the mirror, Measure the distance from where you are
standing to the center of the mirror, and the distance from the
center of the mirror to the base of the tree. Use similar triangles
to calculate the height of the tree.*

*In each of the two methods, explain every step
of this project. Your explanation must contain labeled diagrams,
measurements, and a complete explanation of the steps in this
project, including any and all theorems used.*

*Compare the answers you got. If you found
somewhat different heights from you the two methods, explain why this
might have happened. Explain why the triangles are similar, in each
case. How might these methods be useful in other situations? Do you
think this might be how people in earlier times might have measured
objects too tall to measure directly?*

Later in the school year, we did a similar project, this time using trigonometry. If you are interested in reading more about these projects, you will find more information at the following web site: http://mathforum.org/~sanders/creativegeometry/

I also asked them the following question: "We have found that there are certain postulates that we can use to prove two triangles congruent. More recently, we have studied ways to prove two triangles are similar. Let's compare these two methods. How do the methods for proving triangles similar differ from the methods use to prove triangles congruent? How are they the same?" In my writing-intensive geometry class, my students' explanations and reflections were an important part of these projects. Here are some of their answers and comments:

"*The conditions that determine similar
triangles differ from the conditions that determine congruent
triangles, in some ways. One way is that for the similarity theorems.
The corrresponding sides must be proportional instead of congruent.
However, in both the similarity theorems and the congruence theorems,
corresponding angles must me congruent. So it is the angles that give
the triangle it's shape, and the side lengths that give the triangle
it's size, to a certain extent. Another difference is that it always
takes 3 pieces of information to prove triangles congruent, but you
can prove two triangles similar with just two pieces of information
if you know 2 angle pairs are congruent; you don't need to know
anything about the sides at all." *Adam

Janine had this to say about this project: "*I
am proud of the work I did on similar triangles because I feel that I
did a good job of explaining it and seeing connnections. It was
interesting to notice how the conditions for proving triangles
similar are related in some ways to proving triangles congruent. It
was fun to know that I can figure these things out for
ourselves!*"

Nathanael West

**Go To Homepage** **Go To Introduction**
**1) Constructions** **2) Clock Problem** **3) Test Corrections** **4) ASN Explain** **5) Thoughts About Slope** **6) What is Proof?**

**7) Similar Triangles** **8) Homework Corrections** **9) Quads Midpoints** **10) Quads Congruence** **11) Polygons**

**12) Polygons Into Circles** **13) Area and Perimeter** **14) Writing About Grading** **15) Locus** **16) Extra Credit Projects**

**17) Homework Reflections** **18) Students' Overall Reflections** **19) Parents' Evaluate Method** **20) In Conclusion**