Lesson 1.1 Skills Practice
1
Name
Date
Let’s Get This Started!
Points, Lines, Planes, Rays, and Line Segments
Vocabulary
Write the term that best completes each statement.
1. A geometric figure created without using tools is a(n)
2. .
are two or more lines that are not in the same plane.
3. A(n)
is a location in space.
4. The points where a line segment begins and ends are the
5. A(n)
.
is a straight continuous arrangement of an infinite number of points.
6. Two or more line segments of equal measure are
7. Y
ou
and straightedge.
.
a geometric figure when you use only a compass
8. Points that are all located on the same line are
.
9. A
(n)
is a portion of a line that includes two points and all of the
collinear points between the two points.
© Carnegie Learning
10. A flat surface is a(n)
11. A
(n)
one direction.
.
is a portion of a line that begins with a single point and extends infinitely in
12. Two or more lines located in the same plane are
13. W
hen you
compass, or protractor.
.
a geometric figure, you use tools such as a ruler, straightedge,
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Lesson 1.1 Skills Practice
1
page 2
Problem Set
Identify the point(s), line(s), and plane(s) in each figure.
1.
A
Points: A, B, and C
___›
‹
C
‹___›
Lines: ​AB
    
    and ​BC
Plane: m
B
m
2.
Z
X
Y
a
3.
p
R
S
4.
L
M
N
x
346 © Carnegie Learning
Q
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Lesson 1.1 Skills Practice
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page 3
Name
Date
Draw a figure for each description. Label all points mentioned in the description.
5. P
oints R, S, and T are collinear such that point T is located halfway between
points S and R.
T
S
R
6. P
oints A, D, and X are collinear such that point A is located halfway between
points D and X.
7. P
oints A, B, and C are collinear such that point B is between points A and C and the distance
between points A and B is twice the distance between points B and C.
8. P
oints F, G, and H are collinear such that point F is between points G and H
and the distance between points F and G is one third the distance between
points G and H.
© Carnegie Learning
Identify all examples of coplanar lines in each figure.
9. m
p
10.
c
q
d
n
a
b
Lines m and p are coplanar.
Lines n and q are coplanar.
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Lesson 1.1 Skills Practice
1
11. w
z
page 4
12.
q
x
p
y
r
u
t
s
Identify all skew lines in each figure.
13. 14.
g
h
a
c
f
b
Lines f and g are skew.
Lines f and h are skew.
15. 16.
n
m
x
y
348 l
© Carnegie Learning
w
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Lesson 1.1 Skills Practice
1
page 5
Name
Date
Draw and label an example of each geometric figure.
‹___›
___
17. ​XY    ​ 
18. CD ​
​  
Y
X
___›
___
‹
20. ​FG ​
    
19. PR ​
​  
‹____›
21. ​HM ​
  
 
© Carnegie Learning
 ___›
22. K
​  J ​  
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Lesson 1.1 Skills Practice
1
page 6
Use symbols to write the name of each geometric figure.
23.
R
T
24.
A
B
___
RT ​
​  
25.
26.
M
X
N
Y
27.
28.
C
S
D
R
Use a ruler to measure each segment to the nearest centimeter. Then use symbols to express the
measure of each segment.
29. A
___B
AB 5 4 centimeters or m AB ​
​  5 4 centimeters
30. A
B
32. A
350 © Carnegie Learning
31. A
B
B
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Lesson 1.2 Skills Practice
1
Name
Date
Attack of the Clones
Translating and Constructing Line Segments
Vocabulary
Choose the term from the box that best completes each statement.
Distance Formula
transformation
pre-image
rigid motion
translation
arc
copying (duplicating) a line segment
image
1. A(n)
is a transformation of points in space.
2. The new figure created from a translation is called the
3. A(n)
between two points on a circle.
.
is a part of a circle and can be thought of as the curve
4. A(n)
is the mapping, or movement, of all the points of a
figure in a plane according to a common operation.
© Carnegie Learning
5. The
points on a coordinate place.
can be used to calculate the distance between two
6. In a translation, the original figure is called the
7. A(n)
the same distance and direction.
.
is a rigid motion that “slides” each point of a figure
8. A basic geometric construction called
translate a line segment when measurement is not possible.
can be used to
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Lesson 1.2 Skills Practice
1
page 2
Problem Set
Calculate the distance between each given pair of points. Round your answer to the nearest tenth,
if necessary.
1. (3, 1) and (6, 5)
2. (2, 8) and (4, 3)
x1 5 3, y1 5 1, x2 5 6, y2 5 5
___________________
d 5 ​√(x2 2 x1​)2​ ​1 (y2  
2 y1​)2​ ​ ​
_________________
d 5 ​√(6 2 3​)2​ ​1 (5   
2 1​)2​ ​ ​
_______
d5√
​ ​32​ ​1 ​42​ ​ ​ 
_______
d5√
​ 9 1 16 ​ 
___
d 5 ​√25 ​ 
352 3. (26, 4) and (5, 21)
4. (9, 22) and (2, 29)
5. (0, 26) and (8, 0)
6. (25, 28) and (22, 29)
© Carnegie Learning
d55
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Lesson 1.2 Skills Practice
1
page 3
Name
Date
Calculate the distance between each given pair of points on the coordinate plane. Round your answer to
the nearest tenth, if necessary.
y
7.
x1 5 2, y1 5 8, x2 5 7, y2 5 3
___________________
d 5 ​√(x2 2 x1​)2​ ​1 (y2  
2 y1​)2​ ​ ​
8
_________________
d 5 ​√(7 2 2​)2​ ​1 (3   
2 8​)2​ ​ ​
6
__________
d5√
​ ​52​ ​1 ​(25)​2​ ​ 
4
________
d5√
​ 25 1 25 ​ 
2
___
0
28 26 24 22
2
4
6
8
2
4
6
8
x
22
d 5 ​√50 ​ 
d ¯ 7.1
24
26
28
y
8.
8
6
4
2
0
28 26 24 22
x
22
© Carnegie Learning
24
26
28
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Lesson 1.2 Skills Practice
1
page 4
y
9.
8
6
4
2
0
28 26 24 22
2
4
6
8
2
4
6
8
x
22
24
26
28
10.
y
8
6
4
2
0
28 26 24 22
x
22
24
26
© Carnegie Learning
28
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Lesson 1.2 Skills Practice
1
page 5
Name
Date
11.
y
8
6
4
2
0
28 26 24 22
2
4
6
8
2
4
6
8
x
22
24
26
28
12.
y
8
6
4
2
0
28 26 24 22
x
22
24
© Carnegie Learning
26
28
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Lesson 1.2 Skills Practice
1
page 6
Translate each given line segment on the coordinate plane as described.
___
___
14. Translate CD ​
​  9 units down.
13. Translate AB ​
​  8 units to the left.
y
A9
y
8
8
A
6
6
4
B9
B
0
2
4
2
6
8
x
22
24
24
26
26
28
2
4
6
8
4
6
8
x
28
____
___
15. Translate EF ​
​  7 units to the right.
16. Translate GH ​
​  12 units up.
8
8
6
6
4
4
2
2
0
28 26 24 22
E
F
y
2
4
6
8
x
0
28 26 24 22
22
22
24
24
26
26
28
28
2
x
G
H
© Carnegie Learning
y
356 0
28 26 24 22
22
D
4
2
28 26 24 22
C
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Lesson 1.2 Skills Practice
1
page 7
Name
Date
___
17. Translate JK ​
​  12 units down and 7 units to
the left.
____
18. Translate MN ​
​  5 units down and 10 units to
the right.
y
y
8
8
J
6
6
4
2
0
28 26 24 22
4
K
2
M
4
6
8
x
N
2
0
28 26 24 22
22
22
24
24
26
26
28
28
2
4
6
8
x
Construct each line segment described.
___
___
19. Duplicate AB ​
​  
.
20. Duplicate CD ​
​  
.
A
B
A9
B9
C
____
___
© Carnegie Learning
21. Duplicate EF ​
​  .
E
D
22. Duplicate GH ​
​  
.
F
G
H
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Lesson 1.2 Skills Practice
1
page 8
___
23. Construct a line segment twice the length of JK ​
​  .
J
K
 ​
____
24. Construct a line segment twice the length of MN ​
​  
.
M
N
© Carnegie Learning
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Lesson 1.3 Skills Practice
1
Name
Date
Stuck in the Middle
Midpoints and Bisectors
Vocabulary
Match each definition to the corresponding term.
1. midpoint
a. a line, line segment, or ray that divides a line
segment into two line segments of equal measure
2. Midpoint Formula
b. a basic geometric construction used to locate the
midpoint of a line segment
3. segment bisector
c. a point exactly halfway between the endpoints of
a line segment
4. bisecting a line segment
( 
)
y 1 y2
x1 1 x2 _______
 ​ 
, ​  1  ​ 
 
 
 ​
d. ​​ _______
2
2
Problem Set
Determine the midpoint of a line segment with each set of given endpoints.
2. (3, 8) and (9, 10)
1. (8, 0) and (4, 6)
x1 5 8, y1 5 0
© Carnegie Learning
x2 5 4, y2 5 6
y 1y
x 1 x _______
0 1 ​ 
6 
4 
 ​ 
, ​ 
 ​  
, ​ ______
 
 ​5 ​ ______
​ 8 1 ​
 ​
(​ ​ _______
2
2 ) ( 2
2 )
1
2
1
2
( ___  __ )
5 ​ ​ 12 ​ , ​ 6 ​   ​
2 2
5 (6, 3)
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Lesson 1.3 Skills Practice
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3. (27, 2) and (3, 6)
4. (6, 23) and (24, 5)
5. (210, 21) and (0, 4)
6. (22, 7) and (28, 29)
Determine the midpoint of the given line segment on each coordinate plane using the Midpoint Formula.
x1 5 3, y1 5 2
y
x2 5 7, y2 5 8
8
y
x 1 x _______
2 1 ​ 
8 
7 
 ​ 
, ​ ______
 , ​ y 12 ​ 
 
 )​5 (​ ______
​ 3 1 ​
 ​
(​ ​ _______
2
2
2 )
1
6
1
2
( ___  ___ )
5 ​ ​ 10 ​ , ​ 10 ​  ​
2 2
4
2
0
28 26 24 22
2
2
22
24
4
6
8
x
5 (5, 5)
© Carnegie Learning
7.
26
28
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Lesson 1.3 Skills Practice
1
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Name
Date
8.
y
8
6
4
2
0
28 26 24 22
2
4
6
8
2
4
6
8
x
22
24
26
28
9.
y
8
6
4
2
0
28 26 24 22
x
22
24
© Carnegie Learning
26
28
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Lesson 1.3 Skills Practice
1
10.
page 4
y
8
6
4
2
0
28 26 24 22
2
4
6
8
2
4
6
8
x
22
24
26
28
11.
y
8
6
4
2
0
28 26 24 22
x
22
24
26
© Carnegie Learning
28
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Lesson 1.3 Skills Practice
1
page 5
Name
Date
12.
y
8
6
4
2
0
28 26 24 22
2
4
6
8
x
22
24
26
28
Locate the midpoint of each line segment using construction tools and label it point M.
13.
M
B
© Carnegie Learning
A
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Lesson 1.3 Skills Practice
1
14.
page 6
C
D
15.
E
© Carnegie Learning
F
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Lesson 1.3 Skills Practice
1
page 7
Name
Date
16.
G
17.
H
K
© Carnegie Learning
J
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Lesson 1.3 Skills Practice
1
page 8
18.
Q
© Carnegie Learning
P
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Lesson 1.4 Skills Practice
1
Name
Date
What’s Your Angle?
Translating and Constructing Angles and Angle Bisectors
Vocabulary
Define each term in your own words.
1. angle
2. angle bisector
Describe how to perform each construction in your own words.
© Carnegie Learning
3. copying or duplicating an angle
4. bisecting an angle
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Lesson 1.4 Skills Practice
1
page 2
Problem Set
Translate each given angle on the coordinate plane as described.
2. Translate /DEF 12 units down.
1. Translate /ABC 9 units to the left.
y
D
A
8
8
6
6
4
B9
B
C9 2
0
28 26 24 22
E
C
2
4
6
8
F
2
x
0
28 26 24 22
22
22
24
24
26
26
28
28
2
4
3. Translate /GHJ 10 units to the right.
4. Translate /KLM 13 units up.
y
8
6
6
4
4
2
2
0
2
4
6
8
x
24
26
J
8
6
8
x
2
4
x
22
K
24
G
0
28 26 24 22
22
H
6
y
8
28 26 24 22
368 4
26
28
28
L
M
© Carnegie Learning
A9
y
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Lesson 1.4 Skills Practice
1
page 3
Name
Date
5. Translate /NPQ 8 units to the left and
11 units down.
6. Translate /RST 15 units to the left and
9 units up.
y
y
8
8
P
6
6
4
4
N
2
0
28 26 24 22
2
Q
2
4
6
8
x
0
28 26 24 22
22
22
24
24
26
26
28
28
2
4
6
x
8
T
R
S
Construct each angle as described.
7. Copy /B.
8. Copy /D.
C
B
D
D
© Carnegie Learning
S
R
T
/CBD > /SRT
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Lesson 1.4 Skills Practice
1
page 4
9. Copy /P.
10. Copy /Z.
P
Z
11. Construct an angle that is twice the measure of /K.
K
© Carnegie Learning
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Lesson 1.4 Skills Practice
1
page 5
Name
Date
12. Construct an angle that is twice the measure of /M.
M
Construct the angle bisector of each given angle.
13.
14.
A
D
P
B
B
___
© Carnegie Learning
PD ​
​  is the angle bisector of /P.
16.
15.
S
X
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Lesson 1.4 Skills Practice
1
page 6
17. Construct an angle that is one-fourth the measure of /F.
F
18. Construct an angle that is one-fourth the measure of /X.
X
© Carnegie Learning
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Lesson 1.5 Skills Practice
1
Name
Date
If You Build It . . .
Constructing Perpendicular Lines, Parallel Lines, and Polygons
Problem Set
Construct a line perpendicular to each given line and through the given point.
‹___›
1. Construct a line that is perpendicular to ​CD​
    and passes through point T.
T
C
D
‹___›
© Carnegie Learning
2. Construct a line that is perpendicular to ​AB​
    and passes through point X.
A
X
B
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Lesson 1.5 Skills Practice
1
page 2
‹___›
3. Construct a line that is perpendicular to ​RS​
    and passes through point W.
R
W
S
‹___›
4. Construct a line that is perpendicular to ​YZ​
    and passes through point G.
Y
Z
© Carnegie Learning
G
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Lesson 1.5 Skills Practice
1
page 3
Name
Date
‹____›
5. Construct a line that is perpendicular to ​MN​
    and passes through point J.
J
N
M
‹___›
6. Construct a line that is perpendicular to ​PQ​
    and passes through point R.
P
R
© Carnegie Learning
Q
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Lesson 1.5 Skills Practice
1
page 4
Construct a line parallel to each given line and through the given point.
‹___›
7. Construct a line that is parallel to ​AB​
    and passes through point C.
C
q
A
B
‹___›
Line q is parallel to ​AB​
.
    
‹___›
8. Construct a line that is parallel to ​DE​
    and passes through point F.
E
F
376 © Carnegie Learning
D
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Lesson 1.5 Skills Practice
1
page 5
Name
Date
‹___›
9. Construct a line that is parallel to ​GH​
    and passes through point J.
G
J
H
‹___›
10. Construct a line that is parallel to ​KL​
    and passes through point M.
K
© Carnegie Learning
M
L
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Lesson 1.5 Skills Practice
1
page 6
‹___›
11. Construct a line that is parallel to ​NP​
    and passes through point Q.
N
Q
P
‹___›
12. Construct a line that is parallel to ​RT​
    and passes through point W.
R
T
378 © Carnegie Learning
W
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Lesson 1.5 Skills Practice
Name
1
page 7
Date
Construct each geometric figure.
13. Construct an equilateral triangle. The length of one side is given.
© Carnegie Learning
14. Construct an equilateral triangle. The length of one side is given.
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Lesson 1.5 Skills Practice
1
page 8
15. Construct an isosceles triangle that is not an equilateral triangle such that each leg is longer than
the base. The length of the base is given.
© Carnegie Learning
16. Construct an isosceles triangle that is not an equilateral triangle such that each leg is shorter than
the base. The length of the base is given.
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Lesson 1.5 Skills Practice
Name
1
page 9
Date
17. Construct a square. The perimeter of the square is given.
© Carnegie Learning
18. Construct a square. The perimeter of the square is given.
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Lesson 1.5 Skills Practice
1
page 10
19. Construct a rectangle that is not a square. The perimeter of the rectangle is given.
© Carnegie Learning
20. Construct a rectangle that is not a square. The perimeter of the rectangle is given.
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Lesson 1.6 Skills Practice
Name
1
Date
What’s the Point?
Points of Concurrency
Vocabulary
Describe similarities and differences between each pair of terms.
1. concurrent and point of concurrency
2. incenter and orthocenter
© Carnegie Learning
3. centroid and circumcenter
4. altitude and median
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Lesson 1.6 Skills Practice
1
page 2
Problem Set
Draw the incenter of each triangle.
1. 2.
3. 5. 7. 4.
6.
8.
Draw the circumcenter of each triangle.
10.
© Carnegie Learning
9. 384 Chapter 1 Skills Practice
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Name
Date
11. 12.
13. 14.
15. 16.
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Draw the centroid of each triangle.
17. 18.
19. 20.
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21. 22.
23. 24.
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386 25. 26.
27. 28.
29. 30.
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Draw the orthocenter of each triangle.
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31. 1
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32.
Answer each question about points of concurrency. Draw an example to illustrate your answer.
33. For which type of triangle are the incenter, circumcenter, centroid, and orthocenter the same point?
equilateral triangles
34. For which type of triangle are the orthocenter and circumcenter outside of the triangle?
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35. For which type of triangle are the circumcenter and orthocenter on the triangle?
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36. F
or which type of triangle are the incenter, circumcenter, centroid, and orthocenter
all inside the triangle?
37. F
or what type(s) of triangle(s) do the centroid, circumcenter, and orthocenter all lie
on a straight line?
388 © Carnegie Learning
38. For what type of triangle is the orthocenter a vertex of the triangle?
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Name
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Given the coordinates of the vertices of a triangle, classify the triangle using algebra.
39. A(25, 5), B(5, 5), C(0, 25)
segment AB
segment AC
segment BC
d5
d5
d 5 ​√(0 2 5​)2​ ​1 (2 5  
2 5​)2​ ​ ​
d5
d5
_____________________
2
​ [5 2 (2 5)​]   
​ ​1 (5 2 5​)2​ ​ ​
√
________
​√1​02​ ​1 ​02​ ​ ​ 
____
√
​ 100 ​ 
d 5 10
d5
d5
______________________
​√[0 2 (25)​]2​ ​   
1 (25 2 5​)2​ ​ ​
____________
  
0)​2 ​ ​
√​ ​52​ ​1 (2 1​
____
​√125 ​
 
d ¯ 11.18
____________________
______________
d5√
​ (25​)2​ ​1 (210​
  
)2​ ​ ​
____
d 5 ​√125 ​ 
d ¯ 11.18
The lengths of two of the segments are equal, so the triangle is isosceles.
40. R(23, 21), S(1, 2), T(4, 22)
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41. F(22, 5), G(1, 6), H(5, 24)
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42. M(5, 21), N(3, 25), P(21, 23)
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43. K(22, 1), L(4, 23), M(21, 5)
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1
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44. E(25, 7), F(3, 4), G(28, 21)
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1
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