Lesson 5.1   Skills Practice
Name
Date
We Like to Move It!
Translating, Rotating, and Reflecting Geometric Figures
Problem Set
Transform each given geometric figure on the coordinate plane as described.
1. Translate trapezoid ABCD 11 units to the right.
y
A
A9
B
8
B9
6
C
D
4
C9
2
D9
0
28 26 24 22
2
4
6
8
x
22
24
26
28
5
2. Translate triangle EFG 8 units up.
y
8
В© Carnegie Learning
6
4
2
0
28 26 24 22
22
2
4
6
E
8
x
F
24
26
28
G
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Lesson 5.1   Skills Practice
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3. Rotate rectangle HJKL 90В° counterclockwise about the origin.
y
8
H
J
L
K
6
4
2
0
28 26 24 22
2
4
6
8
x
22
24
26
28
4. Rotate triangle MNP 180В° counterclockwise about the origin.
y
8
6
5
4
2
0
22
2
4
26
514 
8
x
N
24
28
6
M
P
В© Carnegie Learning
28 26 24 22
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Lesson 5.1   Skills Practice
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Name
Date
5. Rotate trapezoid QRST 90В° counterclockwise about the origin.
y
8
6
4
2
0
28 26 24 22
2
4
6
8
x
22
R
24
S
26
28
Q
T
6. Rotate parallelogram WXYZ 180В° counterclockwise about the origin.
y
X
8
5
Y
6
4
W
В© Carnegie Learning
2
Z
0
28 26 24 22
2
4
6
8
x
22
24
26
28
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Lesson 5.1   Skills Practice
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7. Reflect triangle ABC over the y-axis.
y
B
8
A
6
4
2
C
0
28 26 24 22
2
4
6
8
2
4
6
8
x
22
24
26
28
8. Reflect parallelogram DEFG over the x-axis.
y
8
6
5
4
2
0
28 26 24 22
x
22
D
F
G
24
26
28
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Lesson 5.1   Skills Practice
page 5
Name
Date
9. Reflect trapezoid HJKL over the x-axis.
y
J
K
8
6
4
H
2
0
28 26 24 22
L
2
4
6
x
8
22
24
26
28
10. Reflect quadrilateral MNPQ over the y-axis.
y
5
8
6
4
В© Carnegie Learning
2
0
28 26 24 22
22
2
4
6
N
x
8
P
24
26
28
M
Q
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Lesson 5.1   Skills Practice
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Determine the coordinates of each translated image without graphing.
11. The vertices of triangle ABC are A (5, 3), B (2, 8), and C (24, 5). Translate the triangle 6 units to the
left to form triangle AоЂ№ BоЂ№ CоЂ№.
The vertices of triangle AоЂ№ BоЂ№ CоЂ№ are AоЂ№ (21, 3), BоЂ№ (24, 8), and CоЂ№ (210, 5).
12. The vertices of rectangle DEFG are D (27, 1), E (27, 8), F (1, 8), and G (1, 1). Translate the rectangle
10 units down to form rectangle DоЂ№ EоЂ№ FоЂ№ GоЂ№.
13. The vertices of parallelogram HJKL are H (2, 26), J (3, 21), K (7, 21), and L (6, 26). Translate the
parallelogram 7 units up to form parallelogram HоЂ№ JоЂ№ KоЂ№ LоЂ№.
14. The vertices of trapezoid MNPQ are M (26, 25), N (0, 25), P (21, 2), and Q (24, 2). Translate the
trapezoid 4 units to the right to form trapezoid MоЂ№ NоЂ№ PоЂ№ QоЂ№.
15. The vertices of triangle RST are R (0, 3), S (2, 7), and T (3, 21). Translate the triangle 5 units to the left
and 3 units up to form triangle RоЂ№ SоЂ№ TоЂ№.
5
16. The vertices of quadrilateral WXYZ are W (210, 8), X (22, 21), Y (0, 0), and Z (3, 7). Translate the
quadrilateral 5 units to the right and 8 units down to form quadrilateral WоЂ№ XоЂ№ YоЂ№ ZоЂ№.
17. The vertices of triangle ABC are A (5, 3), B (2, 8), and C (24, 5). Rotate the triangle about the origin
90В° counterclockwise to form triangle AоЂ№ BоЂ№ CоЂ№.
The vertices of triangle AоЂ№ BоЂ№ CоЂ№ are AоЂ№ (23, 5), BоЂ№ (28, 2), and CоЂ№ (25, 24).
18. The vertices of rectangle DEFG are D (27, 1), E (27, 8), F (1, 8), and G (1, 1). Rotate the rectangle
about the origin 180В° counterclockwise to form rectangle DоЂ№ EоЂ№ FоЂ№ GоЂ№.
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В© Carnegie Learning
Determine the coordinates of each rotated image without graphing.
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Lesson 5.1   Skills Practice
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Name
Date
19. The vertices of parallelogram HJKL are H (2, 26), J (3, 21), K (7, 21), and L (6, 26). Rotate the
parallelogram about the origin 90В° counterclockwise to form parallelogram HоЂ№ JоЂ№ KоЂ№ LоЂ№.
20. The vertices of trapezoid MNPQ are M (26, 25), N (0, 25), P (21, 2), and Q (24, 2). Rotate the
trapezoid about the origin 180В° counterclockwise to form trapezoid MоЂ№ NоЂ№ PоЂ№ QоЂ№.
21. The vertices of triangle RST are R (0, 3), S (2, 7), and T (3, 21). Rotate the triangle about the origin
90В° counterclockwise to form triangle RоЂ№ SоЂ№ TоЂ№.
22. The vertices of quadrilateral WXYZ are W (210, 8), X (22, 21), Y (0, 0), and Z (3, 7). Rotate the
quadrilateral about the origin 180В° counterclockwise to form quadrilateral WоЂ№ XоЂ№ YоЂ№ ZоЂ№.
Determine the coordinates of each reflected image without graphing.
23. The vertices of triangle ABC are A (5, 3), B (2, 8), and C (24, 5). Reflect the triangle over the x-axis
to form triangle AоЂ№ BоЂ№ CоЂ№.
The vertices of triangle AоЂ№ BоЂ№ CоЂ№ are AоЂ№ (5, 23), BоЂ№ (2, 28), and CоЂ№ (24, 25).
5
В© Carnegie Learning
24. The vertices of rectangle DEFG are D (27, 1), E (27, 8), F (1, 8), and G (1, 1). Reflect the rectangle
over the y-axis to form rectangle DоЂ№ EоЂ№ FоЂ№ GоЂ№.
25. The vertices of parallelogram HJKL are H (2, 26), J (3, 21), K (7, 21), and L (6, 26). Reflect the
parallelogram over the x-axis to form parallelogram HоЂ№ JоЂ№ KоЂ№ LоЂ№.
26. The vertices of trapezoid MNPQ are M (26, 25), N (0, 25), P (21, 2), and Q (24, 2). Reflect the
trapezoid over the y-axis to form trapezoid MоЂ№ NоЂ№ PоЂ№ QоЂ№.
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Lesson 5.1   Skills Practice
page 8
27. The vertices of triangle RST are R (0, 3), S (2, 7), and T (3, 21). Reflect the triangle over the x-axis to
form triangle RоЂ№ SоЂ№ TоЂ№.
28. The vertices of quadrilateral WXYZ are W (210, 8), X (22, 21), Y (0, 0), and Z (3, 7). Reflect the
quadrilateral over the y-axis to form quadrilateral WоЂ№ XоЂ№ YоЂ№ ZоЂ№.
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Lesson 5.2   Skills Practice
Name
Date
Hey, Haven’t I Seen You Before?
Congruent Triangles
Problem Set
Identify the transformation used to create nXYZ on each coordinate plane. Identify the congruent angles
and the congruent sides. Then, write a triangle congruence statement.
1.
Triangle BCA was reflected over the x-axis to
create triangle XYZ.
y
8
___ ___ ___
A
4
B
2
0
28 26 24 22
C
2
22
___ ___
BC > XY 
​  
​, CA ​
​  > YZ ​
​  
, and BA ​
​  > XZ ​
​  
; /B > /X,
/CВ > /Y, and /A > /Z.
6
4
6
X
8
nBCA > nXYZ
x
Y
24
26
Z
28
2.
5
y
X
8
В© Carnegie Learning
6
4
Z
Y
2
0
28 26 24 22
22
2
4
6
8
x
D
24
26
28
E
F
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Lesson 5.2   Skills Practice
3.
page 2
y
P
8
6
4
2
28 26 24 22
Z
X
M
0
2
T
4
6
8
x
22
24
26
28
Y
4.
y
8
B
D
X
6
Z
4
5
N
2
Y
0
28 26 24 22
2
4
6
8
x
22
24
28
522 
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Lesson 5.2   Skills Practice
page 3
Name
Date
5.
y
Y
8
6
X
4
Z
2
0
28 26 24 22
F
2
4
6
x
8
22
24
A
26
28
W
6.
y
8
Z
6
4
X
2
0
28 26 24 22
2
22
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24
5
Y
4
6
8
x
Q
M
26
28
R
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Lesson 5.2   Skills Practice
7.
page 4
y
Y
8
Z
N
6
4
R
2 G
X
28 26 24 22
0
2
4
6
8
2
4
6
8
x
22
24
26
28
8.
y
8
Y
6
4
5
X
Z
2
0
28 26 24 22
22
x
W
24
H
28
524 
F
В© Carnegie Learning
26
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Lesson 5.2   Skills Practice
page 5
Name
Date
9.
y
T
Y
A
8
Z
6
4
V
2
X
0
28 26 24 22
2
4
6
x
8
22
24
26
28
10.
y
M
8
6
5
4
G
2
28 26 24 22
Y
В© Carnegie Learning
X
B
0
2
4
6
8
x
22
24
26
28
Z
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Lesson 5.2   Skills Practice
page 6
List the corresponding sides and angles, using congruence symbols, for each pair of triangles
represented by the given congruence statement.
11. nJPM > nTRW
___ ___ ____ ____
___ ____
JP ​
​  
, PM ​
​  > RW ​
​  
, and JM ​
​  > TW ​
​  
; /J > /T, /P > /R, and /M > /W.
​  > TR ​
12. nAEU > nBCD
13. nLUV > nMTH
14. nRWB > nVCQ
15. nTOM > nBEN
16. nJKL > nRST
17. nCAT > nSUP
18. nTOP > nGUN
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Lesson 5.3   Skills Practice
Name
Date
It’s All About the Sides
Side-Side-Side Congruence Theorem
Vocabulary
Define the term in your own words.
1. Side-Side-Side (SSS) Congruence Theorem
Problem Set
Determine whether each pair of given triangles are congruent by SSS. Use the Distance Formula and
a protractor when necessary.
1.
AB 5 DE 5 3
y
AC 5 DF 5 7
8
B
___________________
6
    d 5 √
​ ​​( x2 2 x1 )2​​ ​1 (​​ y2  
2 y1 )2​​ ​ ​
4
BC 5 ​√(​​ 9 2 2 )2​​ ​1 (​​ 4   
2 7 )2​​ ​ ​
_________________
A
C
__________
BC 5 в€љ
​ ​72​ ​1 (2​3)​2​ ​ 
2
0
28 26 24 22
22
F
2
4
6
8
D
24
26
В© Carnegie Learning
28
x
_______
BC 5 в€љ
​ 49 1 9 ​ 
5
___
BC 5 ​√58 ​ < 7.62
___________________
E
    d 5 ​√​​( x2 2 x1 )2​​ ​1 (​​ y2  
2 y1 )2​​ ​ ​
______________________
  EF 5 ​√​​( 1 2 8 )​​2​1 ( ​​   
23 2 (26) )2​​ ​ ​
__________
  EF 5 √
​ ​​( 27 )2​​ ​1 ​32​ ​ ​ 
_______
  EF 5 √
​ 49 1 9 ​ 
___
  EF 5 ​√58 ​ < 7.62
BC 5 EF
The triangles are congruent by the
SSS Congruence Theorem.
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Lesson 5.3   Skills Practice
2.
page 2
y
J
8
6
4
G
H
2
0
28 26 24 22
2
22
L
4
6
x
8
K
24
26
28
M
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Lesson 5.3   Skills Practice
page 3
Name
Date
3.
y
B
8
6
4
R
G
2
0
28 26 24 22
2
4
6
8
x
22
D
M
24
26
28
A
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Lesson 5.3   Skills Practice
4.
page 4
y
8
P
6
4
M
N
2
0
28 26 24 22
Y
2
22
4
6
8
x
X
24
26
Z
28
В© Carnegie Learning
5
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Lesson 5.3   Skills Practice
page 5
Name
Date
5.
y
C
W
8
M
6
4
2
0
28 26 24 22
2
4
6
8
x
22
Q
24
P
26
S
28
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Lesson 5.3   Skills Practice
6.
page 6
y
8
N
6
P
4
2
Q
0
28 26 24 22
2
4
6
8
x
22
T
24
26
R
S
28
В© Carnegie Learning
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Lesson 5.3   Skills Practice
page 7
Name
Date
Perform the transformation described on each given triangle. Then, verify that the triangles are congruent
by SSS. Use the Distance Formula and a protractor when necessary.
7. Reflect nABC over the y-axis to form nXYZ. Verify that nABC > nXYZ by SSS.
  AB 5 XY 5 12
y
  BC 5 YZ 5 5
8
A
___________________
X
6
     d 5 √
​ ​​( x2 2 x1 )2​​ ​1 ​​( y2  
2 y1 )2​​ ​ ​
_______________________
4
  AC 5 √
​ ​​( 24 2 (29) )   
​​2​1 (​​ 26 2 6 )​​2​ ​
2
)2​​ ​ ​
  AC 5 √
​ ​52​ ​1 ​​( 212 
  
___________
0
28 26 24 22
2
4
6
x
8
_________
  AC 5 √
​ 25 1 144 ​ 
____
22
  AC 5 ​√ 169 ​ 5 13
24
___________________
B
C
26
28
Z
Y
    d 5 ​√​​( x2 2 x1 )​​2​1 (​​ y2  
2 y1 )​​2​ ​
___________________
   XZ 5 ​√​​( 4 2 9 )2​​ ​1 ​​( 26  
2 6 )2​​ ​ ​
______________
)2​​ ​ ​
   XZ 5 ​√​​( 25 )2​​ ​1 ​​( 212 
  
_________
   XZ 5 ​√25 1 144 ​ 
____
   XZ 5 ​√169 ​ 5 13
 AC 5 XZ
5
В© Carnegie Learning
The triangles are congruent by the
SSS Congruence Theorem.
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Lesson 5.3   Skills Practice
page 8
8. Rotate nDEF 180В° clockwise about the origin to form nQRS. Verify that nDEF > nQRS by SSS.
y
8
D
6
4
E
F
2
0
28 26 24 22
2
4
6
8
x
22
24
26
28
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Lesson 5.3   Skills Practice
page 9
Name
Date
9. Reflect nJKL over the x-axis to form nMNP. Verify that nJKL > nMNP by SSS.
y
8
6
4
2
0
28 26 24 22
2
4
6
x
8
22
24
J
K
26
28
L
В© Carnegie Learning
5
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Lesson 5.3   Skills Practice
page 10
10. Translate nHMZ 10 units to the left and 1 unit down to form nBNY. Verify that nHMZ > nBNY
by SSS.
y
8
6
M
4
H
2
0
28 26 24 22
2
4
6
8
x
22
24
26
Z
28
В© Carnegie Learning
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Lesson 5.3   Skills Practice
page 11
Name
Date
11. Rotate nAFP 90В° counterclockwise about the origin to form nDHW. Verify that nAFP > nDHW
by SSS.
y
A
8
6
4
P
2
F
0
28 26 24 22
2
4
6
8
x
22
24
26
28
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5
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Lesson 5.3   Skills Practice
page 12
12. Translate nACE 3 units to the right and 9 units up to form nJKQ. Verify that nACE > nJKQ by SSS.
y
8
6
4
2
0
28 26 24 22
2
4
6
8
x
22
A
24
26
C
28
E
В© Carnegie Learning
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Lesson 5.4   Skills Practice
Name
Date
Make Sure the Angle Is Included
Side-Angle-Side Congruence Theorem
Vocabulary
Describe how to prove the given triangles are congruent. Use the Side-Angle-Side Congruence Theorem
in your answer.
1.
y
A
8
6
C
B
4
2
0
28 26 24 22
22
2
4
X
6
8
x
Z
24
26
28
Y
В© Carnegie Learning
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Lesson 5.4   Skills Practice
page 2
Problem Set
Determine whether each pair of given triangles are congruent by SAS. Use the Distance Formula and a
protractor whenВ necessary.
1. Determine whether nABC is congruent to nDEF by SAS.
AB 5 DE 5 5
y
BC 5 EF 5 7
m/B 5 m/E 5 90В°
The triangles are congruent by the SAS
Congruence Theorem.
A
8
6
4
B
2
28 26 24 22
F
0
2
C
4
6
8
x
22
24
26
28
E
D
2. Determine whether nCKY is congruent to nDLZ by SAS.
y
C
8
5
L
Z
6
4
2
2
4
6
22
Y
K
24
26
28
540 
D
8
x
В© Carnegie Learning
0
28 26 24 22
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Lesson 5.4   Skills Practice
page 3
Name
Date
3. Determine whether nFMR is congruent to nJQW by SAS.
y
8
M
6
F
4
W
2
0
28 26 24 22
2
4
6
8
x
22
R
24
J
26
Q
28
4. Determine whether nQRS is congruent to nXYZ by SAS.
y
8
Q
5
R
6
4
В© Carnegie Learning
S
2
0
28 26 24 22
2
4
6
8
x
22
24
Z
26
Y
28
X
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Lesson 5.4   Skills Practice
page 4
5. Determine whether nJKL is congruent to nMNP by SAS.
y
K
8
J
6
4
L
2
0
28 26 24 22
2
4
6
8
x
22
P
24
M
26
28
N
В© Carnegie Learning
5
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Lesson 5.4   Skills Practice
page 5
Name
Date
6. Determine whether nATV is congruent to nDNP by SAS.
y
T
8
A
6
4
2
V
0
28 26 24 22
2
4
6
8
22
24
x
N
D
26
28
P
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Lesson 5.4   Skills Practice
page 6
Perform the transformation described on each given triangle. Then, verify that the triangles are congruent
by SAS. Use the Distance Formula and a protractor when necessary.
7. Reflect nABC over the y-axis to form nXYZ. Verify that nABC > nXYZ by SAS.
AB 5 XY 5 5
y
A
B
Y
8
AC 5 XZ 5 5
m/A 5 m/X 5 90В°
The triangles are congruent by the SAS
CongruenceВ Theorem.
X
6
4
C
Z
2
0
28 26 24 22
2
4
6
x
8
22
24
26
28
8. Translate nDEF 11 units to the left and 10 units down to form nQRS. Verify that nDEF > nQRS by SAS.
y
8
D
6
5
4
F
E
2
0
28 26 24 22
2
4
6
8
x
22
26
28
544 
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Lesson 5.4   Skills Practice
page 7
Name
Date
9. Rotate nJKL 180В° counterclockwise about the origin to form nMNP. Verify that nJKL > nMNP by SAS.
y
8
6
4
2
0
28 26 24 22
2
4
6
8
22
x
L
24
J
K
26
28
10. Reflect nAFP over the y-axis to form nDHW. Verify that nAFP > nDHW by SAS.
y
A
8
5
F
6
4
В© Carnegie Learning
2
0
28 26 24 22
2
4
6
x
8
22
24
26
28
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Lesson 5.4   Skills Practice
page 8
11. Translate nACE 4 units to the right and 4 units down to form nJKQ. Verify that nACE > nJKQ by SAS.
y
A
8
E
C
6
4
2
0
28 26 24 22
2
4
6
8
x
22
24
26
28
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Lesson 5.4   Skills Practice
page 9
Name
Date
12. Rotate nBMZ 90В° counterclockwise about the origin to form nDRT. Verify that nBMZ > nDRT by SAS.
y
B
8
M
6
4
Z
2
0
28 26 24 22
2
4
6
8
x
22
24
26
28
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Lesson 5.4   Skills Practice
page 10
Determine the angle measure or side measure that is needed in order to prove that each set of triangles
are congruent by SAS.
13. In nART, AR 5 12, RT 5 8, and m/R 5 70В°. In nBSW, BS 5 12, and m/S 5 70В°.
SW 5 8
14. In nCDE, CD 5 7, and DE 5 11. In nFGH, FG 5 7, GH 5 11, and m/G 5 45В°.
15. In nJKL, JK 5 2, KL 5 3, and m/K 5 60В°. In nMNP, NP 5 3, and m/N 5 60В°.
16. In nQRS, QS 5 6, RS 5 4, and m/S 5 20В°. In nTUV, TV 5 6, and UV 5 4.
17.
T
Z
11 in.
15 in.
5
D
15 in.
65В°
18.
11 in.
W
R
K
L
8 ft
T
548 
8 ft
M
X
20 ft
N
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Lesson 5.4   Skills Practice
page 11
Name
Date
19.
A
O
7m
C
50В°
7m
6m
G
6m
D
T
20.
V
E
14 ft
R
35В°
P
5
12 ft
12 ft
В© Carnegie Learning
W
35В°
M
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Lesson 5.4   Skills Practice
page 12
Determine whether there is enough information to prove that each pair of triangles are congruent by SSS
or SAS. Write the congruence statements to justify your reasoning.
?
?
21. nMNP >  nPQM
22. nWXY >  nZYX
W
N
X
30В°
18 in.
10 in.
7 ft
M
7 ft
P
10 in.
30В°
Y
18 in.
Z
Q
The triangles are congruent by SSS.
____ ___
___ ____
NP ​
​  > QM ​
​  
____ ____
MP ​
​  > PM ​
​  
MN ​
​  > PQ ​
​  
?
?
23. nBCE >  nDAF
24. nHJM >  nMKH
J
B
A
F
2 cm
2 cm
E
C
H
D
550 
M
В© Carnegie Learning
5
K
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Lesson 5.4   Skills Practice
page 13
Name
Date
?
?
25. nPQR >  nSTW
26. nMAT >  nMHT
S
Q
A
T
R
M
P
T
W
?
?
27. nBDW >  nBRN
5m
H
28. nABC >  nEDC
D
3m
N
A
B
B
C
5m
D
3m
W
E
В© Carnegie Learning
R
5
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5
552 
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Lesson 5.5   Skills Practice
Name
Date
Angle to the Left of Me, Angle to the Right of Me
Angle-Side-Angle Congruence Theorem
Vocabulary
Describe how to prove the given triangles are congruent. Use the Angle-Side-Angle Congruence Theorem
in your answer.
1.
y
8
P
N
6
4
2
M
0
28 26 24 22
A
2
4
6
8
x
22
24
26
C
28
B
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Lesson 5.5   Skills Practice
page 2
Problem Set
Determine whether each pair of given triangles are congruent by ASA. Use the Distance Formula and a
protractor when necessary.
1. Determine whether nABC is congruent to nDEF by ASA.
m/B 5 m/E 5 90В°
m/C 5 m/F 5 45В°
y
A
8
BC 5 EF 5 5
The triangles are congruent by the ASA
Congruence Theorem.
6
4
B
C
2
0
28 26 24 22
2
22
4
6
x
8
E
D
24
26
28
F
2. Determine whether nNPQ is congruent to nRST by ASA.
y
5
N
8
6
Q
P
2
0
28 26 24 22
S
2
22
24
26
4
6
8
x
В© Carnegie Learning
4
28
R
554 
T
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Lesson 5.5   Skills Practice
page 3
Name
Date
3. Determine whether nAGP is congruent to nBHQ by ASA.
y
P
8
6
4
A
G
2
0
28 26 24 22
2
4
6
8
x
22
Q
24
26
28
H
B
4. Determine whether nCKY is congruent to nDLZ by ASA.
y
8
5
C
K
6
Z
4
В© Carnegie Learning
2
0
28 26 24 22
2
4
6
8
x
22
24
Y
26
D
L
28
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Lesson 5.5   Skills Practice
page 4
5. Determine whether nFMR is congruent to nJQW by ASA.
y
M
8
6
R
4
2
F
28 26 24 22
0
J
22
2
24
4
6
x
8
W
26
28
Q
6. Determine whether nGHJ is congruent to nKLM by ASA.
y
G
8
5
H
6
4
L
2
0
28 26 24 22
4
6
x
8
22
24
26
M 28
K
556 
В© Carnegie Learning
J
2
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Lesson 5.5   Skills Practice
page 5
Name
Date
Perform the transformation described on each given triangle. Then, verify that the triangles are congruent
by ASA. Use the Distance Formula and a protractor when necessary.
7. Reflect nABC over the y-axis to form nXYZ. Verify that nABC > nXYZ by SAS. 
m/C 5 m/Z 5 90В°
m/A 5 m/X 5 63В°
y
8
AC 5 XZ 5 3
6
The triangles are congruent by the ASA
Congruence Theorem.
4
2
0
28 26 24 22
2
22
X
4
6
x
8
A
24
Y
26
Z
C
B
28
8. Rotate nDEF 90° clockwise about the origin to form nQRS. Verify that nDEF > nQRS by SAS. 
y
5
D
8
В© Carnegie Learning
6
4
E
F
2
0
28 26 24 22
2
4
6
8
x
22
24
26
28
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Lesson 5.5   Skills Practice
page 6
9. Translate nHMZ 6 units to the right and 10 units up to form nBNY. Verify that nHMZ > nBNY
by ASA. 
y
8
6
4
2
0
28 26 24 22
2
4
6
8
x
22
H
24
26
M
Z
28
10. Reflect nAFP over the y-axis to form nDHW. Verify that nAFP > nDHW by ASA.
y
5
A
F
8
6
4
0
28 26 24 22
2
22
P
24
26
28
558 
4
6
8
x
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Lesson 5.5   Skills Practice
page 7
Name
Date
11. Rotate nACE 180В° counterclockwise about the origin to form nJKQ. Verify that nACE > nJKQ by SAS.
y
C
8
6
4
A
2
E
0
28 26 24 22
2
4
6
8
x
22
24
26
28
12. Reflect nJKL over the x-axis to form nMNP. Verify that nJKL > nMNP by ASA.
y
J
5
8
K
6
L
4
В© Carnegie Learning
2
0
28 26 24 22
2
4
6
8
x
22
24
26
28
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Lesson 5.5   Skills Practice
page 8
Determine the angle measure or side measure that is needed in order to prove that each set of triangles
are congruent by ASA.
13. In nADZ, m/A 5 20В°, AD 5 9, and m/D 5 70В°. In nBEN, BE 5 9, and m/E 5 70В°.
m/B 5 20В°
14. In nCUP, m/U 5 45В°, and m/P 5 55В°. In nHAT, AT 5 14, m/A 5 45В°, and m/T 5 55В°.
15. In nHOW, m/H 5 10В°, HW 5 3, and m/W 5 60В°. In nFAR, FR 5 3, and m/F 5 10В°.
16. In nDRY, m/D 5 100В°, DR 5 25, and m/R 5 30В°. In nWET, m/W 5 100В°, and m/E 5 30В°.
17. B
D
W
T
8 ft
40В°
R
Z
8 ft
18. K
M
30В°
7 in.
60В°
X
30В°
60В°
T
560 
L
N
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Lesson 5.5   Skills Practice
page 9
Name
Date
19.
G
R
F
18 m
60В°
18 m
W
70В°
70В°
Z
S
20. A
T
P
40В°
5 in.
85В°
5
85В°
40В°
M
В© Carnegie Learning
X
D
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5
562 
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Lesson 5.6   Skills Practice
Name
Date
Sides Not Included
Angle-Angle-Side Congruence Theorem
Vocabulary
Describe how to prove the given triangles are congruent. Use the Angle-Angle-Side Congruence Theorem
in your answer.
y
1.
B
C
8
6
4
A
2
0
28 26 24 22
22
2
4
6
8
x
X
24
26
28
Z
Y
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Lesson 5.6   Skills Practice
page 2
Problem Set
Determine whether each set of given triangles are congruent by AAS. Use the Distance Formula and a
protractor when necessary.
1. Determine whether nABC is congruent to nDEF by AAS.
m/A 5 m/D 5 45В°
y
A
m/B 5 m/E 5 45В°
8
BC 5 EF 5 7
6
The triangles are congruent by the AAS
Congruence Theorem.
4
C
B
2
0
28 26 24 22
22
2
D
4
6
8
x
F
24
26
28
E
2. Determine whether nGHJ is congruent to nKLM by AAS.
5
y
G
8
6
2
J
0
28 26 24 22
L
H
2
22
4
M
24
6
8
x
В© Carnegie Learning
4
26
28
K
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Lesson 5.6   Skills Practice
page 3
Name
Date
3. Determine whether nAGP is congruent to nBHQ by AAS.
y
8
G
Q
6
B
4
2
0
28 26 24 22
2
4
6
8
x
22
24
A
P
26
H
28
4. Determine whether nCKY is congruent to nDLZ by AAS.
y
C
8
5
6
4
В© Carnegie Learning
Y
0
28 26 24 22
L
K
2
22
2
4
6
8
x
Z
24
26
28
D
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Lesson 5.6   Skills Practice
page 4
5. Determine whether nFMR is congruent to nJQW by AAS.
y
F
M
8
6
4
R
2
0
28 26 24 22
2
4
6
8
x
22
W
24
26
J
28
Q
6. Determine whether nNPQ is congruent to nRST by AAS.
y
P
T
8
5
6
N
4
2
0
28 26 24 22
2
4
6
8
x
24
R
26
Q
566 
28
S
В© Carnegie Learning
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Lesson 5.6   Skills Practice
page 5
Name
Date
Perform the transformation described on each given triangle. Then, verify that the triangles are congruent
by AAS. Use the Distance Formula and a protractor when necessary.
7. Reflect nABC over the y-axis to form nXYZ. Verify that nABC > nXYZ by AAS.
m/B 5 m/Y 5 76В°
y
m/C 5 m/Z 5 90В°
8
X
AC 5 XZ 5 12
6
A
The triangles are congruent by the AAS
Congruence Theorem.
4
2
0
28 26 24 22
2
4
6
8
x
22
24
26
Y
Z
C
28
B
8. Translate nDEF 11 units to the left and 11 units down to form nQRS. Verify that nDEF > nQRS by AAS.
y
5
D
8
6
В© Carnegie Learning
4
2
F
0
28 26 24 22
2
E
4
6
8
x
22
24
26
28
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Lesson 5.6   Skills Practice
page 6
9. Rotate nJKL 180В° counterclockwise about the origin to form nMNP. Verify that nJKL > nMNP by AAS.
y
8
6
4
2
0
28 26 24 22
K
2
4
22
6
8
x
L
24
26
J
28
10. Translate nCUP 9 units to the left and 4 units up to form nJAR. Verify that nCUP > nJAR by AAS.
y
8
6
5
4
U
2
0
28 26 24 22
2
4
6
8
x
24
26
28
568 
C
P
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22
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Lesson 5.6   Skills Practice
page 7
Name
Date
11. Reflect nAFP over the x-axis to form nDHW. Verify that nAFP > nDHW by AAS.
y
8
6
4
2
0
28 26 24 22
2
4
6
8
x
22
F
24
26
A
28
P
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5
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Lesson 5.6   Skills Practice
page 8
12. Rotate nACE 270В° counterclockwise about the origin to form nJKQ. Verify that nACE > nJKQ
by AAS.
y
8
C
6
A
4
2
0
28 26 24 22
2
4
6
8
x
22
24
E
26
28
Determine the angle measure or side measure that is needed in order to prove that each set of triangles
are congruent by AAS.
5
13. In nANT, m/A 5 30В°, m/N 5 60В°, and NT 5 5. In nBUG, m/U 5 60В°, and UG 5 5.
m/B 5 30В°
15. In nEMZ, m/E 5 40В°, EZ 5 7, and m/M 5 70В°. In nDGP, DP 5 7, and m/D 5 40В°.
В© Carnegie Learning
14. In nBCD, m/B 5 25В°, and m/D 5 105В°. In nRST, RS 5 12, m/R 5 25В°, and m/T 5 105В°.
16. In nBMX, m/M 5 90В°, BM 5 16, and m/X 5 15В°. In nCNY, m/N 5 90В°, and m/Y 5 15В°.
570 
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Lesson 5.6   Skills Practice
page 9
Name
Date
17.
B
D
18 in.
18 in.
T
40В°
R
40В°
70В°
W
Z
18.
K
X
60В°
30В°
T
3 ft
N
30В°
60В°
M
L
5
19.
F
S
В© Carnegie Learning
50В°
60В°
W
25 m
Z
60В°
50В°
R
G
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Lesson 5.6   Skills Practice
page 10
20. A
D
20 in.
30В°
110В°
P
20 in.
30В°
M
X
T
Determine whether there is enough information to prove that each pair of triangles are congruent by ASA
or AAS. Write the congruence statements to justify your reasoning.
?
?
22. nEFG > nHJK
21. nABD > nCBD
F
B
A
J
C
D
The triangles are congruent by AAS.
/BAD > /BCD
E
K
H
G
/ADB > /CDB
___ ___
BD ​
​  > BD ​
​  
?
?
24. nRST > nWZT
23. nMNQ > nPQN
N
W
P
S
M
Q
R
572 
T
Z
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Lesson 5.6   Skills Practice
page 11
Name
Date
?
?
25. nBDM > nMDH
26. nFGH > nJHG
D
F
J
G
H
55В° 55В°
B
H
60В°
60В°
M
?
?
28. nRST > nWXY
27. nDFG > nJMT
D
M
65В°
W
S
T
50В°
R
4 ft
4 ft
3 cm
70В°
70В°
50В°
В© Carnegie Learning
65В°
F
G
T
J
5
3 cm
X
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Y
  573
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5
574 
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Lesson 5.7   Skills Practice
Name
Date
Any Other Theorems You Forgot to Mention?
Using Congruent Triangles
Problem Set
Construct a perpendicular bisector to each line segment. Connect points on the bisector on either side of
the line segment to form the new line segment indicated.
____
___
1. MN​
​  bisected by PR​
​  at point Q
P
Q
M
N
R
___
5
____
2. HJ​
​  bisected by KM​
​  at point L
В© Carnegie Learning
H
J
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Lesson 5.7   Skills Practice
___
___
___
___
4.​PQ​ bisected by RT​
​  at point S
​  at point D
3. AB​
​  bisected by CE​
page 2
A
P
B
___
Q
___
___
____
6.​ST ​ bisected by UW​
​  at point V
5. FG​
​  bisected by HK​
​  at point J
F
T
G
576 
S
В© Carnegie Learning
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Lesson 5.7   Skills Practice
page 3
Name
Date
Use a triangle congruence theorem to complete each proof. Some of the statements and reasons are
provided for you.
___
___
7. Given: HK​
​  is a perpendicular bisector of FJ​
​  at point J
Prove: Point H is equidistant to points F and G
H
J
F
G
K
___ ___ ___
___
1.  FG ​
​  ' HK ​
​  
, HK ​
​  bisects FG 
​  
​
1. Definition of perpendicular bisector
2.  /FJH and /GJH are right angles.
2.  Definition of perpendicular lines
3.  /FJH  /GJH
3.  All right angles are congruent.
___
___
4.  FJ ​
​   GJ ​
​  
4.  Definition of bisect
5.  HJ 
​  
​ HJ 
​  
​
5.  Reflexive Property
6.  nFJH  nGJH
6.  SAS Congruence Theorem
В© Carnegie Learning
___ ___
___ ____
7. ​FH  
​ GH ​
​  
8. Point H is equidistant to points
F and G
7. Corresponding sides of congruent
triangles are congruent
8.  Definition of equidistant
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Lesson 5.7   Skills Practice
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___
____
​  at point N
8. Given: PM ​
​  is a perpendicular bisector of ST ​
Prove: Point M is equidistant to points S and T
P
S
N
T
M
1. Definition of perpendicular bisector
2. 
2.  Definition of perpendicular lines
3. 
3.  All right angles are congruent.
4.
4. 
5.
5. 
6. 
6. 
7.
7. 
8. Point M is equidistant to points
S and T
8.  Definition of equidistant
В© Carnegie Learning
5
1. 
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Name
Date
___
____
9. Given: CA​
​  is a perpendicular bisector of UW​
​  at point B
Prove: Point C is equidistant to points U and W
U
C
B
A
W
____
___ ___
____
1. Definition of perpendicular bisector
2. 
2. 
3. 
3.  All right angles are congruent.
4.
4.  Definition of bisect
5.
5. 
6. 
6. 
7.
7.  Corresponding sides of congruent
triangles are congruent
8. 
8. 
5
В© Carnegie Learning
1.  UW ​
​  ' CA ​
​  
, CA ​
​  bisects UW ​
​  
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Lesson 5.7   Skills Practice
___
page 6
___
​  at point R
10. Given: TK​
​  is a perpendicular bisector of LM​
Prove: Point T is equidistant to points L and M
L
T
R
K
M
1.
1. 
2. 
2.  Definition of perpendicular lines
3. 
3.  All right angles are congruent.
4.
4. 
___
5.  Reflexive Property
6. 
6. 
7.
7.  8. 
8.  Definition of equidistant
В© Carnegie Learning
5
___
5. ​TR ​  TR ​
​  
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Name
Date
___
___
11. Given: HT​
​  is a perpendicular bisector of UV​
​  at point P
Prove: Point H is equidistant to points U and V
V
H
P
T
U
1.
1. 
2.  /UPH and /VPH are right angles.
2.  Definition of perpendicular lines
3. 
3. 
4.
4.  Definition of bisect
5.
5. 
6. 
6. 
7.
7.  8. 
8. 
В© Carnegie Learning
5
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Lesson 5.7   Skills Practice
page 8
___
____
​  at point V
12. Given: UW​
​  is a perpendicular bisector of CD​
Prove: Point U is equidistant to points C and D
C
U
V
W
D
1. Definition of perpendicular bisector
2. 
2. 
3.  /CVU  /DVU
3.  All right angles are congruent.
4.
4. 
5.
5. 
6. 
6. 
7.
7.  Corresponding sides of congruent
triangles are congruent
8. 
8. 
В© Carnegie Learning
5
1.
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Lesson 5.7   Skills Practice
page 9
Name
Date
Complete each diagram to provide a counterexample that proves the indicated theorem does not work for
congruent triangles. Explain your reasoning. A hint is provided in each case.
13. Angle-Angle-Angle (Hint: Use vertical angles.)
D
A
C
B
___
___
xtend AC ​
E
​  and
​  
, and connect
___BC ​
___points D
and E so that AB ​
​  is parallel to DE ​
​  
. Since
vertical angles are congruent, all three
corresponding angles of the two triangles
are congruent. The side lengths, however,
are different, so nABC is not congruent
to nDEC.
E
14. Side-Side-Angle (Hint: Draw a triangle that shares /P with the given triangle.)
Q
5
P
R
15. Angle-Angle-Angle (Hint: Draw a triangle that shares /T with the given triangle.)
В© Carnegie Learning
T
S
U
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Lesson 5.7   Skills Practice
page 10
16. Side-Side-Angle (Hint: Draw a triangle that shares /L with the given triangle.)
L
K
M
17. Angle-Angle-Angle (Hint: Draw a triangle that has an angle with the same measure as /E.)
E
D
18. Side-Side-Angle (Hint: Draw a triangle that shares /U with the given triangle.)
T
U
V
584 
В© Carnegie Learning
5
F
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Lesson 5.7   Skills Practice
page 11
Name
Date
State the congruence theorem that proves the triangles in each diagram are congruent. If not enough
information is given, name an example of information that could be given that you could use to prove
congruency. Explain your reasoning.
____
____
W
19. Given: V W ​
​   XW ​
​  
Prove: nVYW оЂѕ nXYW
Not enough information is given.
If /VWY оЂѕ /XWY is given, then nVYW оЂѕ nXYW
by the SAS Triangle Congruence Theorem.
V
___
___
X
Y
C
20. Given: CE​
​  is a perpendicular bisector of FD​
​  
D
Prove: nFEC оЂѕ nDEC
E
5
F
В© Carnegie Learning
___
___
21. Given: /WST  /VUT, ST ​
​   UT ​
​  
S
U
Prove: nWST оЂѕ nVUT
T
W
V
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Lesson 5.7   Skills Practice
___
page 12
___
​  
22. Given: nABD is isosceles with AB ​
​   AD ​
B
Prove: nABC оЂѕ nADC
C
A
____
___ ___
___
D
H
23. Given: GH ​
​   JK ​
​  , HJ ​
​   KG ​
​  
Prove: nGHK оЂѕ nJKH
G
J
K
5
___
___
24. Given: PT ​
​   SR ​
​  
, /PQT оЂѕ /RQS P
R
T
586 
Q
S
В© Carnegie Learning
Prove: nTPQ оЂѕ nSRQ
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