MODULE
3
Study Guide Review
ASSESSMENT AND INTERVENTION
MODULE
STUDY GUIDE REVIEW
Congruent Figures
Essential Question: How can you use congruency to solve
real-world problems?
KEY EXAMPLE
3
(Lesson 3.1)
Write the vertices of the image of the figure given by
A (2, 1), B (3, 3), C (2, 4) after the transformations.
(x, y) → (x + 1, y + 2) → (3x, y)
A (2, 1) → A' (3, 3)
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MODULE
PERFORMANCE TASK
B (3, 3) → B' (4, 5)
Apply the transformations in
order to each point. Apply the first
transformation.
A' (3, 3) → A" (9, 3)
Apply the second transformation.
C (2, 4) → C' (3, 6)
B' (4, 5) → B" (12, 5)
C' (3, 6) → C" (9, 6)
The image of the transformed figure is determined by the points
A" (9, 3), B" (12, 5), C" (9, 6).
COMMON
CORE
Mathematical Practices: MP.1, MP.3, MP.4, MP.6
G-CO.A.5, G-CO.B.6
SUPPORTING STUDENT REASONING
Students should begin this problem by focusing on
the transformations needed to move one figure to a
congruent figure in the plane. Here are some issues
they might bring up.
• How to identify an open space on the puzzle
that is congruent to one of the available puzzle
pieces: The open space on the puzzle has at least
two sides that can be matched to the available
puzzle pieces.
• The rotation(s) needed to position the
available puzzle piece: The center of
the rotation will be close to the center of the
available puzzle piece.
• The translation(s) needed to move the
available puzzle piece into position: If the
rotation to a vertical position is done first, then
the translations will be vertical and horizontal.
• If there are pieces that will not fit anywhere
into the pieces already assembled: Yes, the piece
at the upper right appears not to fit anywhere.
151
Module 3
KEY EXAMPLE
(Lesson 3.2)
Determine whether a triangle △ABC is congruent to its image after the
transformations (x, y) → (x + 1, y + 2) → (2x, y).
The transformation (x, y) → (x + 1, y + 2) is a translation, which is a rigid motion, so after this
transformation the image is congruent. The transformation (x, y) → (2x, y) is a dilation, which is not
a rigid motion, so the image from this transformation is not congruent.
After the transformations, the image is not congruent to △ABC because one of the transformations
is not a rigid motion.
KEY EXAMPLE
(Lesson 3.3)
Find the angle in △DFE congruent to ∠A and the side congruent to ¯
BC when
△ABC ≅ △DFE.
Since_
△ABC
_≅ △DFE, and corresponding parts of congruent figures are congruent, ∠A ≅ ∠D
and BC ≅ FE.
Module 3
151
Study Guide Review
SCAFFOLDING SUPPORT
GE_MNLESE385795_U1M03MC 151
• Show students how to use a protractor to approximate the angle of rotation of
a puzzle piece. Remind students to specify whether the rotation is clockwise or
counterclockwise.
• Suggest that students draw a rough sketch of the pieces that have not yet been
fitted into the puzzle and number them. Have them also draw a rough sketch
of the edge of the puzzle showing the empty spaces and letter the spaces. Then
students can begin their descriptions with phrases such as “To move piece 2
into slot E...”.
5/14/14 6:51 PM
EXERCISES
Write the vertices of the image of the figure after the transformations. (Lesson 3.1)
1.
SAMPLE SOLUTION
The figure given by A(1, -2), B(2, 5), C(-3, 7), and the transformations
A' (3, 2), B' (-4, 4), C' (-6, -6) .
x, y → (x, y - 1) → (-y, 2x)
(
I made a sketch of the pieces that had not yet
been fitted into the puzzle and numbered them
from 1 to 5. I also made a sketch of the empty
spaces at the edge of the puzzle and lettered them
A to F. Here are the moves I used to transfer the
three pieces into the puzzle:
)
Find the rigid motions to transform one figure into its congruent figure. (Lesson 3.2)
F
2. In the figure, △ABC ≅ △DEF.
The rigid motions to transform from △ABC to △DEF are
(x, y) → (-y, x) → (x -2, y +2) .
Find the congruent parts. (Lesson 3.3)
3. Given △ABC ≅ △DEF, ∠A ≅
∠D .
_
4. Given △ABC ≅ △DEF, CA ≅
¯
FD .
E
D
-8
-4
8
y
B
4 A
C
x
0
-4
4
To move Piece 4 into Space C, rotate the piece
approximately 75 degrees clockwise, then move it
vertically upward and horizontally to the left.
8
-8
To move Piece 1 into Space D, rotate the piece
approximately 150 degrees counterclockwise, then
move it vertically downward and horizontally
to the left.
To move Piece 2 into Space A, rotate the piece
approximately 100 degrees counterclockwise, then
move it vertically upward and horizontally
to the left.
MODULE PERFORMANCE TASK
Jigsaw Puzzle
A popular pastime, jigsaw puzzles are analogous to the series of
transformations that can be performed to move one figure onto another
congruent figure.
© Houghton Mifflin Harcourt Publishing Company • Image Credits:
©Hitdelight/Shutterstock
In the photo, identify at least three pieces that would likely fit into one of the
empty spaces in the puzzle. Describe the rotations and translations necessary
to move the piece to its correct position in the puzzle.
Module 3
152
Study Guide Review
DISCUSSION OPPORTUNITIES
GE_MNLESE385795_U1M03MC 152
6/9/15 12:25 AM
• Can any available piece be moved into position using a single transformation?
• The available pieces look very similar. How will you determine whether you
chose the correct piece to transform to the congruent piece?
Assessment Rubric
2 points: Student correctly solves the problem and explains his/her reasoning.
1 point: Student shows good understanding of the problem but does not fully
solve or explain.
0 points: Student does not demonstrate understanding of the problem.
Study Guide Review 152
Ready to Go On?
Ready to Go On?
3.1–3.3 Congruent Figures
ASSESS MASTERY
Use the assessment on this page to determine if
students have mastered the concepts and standards
covered in this module.
• Online Homework
• Hints and Help
• Extra Practice
Predict the results of the transformations. (Lesson 3.1)
1. Triangle △ABC is in the first quadrant and translated along ⟨2, 1⟩ and reflected across the x-axis.
The first quadrant
Which quadrant will the triangle be in after the first transformation?
Which quadrant will the triangle be in after the second transformation? The fourth quadrant
ASSESSMENT AND INTERVENTION
Determine whether the triangles are congruent using rigid motions. (Lesson 3.2)
2. Using the graph with △ABC, △DEF, and △PQR:
A. Determine whether △ABC is congruent to △DEF.
△ABC is not congruent to △DEF.
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Differentiated Instruction Resources
• Reading Strategies
• Success for English Learners
• Challenge Worksheets
Assessment Resources
R
D
-2
P
B
2
0
-2
A
2
Cx
4
Q
△DEF is congruent to △PQR.
Find the congruent parts of the triangles. (Lesson 3.3)
3. List all of the pairs of congruent sides for two congruent triangles △ABC and △DEF.
¯
AB ≅ ¯
DE , ¯
BC ≅ ¯
EF , ¯
CA ≅ ¯
FD
© Houghton Mifflin Harcourt Publishing Company
• Reteach Worksheets
E
-4
B. Determine whether △DEF is congruent to △PQR.
ADDITIONAL RESOURCES
Response to Intervention Resources
F 4 y
ESSENTIAL QUESTION
4. How can you determine whether a figure is congruent to another figure?
Answers may vary. Sample: You can figure out whether a figure is congruent to
another by determining whether a sequence of rigid motions will transform one
figure into the other.
• Leveled Module Quizzes
Module 3
COMMON
CORE
GE_MNLESE385795_U1M03MC 153
153
Module 3
Study Guide Review
153
Common Core Standards
5/14/14 6:51 PM
Content Standards Mathematical Practices
Lesson
Items
3.1
1
G-CO.A.5
MP.4
3.1, 3.2
2
G-CO.B.7
MP.7
3.3
3
G-CO.B.7
MP.4
MODULE
MODULE 3
MIXED REVIEW
MIXED REVIEW
Assessment Readiness
Assessment Readiness
1. A line segment with points R(3, 5) and S(5, 5) is reflected across the line y = -x and
translated 2 units down. Determine whether each choice is a coordinate of the image
of the line segment. Select Yes or No for A–C.
A. R' (-5, -3)
Yes
No
B. R' (-5, -5)
Yes
Yes
C. S' (-5, -7)
3
ASSESSMENT AND INTERVENTION
No
No
2. The polygon ABCD is congruent to PQRS. The measure of angle B is equal to 65°. Choose
True or False for each statement.
A. The supplement of angle Q measures 115°.
True
False
B. Angle Q measures 115°.
True
False
C. The supplement of angle B measures 115°.
True
False
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prepare students for high-stakes tests.
ADDITIONAL RESOURCES
3. Triangle LMN is a right triangle. The measure of angle L is equal to 35°. Triangle LMN is
congruent to △PRQ with right angle R. Choose True or False for each statement.
A. The measure of angle Q is 55°.
True
False
B. The measure of angle R is 90°.
True
False
C. The measure of angle P is 35°.
True
False
¯; ED
¯
FD
Module 3
COMMON
CORE
• Leveled Module Quizzes: Modified, B
AVOID COMMON ERRORS
Item 1 Some students will stop too soon when
faced with a problem with multiple steps. Encourage
students to number each step, and then make sure
they have completed each one before choosing a final
answer to the problem.
© Houghton Mifflin Harcourt Publishing Company
_
4. The two triangles, △ABC and
_△DEF, are congruent. Which side is congruent to CA?
Which side is congruent to BA?
Assessment Resources
Study Guide Review
154
Common Core Standards
GE_MNLESE385795_U1M03MC.indd 154
16/06/14 11:41 AM
Content Standards Mathematical Practices
Lesson
Items
3.1, 2.1, 2.2
1*
G-CO.A.5
MP.5
3.3, 1.2
2*
G-CO.A.1, G-CO.B.6
MP.2
3.3
3
G-CO.B.6
MP.2
3.3
4
G-CO.B.6
MP.7
* Item integrates mixed review concepts from previous modules or a previous course.
Study Guide Review 154