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Sarah Thomas
Lesson Title: Congruent & Similar Triangles
Grade Level: 6th Grade
Number of Class Periods: Two class periods
Text or Resource and Relevant Pages: Teaching and Learning High School
Mathematics by Charlene Beckmann, Denisse Thompson, Rheta Rubenstein Page 99-100
Overview:
Students will learn the theorems of congruent and similar triangles. By the end of
the lesson they should have a basic understanding of the SSS, AAA, SAS, and ASA
theorems. To achieve this, the students will do an activity “How Many Triangles?” with
spaghetti noodles.
Objectives:
G.TR.07.03 Understand that in similar polygons, corresponding angles are congruent and
the ratios of corresponding sides are equal; understand the concepts of similar figures and
scale factor.
G.TR.07.04 Solve problems about similar figures and scale drawings.
G.TR.07.05 Show that two triangles are similar using the criteria: corresponding angles
are congruent (AAA similarity); the ratios of two pairs of corresponding sides are equal
and the included angles are congruent (SAS similarity); ratios of all pairs of
corresponding sides are equal (SSS similarity); use these criteria to solve problems and to
justify arguments.
8.G.4. Understand that a two-dimensional figure is similar to another if the second can be
obtained from the first by a sequence of rotations, reflections, translations, and dilations;
given two similar two-dimensional figures, describe a sequence that exhibits the
similarity between them.
Materials:
Simple classroom materials such as pencil and eraser.
Spaghetti noodles
Tape
Handout
Protractor
Tracing Paper
Use of Space:
The students will start out in their own desk, working on the first part of the launch
independently. For the second part, on through the explore, students be arranged in two
groups, one of 3 and one of 4. When in their small groups, student desks will be clustered
so they are all facing each other.
Launch (Day 1):
Students will work independently on the first part of the launch. This will consist
of a worksheet where they are given two similar triangles with the side lengths provided.
They will use their prior knowledge from the previous lesson of similarity between
shapes to explain how the triangles are similar (proportional sides). Some of the angles
will be provided and students will be asked to measure the remaining angles, to find that
the triangles have the exact same angles as each other. Students will also be given two
congruent triangles and asked to measure the sides/angles in the same fashion. Then, we
will come back together as a class and briefly discuss the relationship between the angles
© 2010 John Wiley & Sons, Inc.
Sarah Thomas
of the triangles, and what the difference is between similarity and congruence, as some
students may not know what congruence is yet.
Explore (Day 1):
Now that students have recognized the relationship between the angles of
congruent and similar triangles, there will be a short lecture format to begin the lesson.
This is necessary to introduce the students to the letters A and S, and how the different
possible combinations of these letters relate to triangles. Otherwise, students may not
know what ASA means, or that the order of the letters matter. The teacher will draw on
the whiteboard examples of the theorems SSS, ASA, and AAS. Specifically, this would
include drawing two triangles and showing how these theorems would relate the triangles
together. Students will then pair up, and the teacher will ask them to write down all the
possibilities using three letters (of A’s and S’s) on the back page of the launch. They will
then talk with their partners and make predictions on whether each theorem is a
congruence theorem, a similarity theorem, or neither. To conclude the period, each pair
will come up to the board and put a tally to indicate what they predict each theorem is.
Students may have difficulty with their predictions, especially when facing
congruence versus similarity because technically, if two triangles are congruent, they are
also similar. This will be a good point for discussion in the share and summarize in day 2
of the lesson.
(Day 2):
Students will work on the activity How Many Triangles. The first page of the
activity will serve as a kind of “re-launch” by making them look at the SAS example.
The students will use spaghetti noodles to create triangles and discover the various
theorems of triangle congruence/similarity in this activity. Students will work with the
same partner as they worked with on Day 1. This activity will help the students check
their predictions for the theorems, distinguishing between congruence, similarity, or
neither. Students will work with SSS, SAS, AAS, ASA, and AAA. They will also work
with SSA, the only theorem which is neither congruence, nor similarity. SSA will most
likely be the biggest struggle for the students if they do not notice that they can make
more than one triangle. It is possible that the students will not move the outer side to
form the second triangle, non-similar triangle. This could be addressed by prompting
questions from the teacher, such as asking a question about the angle in between the two
sides, or by asking another group what they got. Students may also struggle in general
with forming the noodles into actual triangles, and knowing where the angles and sides
go relative to each other. This will hopefully be avoided by allowing the students to
work together, and tape the noodles to the tracing paper. If students are really struggling,
the teacher could guide them through one of the easier examples (SSS). Once students
have finished with the noodle part of the explore section, they will answer the final
questions on the back page, to see how some of the theorems relate to each other,
specifically, with AA similarity. Students will also check their results with their previous
predictions.
Share and Summarize (Day 2):
Students will regroup together as a class (but still sitting by their partner) and
discuss as a larger group their findings from the explore section. Specifically, the
students will be asked if any of their predictions changed, and if they were surprised by
anything they found. They will be also be asked if all theorems resulted in congruent
triangles, which one(s) didn’t, and why they didn’t. Time will especially be allotted to
© 2010 John Wiley & Sons, Inc.
Sarah Thomas
talk about SSA and why it does not always result in a congruent triangle. The teacher
will ask a volunteer to show their example of SSA on the ELMO (if there is one) or
overhead projector, to show two different triangles with the given parts. Students will
also be asked a question from their worksheet – are any of the theorems obvious from
others? This will lead the discussion into AA similarity and how it relates to AAS and
ASA. Specifically, these two are congruence theorems when we have the side with two
angles. Students will then be asked to summarize the difference between similarity and
congruence.
Application or Extension:
Students will move forward from triangle congruency to more work with triangle
similarity by using the noodles to make larger triangles. This will be done by using their
previous knowledge about proportions and ratios of sides. To make similar triangles with
for example, twice as long or three times as long of side lengths. They will use the same
idea of the congruency theorems, but now with making sure sides are proportional to
each other (with SSS and SAS), and also recognizing the sides don’t matter with AA, as
summarized in class.
© 2010 John Wiley & Sons, Inc.
Sarah Thomas
HOW MANY TRIANGLES?
If two triangles are congruent, they are the same – they both have exactly
the same three sides and exactly the same three angles. This means two
congruent triangles have six pairs of congruent parts (three pairs of sides
and three pairs of angles). But, sometimes we can check if two triangles
are congruent when we know fewer than six of these pairs.
The question is: What pairs of congruences are just enough?
Example:
Suppose a triangle has sides with lengths S1 and S3, and the angle between the sides
measures A3.
Trace an angle measuring A3 on tracing paper.
Then break spaghetti noodles to match up with
sides S1 and S3.
Tape these two noodles on your paper so the
angle measure between them is A3.
If we have another triangle with the same
information (S1, S3, A3) – Does it have to be
congruent to our first triangle? Check using
spaghetti noodles.
This relationship is called Side-Angle-Side (SAS). Why do you think this is?
© 2010 John Wiley & Sons, Inc.
Sarah Thomas
Today, you will explore which pairs of congruences between two triangles guarantee
the triangles must be congruent. Use the lengths (S1, S2, S3) and angles (A1, A2, A3) in
the box with the spaghetti and tracing paper to help you.
For each relationship:
a. In the second column, sketch two congruent triangles and show the congruence
relationship (see SAS for an example below)
b. Using the components listed in the third column, see if you can find any
triangles that are not congruent to each other with the 3 listed components.
c. Determine if the relationship is a congruence relationship (always makes the
same triangle). Explain in column four.
Relationship
SSS
Sketch
Components
S1, S2, S3
SSA
S1, S2, A2
SAS
S1, A3, S3
© 2010 John Wiley & Sons, Inc.
Is it a congruence relationship? Explain.
Sarah Thomas
AAS
A1, A2, S1
ASA
A1, S1, A2
AAA
A1, A2, A3
© 2010 John Wiley & Sons, Inc.
Sarah Thomas
(If time) Test some of the relationships using other choices of angle measures or side
lengths not in the table. Did your congruence decisions change? Why or why not?
Are there any other relationships between sides and angles you did not explore? Why
or why not?
Which relationships, if any, are obvious from other relationships? How?
© 2010 John Wiley & Sons, Inc.