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Problem 1
Got It?
Objective To identify congruence transformations
To prove triangle congruence using isometries
Congruence Transformations
9-5
In the Solve It, you may have used the properties of rigid motions to describe why the
wings are identical.
Essential Understanding You can use compositions of rigid motions to
understand congruence.
Lesson Vocabularycongruent
transformation
LV
Identifying Equal Measures
The composition (r(9 0 , P) Rn)(LMNO) GHJ K is shown at the right.
A Which angle pairs have equal measures?
Because compositions of isometries preserve angle measure,
corresponding angles have equal measures.
m L m G, m M m H , m N m J , and
m O m K .
B Which sides have equal lengths?
By definition, isometries preserve distance. So,
corresponding side lengths have equal measures.
LM GH, MN HJ, NO JK, and LO GK.
1. The composition (Rt T 2, 3 )( ABC) XYZ. List all of the pairs of
angles and sides with equal measures.
n
P HG
J
K
L M
NO
How can you use the properties of isometries to find equal angle measures and equal side lengths? Isometries preserve angle measure and distance, so identify corresponding angles and corresponding side lengths.
Content StandardsG.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent . . .Also G.CO.6, G.CO.8
MATHEMATICAL PRACTICES
1 Common Core
CC-20CC-14
Problem 2
Got It?
In Problem 1 you saw that compositions of rigid motions preserve corresponding side
lengths and angle measures. This suggests another way to define congruence.
Identifying Congruent Figures
Which pairs of figures in the grid are congruent? For each pair, what is a sequence of rigid motions that maps one figure to the other?
Figures are congruent if and only if there is a sequence
of rigid motions that maps one figure to the other. So, to
find congruent figures, look for sequences of translations,
rotations, and reflections that map one figure to another.
Because r(180 , O)( DEF) LMN , the triangles are
congruent.
Because (T 1, 5 Ry-axis)(ABCJ) WXYZ, the
trapezoids are congruent.
Because T 2, 9 (HG) PQ, the line segments are
congruent.
2. Which pairs of figures in the grid are congruent? For each pair, what is a
sequence of rigid motions that map one figure to the other?
Key Concept Congruent Figures
Two figures are congruent if and only if there is a sequence of one or more rigid motions that
maps one figure onto the other.
y
xO2 2
2
6 6
6
4
2
6
4
4
HG
D
E
F
A B
CJ
L
M
N
Y
X WQ
P
Z
y
xO 2
2
6 6
6
4
2
6
4
4
4
F
B
A
C
D UV
W
H K
I JQ
N
M
Does one rigid motion count as a sequence? Yes. It is a sequence of length 1.
CC-14 Congruence Transformations 2
Problem 3
Got It?
Because compositions of rigid motions take figures to congruent figures, they are also
called congruence transformations.
Identifying Congruence Transformations
In the diagram at the right, JQV EWT. What is a congruence transformation that maps JQV onto EWT ?
Because EWT lies above JQV on the plane, a translation
can map JQV up on the plane. Also, notice that EWT is on the
opposite side of the y-axis and has the opposite orientation of
JQV. This suggests that the triangle is reflected across the y-axis.
It appears that a translation of JQV up 5 units, followed by a
reflection across the y-axis maps JQV to EWT. Verify by
using the coordinates of the vertices.
T 5, 0 (x, y) (x 5, y)
T 5, 0 (J) (2, 4)
Ry-axis(2, 4) ( 2, 4) E
Next, verify that the sequence maps Q to W and V to T.
T 5, 0 (Q) (1, 1) T 5, 0 (V) (5, 2)
Ry-axis(1, 1) ( 1, 1) W Ry-axis(5, 2) ( 5, 2) T
So, the congruence transformation Ry-axis T 5, 0 maps JQV onto EWT . Note that
there are other possible congruence transformations that map JQV onto EWT .
3. What is a congruence transformation that maps
NAV to BCY ?
y
x
T
E
Q
J
V
WO
2
4
2
4
42
4
Identify the corresponding parts and find a congruence transformation that maps the preimage to the image. Then use the vertices to verify the congruence transformation.
A sequence of rigid motions that maps JQV onto EWT
The coordinates of the vertices of the triangles
y
x
T
E
Q
J
V
WO
2
4 2
2
4
4
4
y
x
A
C
Y
B
N
VO 22
2
4
4
4
4
3 Common Core
Problem 4
Got It?
In Chapter 4, you studied triangle congruence postulates and theorems. You
can use congruence transformations to justify criteria for determining triangle
congruence.
Verifying the SAS Postulate
Given: J P, PA JO, FP SJ
Prove: JOS PAF
Step 1 Translate PAF so that points A and O coincide.
Step 2 Because PA JO, you can rotate JOS about point A
so that PA and JO coincide.
Step 3 Reflect PAF across PA. Because reflections preserve
angle measure and distance, and because J P
because FP SJ , you know that the reflection maps
P to J and FP to SJ . Since points S and F coincide,
PAF coincides with JOS.
There is a congruence transformation that maps PAF onto
SOJ, so PAF JOS.
4. Verify the SSS postulate.
Given: TD EN , YT SE , YD SN
Prove: YDT SNE
Proof
S
O
J
P
F A
O
S
J
A
P
F
P
A
F
O
J
S
OA
FS
PJ
T
D
E
S
N
Y
In Problem 4, you used the transformational approach to prove triangle congruence.
Because this approach is more general, you can use what you know about congruence
transformations to determine whether any two figures are congruent.
How do you show that the two triangles are congruent?Find a congruence transformation that maps one onto the other.
CC-14 Congruence Transformations 4
Problem 5
Got It?
Determining Congruence
Is Figure A congruent to Figure B? Explain how you know.
Figure A can be mapped to Figure B by a sequence of
reflections or a simple translation. So, Figure A is congruent
to Figure B because there is a congruence transformation
that maps one to the other.
5. Are the figures shown at the right congruent?
Explain.
Figure A
Figure B
Do you know HOW?Use the graph for Exercises 1 and 2.
1. Identify a pair of
congruent figures and
write a congruence
statement.
2. What is a congruence
transformation that
relates two congruent
figures?
Do you UNDERSTAND? 3. How can the definition of congruence in terms of
rigid motions be more useful than a definition of
congruence that relies on corresponding angles
and sides?
4. Reasoning Is a composition of a rotation followed by a
glide reflection a congruence transformation? Explain.
5. Open Ended What is an example of a board game in
which a game piece is moved by using a congruence
transformation?
R
x
y
V
A
T B
K
QS
I
O
2
6 4 2
2
6
4
4
Lesson Check
Practice and Problem-Solving Exercises
For each coordinate grid, identify a pair of congruent figures. Then determine a congruence transformation that maps the preimage to the congruent image.
6. 7. 8.
PracticeA See Problem 1 and 2.
x
y
B
J
T
V
Q
E
YL
G
2 44
4
4
G
A
D
F
C
R
y
xO4 2
2
4
4
4x
y
FA
E
K
S
T
B
D
M
I
C
2
4
4
4
How can you determine whether the figures are congruent?You can find a congruence transformation that maps Figure A onto Figure B.
MATHEMATICAL PRACTICES
MATHEMATICAL PRACTICES
5 Common Core
In Exercises 9–11, find a congruence transformation that maps LMN to RST .
9. 10. 11.
12. Verify the ASA Postulate for triangle congruence by using congruence
transformations.
Given: EK LH Prove: EKS HLA
E H
K L
13. Verify the AAS Postulate for triangle congruence by using congruence
transformations.
Given: I V Prove: NVZ CIQ
C N
QC NZ
In Exercises 14–16, determine whether the figures are congruent. If so, describe a congruence transformation that maps one to the other. If not, explain.
14. 15. 16.
Construction The figure at the right shows a roof truss of a new building. Identify an isometry or composition of isometries to justify each of the following statements.
17. Triangle 1 is congruent to triangle 3.
18. Triangle 1 is congruent to triangle 4.
19. Triangle 2 is congruent to triangle 5.
See Problem 3.
x
y
L
M N
S
R
T
O 2
2
4
4
x
yL
M
N
SR
TO 2 4
2
4 2
4
x
y
L
M
N
S
T
RO 2 4
2
4
4
See Problem 4.Proof
L
H
A
E K
S
Proof
ZV
N
C
I Q
See Problem 5.
ApplyB
1
2 5
3 4
CC-14 Congruence Transformations 6
20. Vocabulary If two figures are ________________, then there is an isometry that
maps one figure onto the other.
21. Think About a Plan The figure at the right shows two congruent,
isosceles triangles. What are four different isometries that map
the top triangle onto the bottom triangle?
How can you use the three basic rigid motions to map the top
triangle onto the bottom triangle?
What other isometries can you use?
22. Graphic Design Most companies have a logo that
is used on company letterhead and signs. A graphic
designer sketches the logo at the right. What congruence
transformations might she have used to draw this logo?
23. Art Artists frequently use congruence transformations in their work. The artworks
shown below are called tessellations. What types of congruence transformations
can you identify in the tessellations?
a. b.
24. In the footprints shown below, what congruence transformations can you use to
extend the footsteps?
25. Prove the statements in parts (a) and (b) to show congruence in terms of
transformations is equivalent to the criteria of for triangle congruence you learned
in Chapter 4.
a. If there is a congruence transformation that maps ABC to DEF then
corresponding pairs of sides and corresponding pairs of angles are congruent.
b. In ABC and DEF , if corresponding pairs of sides and corresponding pairs
of angles are congruent, then there is a congruence transformation that maps
ABC to DEF .
x
y
O 2 44 2
Proof
7 Common Core
Mixed Review 33. A triangle has vertices A(3, 2), B(4, 1), and C(4, 3). Find the coordinates
of the images of A, B, and C for a glide reflection with translation
(x, y) (x, y 1) and reflection line x 0.
The lengths of two sides of a triangle are given. What are the possible lengths for the third side?
34. 16 in., 26 in. 35. 19.5 ft, 20.5 ft 36. 9 m, 9 m 37. 412
yd, 8 yd
Get Ready! To prepare for Lessons 9-6, do Exercises 38–40.
Determine the scale drawing dimensions of each room using a scale of 14 in. 1 ft.
38. kitchen: 12 ft by 16 ft 39. bedroom: 8 ft by 10 ft 40. laundry room: 6 ft by 9 ft
See Lesson 9-4.
See Lesson 5-6.
See Lesson 7-2.
26. Baking Cookie makers often use a cookie press so that
the cookies all look the same. The baker fills a cookie
sheet for baking in the pattern shown. What types of
congruence transformations are being used to set each
cookie on the sheet?
27. Use congruence transformations to prove the Isosceles Triangle Theorem.
Given: FG FH
Prove: G H
28. Reasoning You project an image for viewing in a large classroom. Is the projection of
the image an example of a congruence transformation? Explain your reasoning.
Proof
F
H
G
ChallengeC
Standardized Test Prep
29. To the nearest hundredth, what is the value of x in the diagram at the right?
30. In FGH and XYZ , G and Y are right angles. FH XZ and GH YZ .
If GH 7 ft and XY 9 ft, what is the area of FGH in square inches?
31. ACB is isosceles with base AB. Point D is on AB and CD is the bisector of C .
If CD 5 in. and DB 4 in., what is BC to the nearest tenth of an inch?
32. Two angle measures of JKL are 30 and 60. The shortest side measures 10 cm.
What is the length, in centimeters, of the longest side of the triangle?
SAT/ACT
20
xx
CC-14 Congruence Transformations 8