Download Lesson 2: CPCTC

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Title of Lesson: The Triangles are Congruent, Now What?
UFTeach Students’ Names: Rachel Edwards and Jacob Lehenbauer
Teaching Date and Time: October 25, 2012 10:30-12:45
Length of Lesson: 50 minutes
Grade / Topic: 10-11th grade Honors Geometry
Source of the Lesson:
Jurgensen, Ray C., Richard G. Brown, and John W. Jurgensen. Geometry. Evanston, IL: McDougal Littell,
2000. Print.
Cooperative Learning and High School Geometry: Becky Bride; Kagan Publishing; www.KaganOnline.com
Concepts
This concept follows from those learned by the students previously in proving triangles congruent. When
triangles are introduced, the most useful ability, in relation to proving geometric theorems, is proving triangles
congruent to each other. This can be done with various information, the minimum necessary information needed
is three corresponding congruent pieces. These pieces can be all three sides congruent, two pairs of
corresponding congruent angles with a pair of corresponding congruent sides in between, or two pairs of
corresponding congruent sides with a pair of corresponding congruent angles in between. The theorems related,
named Side Angle Side and so on, can be used to prove triangles congruent. After using these theorems, it is
useful to know additional information that follows from these theorems. CPCTC stands for Corresponding Parts
of Congruent Triangles are Congruent. This is useful for concluding much more about the related triangles and
proving other things thereafter. It is important to remember positioning of the triangles, as these triangles may
be oriented different directions.
http://www.mathwarehouse.com/geometry/congruent_triangles/congruent-parts-CPCTC.php
Warren Buck, J. Pahikkala. "CPCTC" (version 4). PlanetMath.org. Freely available at
http://planetmath.org/CPCTC.html.
Florida State Standards (NGSSS):
MA.912.G.4.6: Prove that triangles are congruent or similar and use the concept of corresponding parts of congruent triangles.
Cognitive Complexity: Moderate
Process Skills needed for this lesson:
● solving a problem requiring multiple operations
● retrieving information from a graph, table, or figure and using it to solve a problem
● providing a justification for steps in a solution process
● comparing figures or statements
Performance Objectives
●
Given two congruent triangles, students will be able to state which sides of the two triangles are
corresponding to each other.
●
Given two congruent triangles, students will be able to state which angles of the two triangles are
corresponding to each other.
Step 1/2: Explorations in
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●
Students will be able to prove that corresponding parts of congruent triangles are congruent by using
either SSS, SAS, ASA, or AAS theorems.
Materials List and Student Handouts
●
Appendix A, x24 per class
●
Appendix B, x24 per class
●
Appendix C, x24 per class
●
Appendix D, x 24 per class
Advance Preparations
●
Teachers will have a copy of the appendices they wish to go over with the class for viewing on the
document camera
●
Teachers will have all the worksheets ready to be handed out into groups
Safety
●
There are no significant safety concerns with this lesson.
5E Lesson Templates
ENGAGEMENT
Time: 5 minutes
What the Teacher Will Do
Teacher Directions and
Student Responses and
Probing Questions
Potential Misconceptions
The teacher will give Appendix A Today I want to start with
to the students.
going over what you have
learned before. I want you to
do this worksheet by
yourself and we will go over
the results.
The teacher will encourage the
students to exercise their prior
knowledge of the methods of
proving triangles congruent to
review and give them
confidence for the lesson.
(give students a few minutes
to complete the worksheet)
Do we already know how to
prove two triangles
congruent? How?
[Yes! By using the SAS, SSS,
AAS and ASA theorems.]
Can all triangles be proved
Guide the students to the
congruent using the same
concept of CPCTC by asking for theorem?
additional information about the
[No, we need to use different
theorems depending on what
prior information we have.]
Step 1/2: Explorations in
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triangles.
Even if we have 3 congruent
sides or angles in any order,
is that enough to prove that
triangles are congruent?
[No, for example, problem 3 on
the worksheet gives SSA. This
is not enough information to
prove that two triangles are
congruent.]
Is there anything else we can
now conclude? How would [The students can have various
we prove them?
answers here, hopefully
including the relationships of
the other parts of the triangle.]
EXPLORATION
minutes
What the Teacher Will
Do
Time: 15
Teacher Directions and
Probing/Eliciting Questions
This activity was a good
review for what we are going
to explore next. You all had
great observations!
This next worksheet will be
done in groups. Talk with
your group to reason out
these problems together.
Make sure you give correct
justification for your
reasoning.
The teacher should
How can we prove these two
monitor the groups
triangles congruent? Which
progresses as they work, theorem can we use from
asking guiding/probing before?
questions as they do.
Student Responses and
Misconceptions
Hand out Appendix B.
Teacher will explain that
we are looking to see if
we can find the triangles
congruent using things
we already know.
[Answers will vary, but they should
use one of the SAS, SSS, etc.
theorems.] Students may try to use
other methods still, they should be
guided to the use of the theorems they
learned previously. Students cannot
use AAA or SSA for theorems!
[Shouldn’t those be congruent too?]
Students should be led to the fact that
all the other parts are also congruent.
Step 1/2: Explorations in
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Now we have the triangles
Students may think that they cannot
congruent, good! Does that assume this, but it should be pointed
mean we can say anything
out that since the triangles are proven
else about them? What about to be congruent, every single part of
the other parts we didn’t use? the two triangles needs to be
congruent to its corresponding part.
EXPLANATION
minutes
What the Teacher Will Do
Time: 15
Teacher Directions and
Student Responses and
Probing/Eliciting Questions
Misconceptions
Let’s look at problem 1. (Call
on student) How did you
conclude that triangle AED is [SAS (their reasoning will
congruent to triangle BEC? vary) ]
Why?
The teacher will go over the
results of Appendix B with the
students, hearing how they
proved the triangles congruent
(to give practice with other
material) and question them
about other conclusions that can How did you conclude that
be drawn about the triangles
the measure of angle D is
once they are proved congruent. equal to the measure of angle [Since the two triangles are
C? (Call on another student) congruent, the corresponding
angles should also be
For problem 2, how did you congruent.]
conclude that triangle PQO is
congruent to triangle RSO?
[ASA (their reasoning will
From this, how did you
vary) ]
conclude that O is the
midpoint of QR? (call on
[since the triangles are
another student)
congruent, QO is congruent to
RO. Thus O is the midpoint of
QR by definition of midpoint.]
For problem 3, how did you
conclude that triangle JMO is
congruent to triangle KMO? [AAS, their reasoning will vary)]
(call on a student)
From this, how did you
conclude that M is the
Step 1/2: Explorations in
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midpoint of JK? (call on
another student)
[since the triangles are
congruent, JM is congruent to
KM. Thus O is the midpoint of
JK by definition of midpoint.]
For problem 4, how did you
conclude that triangle NOP is
congruent to triangle SRP? [AAS (their reasoning will
(call on a student)
vary) ]
From this, how did you
conclude that NP is
congruent to SP? (call on
another student)
[since both of the triangles are
congruent, their sides that
match up will be equal.]
Now that we know these two
triangles are congruent, is
there anything else we can
[Yes, their other parts look to
see about the triangles? (call be the same too. Which makes
on another student)
sense, if the triangles are both
congruent.]
After going over these
answers, does the fact that
two triangles are congruent [Yes, because all of the
give us additional information corresponding angles and the
about the other angles and
sides of the two triangles
sides in the figures?
should be equal/congruent.]
What can we use this
information to do?
To use this, we just say the
fact that “Corresponding
parts of Congruent triangles
are Congruent.”
How is this new information
useful?
[We could conclude other
things about the triangles, like
the other corresponding angles
and the other corresponding
sides are congruent.]
[This could be useful because
we can use those congruences
to help us prove other things
about the figure!]
Possible misconceptions
Step 1/2: Explorations in
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using AAA or SSA to prove
congruent triangles is
incorrect.
From this, we can see that
two triangles are congruent if
and only if their vertices can
be matched up so that the
corresponding parts of the
triangles are congruent.
So lets review steps to
proving two corresponding
segments or two
corresponding angles are
congruent (write these steps
on the board)
1) Identify two triangles in
which the two corresponding
segments or corresponding
angles are corresponding
parts
2) Prove that the triangles are
congruent
3) State that the two
corresponding parts are
congruent using the reason
“Corresponding Parts of
Congruent Triangles are
Congruent” (CPCTC)
EVALUATION
minutes
What the Teacher Will
Do
It should be made sure that the
students know these things can
only be concluded after the
triangles have been proven
congruent by one of the
previously used theorems.
Students could also see
different parts of the triangles
congruent based on how they
are arranged. Positioning
should be stressed, so
students are always using the
correct corresponding parts.
Time: 15
Assessment
Student
Responses
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Hand out Appendix C
(one per student).
Tell the class that they will now be doing an
evaluation. They are to do the work on their own.
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Appendix A
Name: ____________________________________________________________________________
Directions: Determine whether the following triangles are congruent. If so, complete the congruence
statement by filling in the blank and write the postulate or theorem that justifies the congruence.
Otherwise, state that the “congruence cannot be proven.”
E
B
D
C
A
Justification:
∆𝐵𝐶𝐴 ≅ ∆______?
M
F
A
R
K
∆𝐴𝐾𝑀 ≅ ∆______?
Justification:
O
L
M
N
P
Step 1/2: Explorations in
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Justification:
∆𝐿𝑀𝑁 ≅ ∆______?
Appendix B
Name: ________________________________________________________________________________________
Using Congruent Triangles to Show Other Congruencies
Directions: In your group, construct two-sided proofs for each of the figures below using the given
statements.
#1
#2
A
B
P
Q
E
O
C
D
Given : AE @ CE
DE @ BE
Pr ove : ÐA @ ÐC
#3
R
S
Given : ÐP @ ÐS
O is the midpoint of PS
Pr ove : O is the midpoint of QR
#4
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O
N
S
1
O
3
1
2
4
K
M
2
R
P
Given : NO ^ OR
SR ^ OR
Given : Ð1 @ Ð2
Ð3 @ Ð4
< 1 @< 2
Pr ove : M is the midpoint of JK
Prove : NP @ SP
Appendix C
NO @ SR
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Name: _________________________________________________________________________________
Given: The two triangles below are congruent.
D
C
O
A
B
Fill in the blank and provide reasons:
3) ̅̅̅̅
𝐴𝑂 ≅ ________?
1) ∆𝐴𝐵𝑂 ≅ ∆__________?
̅̅̅̅ ≅ ________?
4) 𝐵𝑂
2) ∠ 𝐴 ≅ ∠ ______?
5) Using what you have learned in the previous chapters and what you learned today to complete a twosided proof using the given statements below:
Q
3
P
2
1
4
5
R
S
GIVEN:
 4  5,QR  SR
PROVE:
 2  3