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Prove It
How do we create truth?
2:1:27:Triangles, Part 5: Proving Congruent Triangles
TITLE OF LESSON
Geometry Unit 1 Lesson 27– Triangles, Part 5: Proving Congruent Triangles
Prove it! What’s on the outside? What’s on the inside? Of Geometry
TIME ESTIMATE FOR THIS LESSON
One class period
ALIGNMENT WITH STANDARDS
California – Geometry
4.0 Students prove basic theorems involving congruence and similarity.
5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding
parts of congruent triangles.
MATERIALS
Sticks
Envelopes
Paper
Tape or Glue
LESSON OBJECTIVES
To introduce to the following concepts:
• Corresponding angles
• Corresponding sides
• Included angle
• Included side
• Congruent Triangles
EXPLANATION OF LESSON
We will continue with congruence in triangles. You’ll have to prepare sets of sticks before class, according to
the directions below. (You can also use strips of paper.) You know your class best, and if you think you should
divide them into smaller groups, have an appropriate set of envelopes.
You’ll be making six identical sets of three sticks. Each set will go into a separate envelope, so you’ll have
six envelopes total.
SET 1
(GROUP 1)
SET 2
(GROUP 2)
SET 3
(GROUP 1)
SET 4
(GROUP 2)
SSS 1
SAS 1
ASA 1
SSS 1
SAS 1
ASA 1
SSS 2
SAS 2
ASA 2
SSS 2
SAS 2
ASA 2
You might be able to guess from this what the sticks will look like, but here it is:
The first two SSS envelopes (SSS 1 and SSS 1) will each contain three equal-length sticks. So, for example,
the two SSS 1s may each have three sticks that are 3 inches long. The students will use these to work with the
Side-Side-Side postulate for proving triangle congruence. The two SSS 2 envelopes, then, will each also have
three equal-length sticks.
The first two SAS envelopes (SAS 1 and SAS 1) will each contain two equal-length sticks, and one that has a
different length. So, for example, the two SSS 1s may each have two sticks that are 2 inches long, and one stick
that is 3 inches long. The students will use these to visualize the Side-Angle-Side postulate for proving triangle
congruence. The two SAS 2 envelopes, then, will each also have two equal-length sticks and one that has a
different length, say two that are 4 inches and one that is 7.
The first two ASA envelopes (ASA 1 and ASA 1) will each contain three sticks of different lengths. The
students will use these to work with the Angle-Side-Angle postulate. The two ASA 2 envelopes, then, will
each also have three sticks of different lengths.
1
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Prove It
How do we create truth?
2:1:27:Triangles, Part 5: Proving Congruent Triangles
When you cut the sticks, make sure your measurements work, or that you allow students to be flexible. In other
words, if you have two sticks that are 2 inches each, and one stick that is 5 inches, try, yourself to make that into a
triangle. You’ll end up with something like this:
You could actually discuss this if it comes up. This figures into why the SSS postulate works. If three sides are the
same length, they can only perfectly form one triangle.
When you divide the class in half, in Activities #1, the Manager of each half will get two sets of envelopes – one
set of SSS 1, SAS 1, and ASA 1, and one set of SSS 2, SAS 2, and ASA 2. Follow the lesson plan from there.
FOCUS AND MOTIVATE STUDENTS
1) Homework Check – Stamp/initial complete homework assignment. Pass back graded work and have students
place in the appropriate sections of their binders.
2) Agenda – Have students copy the agenda.
3) Homework Review – (8 minutes) Review homework from Lesson 26. Have each student present to the class
part of yesterday’s homework, which was to:
•
•
•
•
•
Draw two triangles that are congruent. Label the equal angles and equal sides with curved lines and slashes
so that congruence is apparent. Repeat this process for two different congruent triangles.
Draw two sides and an included angle.
Draw another set of two sides and included angle in which the corresponding sides are equal and the
included angle is equal. What can we conclude? Which postulate does this demonstrate?
Draw a side between two angles. Do not complete the triangle.
Draw another side of the same size between two angles of same measurement as the previous drawing. What
can we conclude?
Collect the homework assignments.
ACTIVITIES – INDIVIDUAL AND GROUP
1.
Assign Group Roles – (2 minutes) Divide the class in half. Ask all students to have their binders open for notes
because they’ll be taking notes, then doing an activity, then taking notes, then back to the activity, and so on. Ask
for a Manager for each group (see Group Roles below.) Explain to the Managers that you’re going to give them
envelopes for their groups, but that they are responsible for holding onto them and opening them only when you say
to open them. Hand each group Manager the set of envelopes for his group. Also, give the manager tape or glue,
whichever you’ve decided to use for this activity. Now, ask for a Facilitator. These are the only two set group roles
you’ll have today because all students should have the opportunity to act as Arranger. Read the group roles (below)
to the class and make sure they understand them
2.
Definitions – (12 minutes) Explain that today you’re going to learn the three most common postulates for proving
that triangles are congruent. You’ll give them definitions, but once they understand the postulates, they’ll just use
their common abbreviations.
Write the following on the board. Have the students write this postulate into their binders under the section for
Postulates and Theorems. Two triangles are congruent if all three sides are equal. This is known as the Side-SideSide postulate or SSS for short.
3.
Group Work – Ask the Managers to open their envelopes labeled SSS 1. As a group, they are to construct a triangle
using the three sticks. This time, they cannot break any of the sticks. Have them tape or glue the triangle to a piece
of paper. Now compare the results from the two different groups. What do they conclude? (The triangles are
congruent. They may be positioned differently but if positioned the same they should line up exactly.)
Now have them repeat the task with the set of sticks marked SSS 2. Is it possible for the two groups to ever form
2
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Prove It
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2:1:27:Triangles, Part 5: Proving Congruent Triangles
two different triangles when the sides are all equal to each other? (No.) What if only the angles were equal to each
other and you didn’t know the sizes of the sides? Ask if it’s possible to form two different triangles when the angles
are equal to each other and we know nothing about the sides? (Yes.) Can anyone demonstrate this using string or
sticks? (Be prepared to demonstrate if there are no students willing to demonstrate. The easiest way is to construct
two equilateral triangles with different size sides. The angles will always equal 60°.) Ask if any of the students have
ever taken a drawing of theirs from a small size, to a larger size, using a grid. Students who do graffiti (the art, not
the tagging) should be familiar with this. This is a similar idea. The triangle may look exactly the same, but it’s
bigger. Then, in geometry, it is NOT congruent. Congruent means exactly the same. So if the sides are different
lengths, then the triangles are not congruent, but if the sides are all the same length, the triangle has to be
congruent.
4.
Definitions – (12 minutes) Write the following on the board. Have the students write this postulate into their
binders under the section for Postulates and Theorems. Two triangles are congruent if two sides and the included
angle are equal to two sides and the included angle of the other triangle. This is known as the Side-Angle-Side
postulate or SAS for short.
5.
Group Work – Ask the Managers to open the envelopes marked SAS 1. They are to construct a triangle using the
three sticks. They’ll have to manipulate the sticks to form as close to a perfect triangle as they can. It will be easiest
if their included angle is a right angle since they’re easier to verify. If they create another sized angle they’ll have to
verify the size with a protractor. Once they have them complete, have them tape or glue the triangle to a piece of
paper. Now compare the results from the two different groups. What do they conclude? (The triangles are
congruent. They may be positioned differently but if positioned the same they should line up exactly.) Repeat the
process with the envelopes labeled SAS 2. Again, ask if it is possible for the two groups to have formed different
sized triangles? (No.) Ask student to try, by drawing, if they think they can.
6.
Definitions – (12 minutes) Write the following on the board. Have the students write this postulate into their
binders under the section for Postulates and Theorems. Two triangles are congruent if two angles and the included
side are equal to two angles and the included side of the other triangle. This is known as the Angle-Side-Angle
postulate or ASA for short.
7.
Group Work – Have the Managers open the envelopes marked ASA 1. Students are to construct a triangle using the
three sticks. The smaller sticks should be positioned first, and then they can form their angles with the other two
sticks. Because they are testing the ASA postulate, remind them to make sure they form two equal angles BEFORE
they tape or glue anything down. For this one, they may have to break one or two sticks, but make sure neither
group breaks the stick they have in common. You will want to verify the size of the angles with a protractor. Have
them tape or glue the triangle to a piece of paper. Now compare the results from the two different groups. What do
they conclude? (The triangles are congruent. They may be positioned differently but if positioned the same they
should line up exactly.) Now repeat this process with the sticks labeled ASA 2. And once again, is it possible for
the two groups to have formed different sized triangles? (No.)
8.
Review – (4 minutes). Review the three methods of showing congruence. Have each student state one of the
postulates. Go around the class having each student name one not named by the student before him, then explain,
briefly, what it means.
9.
Homework Review – Explain the homework assignment. Field questions.
HOMEWORK
1) Write down the three ways to prove that a triangle is congruent. Label them the SSS, ASA and SAS postulates.
2) Draw two different examples of each.
3) Find five examples of congruent triangles in the world. Answer the following questions:
a) How do you know that they are congruent?
b) Is it, in each case, important that they be congruent? If so, why? As an example, you can use
architecture—think of a familiar building near your school. In this case, it would probably be quite
important for the two to be exactly congruent.
3
© 2001 ESubjects Inc. All rights reserved.
Prove It
How do we create truth?
2:1:27:Triangles, Part 5: Proving Congruent Triangles
GROUP ROLES
Manager – You are responsible for taking care of the envelopes and the glue or tape.
Facilitator – You are responsible for making sure everyone has a turn as Arranger and for keeping your group
focused.
Arranger – When you are acting as Arranger, you are responsible for arranging the sticks to illustrate the
appropriate postulate.
DOCUMENTATION FOR PORTFOLIO
None
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© 2001 ESubjects Inc. All rights reserved.