LESSON 10.1
Skills Practice
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Squares and Rectangles
Properties of Squares and Rectangles
Vocabulary
Define the term in your own words.
1. Explain the Perpendicular/Parallel Line Theorem in your own words.
10
Problem Set
Use the given statements and the Perpendicular/Parallel Line Theorem to identify the pair of parallel lines
in each figure.
1. Given: l ' m and l ' r
2. k ' b and g ' b
l
g
k
h
b
m
p
r
mir
© Carnegie Learning
3. Given: z ' g and z ' p
4. Given: c ' t and t ' e
c
z
v
t
g
e
b
i
p
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LESSON 10.1
Skills Practice
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5. Given: n ' r and r ' q
y
n
6. Given: b ' x and k ' b
b
q v
x
r
e
z
k
Complete each statement for square GKJH.
10
G
K
E
H
J
___
___
____
___
7. GK > KJ > JH > HG
8. /KGH > /
>/
9. /GEK, /
___
10. GK i
___
11. GE >
,/
>/
,/
>/
,/
,/
>/
,/
>/
, and /
>/
are right angles.
____
and GH i
>
>/
>/
>/
>/
>/
>/
© Carnegie Learning
12. / KGE > /
>
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Chapter 10 Skills Practice
LESSON 10.1
Skills Practice
page 3
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Complete each statement for rectangle TMNU.
M
N
L
T
___
____
____
___
U
10
13. MN > TU and MT > NU
14. /NMT > /
>/
15. /MTU, /
____
,/
>/
, and /
are right angles.
___
16. MN i
and MT i
____
17. MU >
___
18. ML >
>
>
Construct each quadrilateral using the given information.
___
___
___
19. Use AB to construct square ABCD with diagonals AC and BD intersecting at point E.
A
B
© Carnegie Learning
D
C
E
A
B
Chapter 10 Skills Practice
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LESSON 10.1
Skills Practice
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page 4
____
___
____
___
20. Use WX to construct square WXYZ with diagonals WY and XZ intersecting at point P.
W
X
10
___
21. Use OP to construct square OPQR with diagonals OQ and PR intersecting at point T.
P
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Chapter 10 Skills Practice
LESSON 10.1
Skills Practice
page 5
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___
___
___
22. Use RE to construct rectangle RECT with diagonals RC and ET intersecting at point A. Do not
construct a square.
R
E
10
___
___
___
23. Use QR to construct rectangle QRST with diagonals QS and RT intersecting at point P. Do not
construct a square.
R
© Carnegie Learning
Q
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LESSON 10.1
Skills Practice
page 6
____
____
___
24. Use MN to construct rectangle MNOP with diagonals MO and NP intersecting at point G. Do not
construct a square.
M
N
10
Create each proof.
___
___
25. Write a paragraph proof to prove AC > BD in square ABCD.
B
D
C
___
___
You are given square ABCD with diagonals AC and DB. By definition of a square, /ADC and
/BCD are right angles. Because all right angles___
are congruent,
you can conclude that
___
___ ___
/ADC > /BCD. Also by definition of a square, AD > BC . By the Reflexive
___ ___Property, DC > DC .
Therefore, n ADC > n BCD by the SAS Congruence Theorem. So, AC > BD by CPCTC.
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LESSON 10.1
Skills Practice
page 7
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___
__
___
__
26. Write a paragraph proof to prove GL > IL and HL > JL in square GHIJ.
H
G
L
I
J
10
___
___
27. Create a two-column proof to prove ML ' NP in square MNLP.
M
N
Q
L
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LESSON 10.1
Skills Practice
___
page 8
___
28. Create a two-column proof to prove RT > SU in rectangle RSTU.
R
S
V
U
T
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Chapter 10 Skills Practice
LESSON 10.1
Skills Practice
page 9
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___
___
___
____
29. Create a two-column proof to prove LK > KN and OK > KM in rectangle LMNO.
M
L
K
O
N
10
30. Write a paragraph proof to prove opposite sides are parallel in rectangle WXYZ.
© Carnegie Learning
W
X
A
Z
Y
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Chapter 10 Skills Practice
LESSON 10.2
Skills Practice
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Parallelograms and Rhombi
Properties of Parallelograms and Rhombi
Vocabulary
1. Explain how the Parallelogram/Congruent-Parallel Side Theorem can be used to determine if a
quadrilateral is a parallelogram.
10
Problem Set
Complete each statement for parallelogram MNPL.
M
N
R
L
____
___
___
P
___
1. MN > LP and ML > NP
2. /NML > /
____
3. MN i
____
___
and ML i
___
and LR >
© Carnegie Learning
4. MR >
and /MLP > /
Chapter 10 Skills Practice
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LESSON 10.2
Skills Practice
page 2
Complete each statement for rhombus UVWX.
U
V
E
X
W
____
___
____
___
5. UV > VW > WX > XU
6. /UVW > /
10
___
and /XUV > /
___
7. UV i
and UX i
___
___
8. UE >
and XE >
Construct each quadrilateral using the given information.
___
___
___
9. Use RS to construct parallelogram RSTU with diagonals RT and SU intersecting at point G.
R
S
U
T
G
S
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LESSON 10.2
Skills Practice
page 3
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___
___
___
10. Use AB to construct parallelogram ABCD with diagonals AC and BD intersecting at point F.
A
B
10
____
____
___
11. Use WX to construct parallelogram WXYZ with diagonals WY and XZ intersecting at point H.
W
X
___
___
___
12. Use AB to construct rhombus ABCD with diagonals AC and BD intersecting at point K. Do not
construct a square.
B
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A
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LESSON 10.2
Skills Practice
____
page 4
____
___
13. Use WX to construct rhombus WXYZ with diagonals WY and XZ intersecting at point V. Do not
construct a square.
w
X
10
___
___
___
14. Use EF to construct rhombus EFGH with diagonals EG and FH intersecting at point T. Do not
construct a square.
F
E
Determine the missing statement needed to prove each quadrilateral is a parallelogram by the
Parallelogram/Congruent-Parallel Side Theorem.
___
____
___
16. PS > QR
X
Y
Q
P
R
Z
W
___ ____
XY > ZW
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15. XY i ZW
LESSON 10.2
Skills Practice
page 5
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___
____
___
17. MN > PQ
___
18. CF i DE
C
M
D
F
E
P
N
Q
___
10
___
___
19. ST i VU
___
20. KG > HL
S
K
T
H
G
L
V
U
Create each proof.
21. Write a paragraph proof to prove /BAD > /DCB in parallelogram ABCD.
© Carnegie Learning
A
D
B
C
___
___
We
parallelogram
ABCD with diagonals AC and DB . By definition of a parallelogram,
___given___
___
___ are
AB i CD and AD i BC . So, /ABD > /CDB
and /ADB > /DBC by the Alternate Interior Angle
___ ___
Theorem. By the Reflexive Property, DB > DB . Therefore, nABD > nCDB by the ASA Congruence
Theorem. So, /BAD > /DCB by CPCTC.
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LESSON 10.2
Skills Practice
___
page 6
__
___
___
22. Create a two-column proof to prove GL > IL and JL > HL in parallelogram GHIJ.
G
H
L
J
I
10
23. Write a paragraph proof to prove opposite sides are congruent in parallelogram HIJK.
H
J
© Carnegie Learning
K
I
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LESSON 10.2
Skills Practice
page 7
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___
___
24. Create a two-column proof to prove RT ' SU in rhombus RSTU.
R
S
M
U
T
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Chapter 10 Skills Practice
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LESSON 10.2
Skills Practice
page 8
25. Create a two-column proof to prove /QPR > /SPR, /QRP > /SRP, /PSQ > /RSQ, and
/PQS > /RQS in rhombus PQRS.
P
Q
A
S
R
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Chapter 10 Skills Practice
LESSON 10.2
Skills Practice
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26. Create a two-column proof to prove rhombus RSTU is a parallelogram.
R
U
S
T
10
Use the given information to answer each question.
27. Tommy drew a quadrilateral. He used a protractor to measure all four angles of the quadrilateral. How
many pairs of angles must be congruent for the quadrilateral to be a parallelogram? Explain.
© Carnegie Learning
Opposite angles are congruent in a parallelogram, so both pairs of opposite angles must be
congruent.
28. Khyree cut a quadrilateral out of a piece of cardstock, but is not sure if the figure is a parallelogram or
a rhombus. He measures the lengths of the opposite sides and determines them to be congruent. He
measures the opposite angles of the quadrilateral and determines them also to be congruent. He
measures one angle and is able to determine that the quadrilateral is a rhombus. What angle did he
measure? Explain.
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LESSON 10.2
Skills Practice
page 10
29. Penny makes the following statement: “Every rhombus is a parallelogram.” Do you agree? Explain.
10
30. Sally plans to make the base of her sculpture in the shape of a rhombus. She cuts out four pieces of
wood to create a mold for concrete. The pieces of wood are the following lengths: 5 inches, 5 inches,
3 inches, and 3 inches. Will the base of Sally’s sculpture be a rhombus? Explain.
32. Three angles of a parallelogram have the following measures: 58°, 122°, and 58°. What is the
measure of the fourth angle? How do you know?
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Chapter 10 Skills Practice
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31. Ronald has a picture in the shape of a quadrilateral that he cut out of a magazine. How could Ronald
use a ruler to prove that the picture is a parallelogram?
LESSON 10.3
Skills Practice
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Kites and Trapezoids
Properties of Kites and Trapezoids
Vocabulary
Write the term from the box that best completes each statement.
base angles of a trapezoid
biconditional statement
midsegment
isosceles trapezoid
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1. The
common side.
are either pair of angles of a trapezoid that share a base as a
2. A(n)
is a trapezoid with congruent non-parallel sides.
3. A(n)
is a statement that contains if and only if.
4. The
of a trapezoid is a segment formed by connecting the midpoints
of the legs of the trapezoid.
Problem Set
Complete each statement for kite PRSQ.
___
___
___
___
1. PQ > QS and PR > SR
2. /QPR > /
P
Q
T
R
© Carnegie Learning
___
3. PT >
4. /PQT > /
S
and /PRT > /
Chapter 10 Skills Practice
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LESSON 10.3
Skills Practice
page 2
Complete each statement for trapezoid UVWX.
___
____
5. The bases are UV and WX .
6. The pairs of base angles are /
and /
and /
.
7. The legs are
8. The vertices are
10
and
U
and /
,
.
,
X
,
, and
V
W
.
Construct each quadrilateral using the given information.
___
___
9. Construct kite QRST with diagonals QS and RT intersecting at point M.
Q
S
R
T
Q
T
M
R
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Chapter 10 Skills Practice
LESSON 10.3
Skills Practice
page 3
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____
___
10. Construct kite LMNO with diagonals MO and LN intersecting at point A.
M
O
L
N
10
____
___
11. Construct kite UVWX with diagonals UW and XV intersecting at point K.
U
V
© Carnegie Learning
X
W
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LESSON 10.3
Skills Practice
page 4
___
12. Construct trapezoid ABCD with AB as a base.
A
B
10
___
13. Construct trapezoid HGYU with HG as a base.
H
G
___
14. Construct trapezoid BNJI with BN as a base.
N
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LESSON 10.3
Skills Practice
page 5
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Use the given figure to answer each question.
15. The figure shown is a kite with /DAB > /DCB. Which sides of the kite are congruent?
A
B
D
C
___
___
___
___
AB and CB are congruent.
10
AD and CD are congruent.
___
___
16. The figure shown is a kite with FG > FE. Which of the kite’s angles are congruent?
E
F
H
G
__
© Carnegie Learning
17. Given that IJLK is a kite, what kind of triangles are formed by diagonal IL ?
I
J
K
L
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LESSON 10.3
Skills Practice
page 6
____
18. Given that LMNO is a kite, what is the relationship between the triangles formed by diagonal MO ?
N
M
O
L
10
19. Given that PQRS is a kite, which angles are congruent?
P
Q
S
R
20. Given that TUVW is a kite, which angles are congruent?
T
W
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Chapter 10 Skills Practice
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LESSON 10.3
Skills Practice
page 7
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Use the given figure to answer each question.
___
___
21. The figure shown is an isosceles trapezoid with AB || CD. Which sides are congruent?
A
B
C
D
___
___
AC and BD are congruent.
10
___
___
22. The figure shown is an isosceles trapezoid with EH > FG. Which sides are parallel?
E
F
H
G
__
___
23. The figure shown is an isosceles trapezoid with IJ > KL. What are the bases?
I
J
K
© Carnegie Learning
L
____
___
24. The figure shown is an isosceles trapezoid with MP > NO. What are the pairs of base angles?
P
M
N
O
Chapter 10 Skills Practice
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LESSON 10.3
Skills Practice
page 8
___
___
___
___
25. The figure shown is an isosceles trapezoid with PQ i RS . Which sides are congruent?
P
Q
R
10
S
26. The figure shown is an isosceles trapezoid with LM i KN . What are the pairs of base angles?
K
L
M
N
Create each proof.
27. Write a paragraph proof to prove /ABC > /ADC in kite ABCD.
A
E
C
D
___
___
You are given ___
kite ABCD
diagonals
___ BD and AC intersecting at point E. By definition
___ ___of a kite,
___ with___
we know that AB > AD and BC > CD . By the Reflexive Property, you know that AC > AC .
Therefore, n ABC > n ADC by the SSS Congruence Theorem. So, /ABC > /ADC by CPCTC.
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LESSON 10.3
Skills Practice
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___
___
28. Write a two-column proof to prove HF > JF in kite GHIJ.
I
H
F
J
10
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LESSON 10.3
Skills Practice
page 10
____
29. Write a paragraph proof to prove WZ bisects /XWY and /XZY in kite WYZX.
W
X
P
Y
Z
10
30. Write a paragraph proof to prove /U > /T in isosceles trapezoid RSTU.
R
X
T
© Carnegie Learning
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LESSON 10.3
Skills Practice
page 11
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31. Write a two-column proof to prove /K > /L in isosceles trapezoid JKLM.
K
J
L
N
M
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LESSON 10.3
Skills Practice
page 12
___
___
32. Write a two-column
proof to prove that diagonals RT and SU in isosceles trapezoid RSTU are
___
___
congruent if RU > ST .
S
R
T
U
10
Construct each isosceles trapezoid using the given information.
___
33. Construct isosceles trapezoid ABCD if PS is the perimeter of the trapezoid.
P
S
B
C
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P
LESSON 10.3
Skills Practice
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34. Construct isosceles trapezoid WXYZ if AD is the perimeter of the trapezoid.
A
D
10
____
35. Construct isosceles trapezoid EFGH if WZ is the perimeter of the trapezoid.
W
Z
___
© Carnegie Learning
36. Construct isosceles trapezoid PQRS if FJ is the perimeter of the trapezoid.
F
J
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LESSON 10.3
Skills Practice
page 14
____
37. Construct isosceles trapezoid JKLM if TW is the perimeter of the trapezoid.
T
W
10
___
38. Construct isosceles trapezoid STUV if EH is the perimeter of the trapezoid.
E
H
39. Alice created a kite out of two sticks and some fabric. The sticks were 10 inches and 15 inches long.
She tied the sticks together so they were perpendicular and attached the fabric. When she measured
the kite, she noticed that the distance from where the sticks meet to the top of the kite was 5 inches.
What is the area of the kite Alice created?
1 (5)(5) 1 2 __
1 (5)(10)
A 5 2 __
2
2
( )
( )
A 5 25 1 50
A 5 75
The area of the kite is 75 square inches.
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Chapter 10 Skills Practice
© Carnegie Learning
Use the given information to answer each question.
LESSON 10.3
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Skills Practice
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40. Simon connected a square and two congruent right triangles together to form an isosceles trapezoid.
Draw a diagram to represent the isosceles trapezoid.
41. Magda told Sam that an isosceles trapezoid must also be a parallelogram because there is a pair of
congruent sides in an isosceles trapezoid. Is Magda correct? Explain.
10
42. Sylvia drew what she thought was an isosceles trapezoid. She measured the base angles and
determined that they measured 81°, 79°, 101° and 99°. Could her drawing be an isosceles
trapezoid? Explain.
© Carnegie Learning
43. Joanne constructed a kite with a perimeter of 38 centimeters so that the sum of the two shorter sides
is 10 centimeters. What are the lengths of each of the two longer sides?
44. Ilyssa constructed a kite that has side lengths of 8 inches and 5 inches. What are the lengths of the
other two sides? Explain.
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Chapter 10 Skills Practice
LESSON 10.4
Skills Practice
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Interior Angles of a Polygon
Sum of the Interior Angle Measures of a Polygon
Vocabulary
Give an example of the term.
1. Draw an example of a polygon. Label the interior angles of the polygon.
10
Problem Set
Draw all possible diagonals from vertex A for each polygon. Then write the number of triangles formed by
the diagonals.
1.
2.
A
A
© Carnegie Learning
2 triangles
3.
4.
A
A
Chapter 10 Skills Practice
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LESSON 10.4
Skills Practice
5.
page 2
6.
A
A
Use the triangles formed by diagonals to calculate the sum of the interior angle measures of
each polygon.
10
7. Draw all of the diagonals that connect to vertex A. What is the sum of the interior angles of
square ABCD?
A
B
D
C
The diagonal divides the figure into two triangles. The sum of the interior angles of each triangle is
180º, so I multiplied 180º by 2 to calculate the sum of the interior angles of the figure:
180º 3 2 5 360º
The sum of the interior angles is 360º.
8. Draw all of the diagonals that connect to vertex E. What is the sum of the interior angles of
figure EFGHI?
E
I
G
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LESSON 10.4
Skills Practice
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9. Draw all of the diagonals that connect to vertex J. What is the sum of the interior angles of
figure JKMONL?
J
K
L
M
N
O
10
10. Draw all of the diagonals that connect to vertex P. What is the sum of the interior angles of the
figure PQRSTUV?
P
Q
V
R
U
T
© Carnegie Learning
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Chapter 10 Skills Practice
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LESSON 10.4
Skills Practice
page 4
11. Draw all of the diagonals that connect to vertex W. What is the sum of the interior angles of the
figure WXYZABCD?
X
W
D
Y
C
Z
B
A
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12. Draw all of the diagonals that connect to vertex H. What is the sum of the interior angles of the
figure HIJKLMNOP?
H
I
P
J
O
K
N
L
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752
Chapter 10 Skills Practice
LESSON 10.4
Skills Practice
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Calculate the sum of the interior angle measures of each polygon.
13. A polygon has 8 sides.
The sum is equal to (n 2 2) ? 180º:
(8 2 2) ? 180º 5 6 ? 180º 5 1080º
The sum of the interior angles of the polygon is 1080º.
14. A polygon has 9 sides.
10
15. A polygon has 13 sides.
16. A polygon has 16 sides.
© Carnegie Learning
17. A polygon has 20 sides.
18. A polygon has 25 sides.
Chapter 10 Skills Practice
753
LESSON 10.4
Skills Practice
page 6
The sum of the measures of the interior angles of a polygon is given. Determine the number of sides for
each polygon.
19. 1080°
20. 1800°
180°(n 2 2) 5 1080°
n2256
n58
8 sides
21. 540°
22. 1260°
23. 3780°
24. 6840°
10
For each regular polygon, calculate the measure of each of its interior angles.
25.
26.
(n 2 2)180° ___________
(8 2 2)180°
___________
5
n
(6)180°
5 _______
8
1080°
5 ______
8
5 135º
The measure of each interior angle is 135º.
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Chapter 10 Skills Practice
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LESSON 10.4
Skills Practice
page 7
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27.
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28.
10
30.
© Carnegie Learning
29.
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LESSON 10.4
Skills Practice
page 8
Calculate the number of sides for each polygon.
31. The measure of each angle of a regular polygon is 108º.
(n 2 2)180°
108º 5 ___________
n
108ºn 5 (n 2 2)(180º)
108ºn 5 180ºn 2 360º
72ºn 5 360º
n55
The regular polygon has 5 sides.
10
32. The measure of each angle of a regular polygon is 156º.
34. The measure of each angle of a regular polygon is 162º.
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Chapter 10 Skills Practice
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33. The measure of each angle of a regular polygon is 160º.
LESSON 10.4
Skills Practice
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35. The measure of each angle of a regular polygon is 144°.
10
© Carnegie Learning
36. The measure of each angle of a regular polygon is 165.6°.
Chapter 10 Skills Practice
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Chapter 10 Skills Practice
LESSON 10.5
Skills Practice
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Exterior and Interior Angle Measurement Interactions
Sum of the Exterior Angle Measures of a Polygon
Vocabulary
Identify the term in the diagram.
1. Identify the term that is illustrated by the arrow in the diagram below.
10
Problem Set
© Carnegie Learning
Extend each vertex of the polygon to create one exterior angle at each vertex.
1.
2.
3.
4.
Chapter 10 Skills Practice
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LESSON 10.5
Skills Practice
5.
page 2
6.
Calculate the sum of the measures of the exterior angles for each polygon.
10
7. pentagon
360°
8. hexagon
9. triangle
10. nonagon
11. 20-gon
Given the measure of an interior angle of a polygon, calculate the measure of the adjacent exterior angle.
13. What is the measure of an exterior angle if it is adjacent to an interior angle of a polygon that
measures 90º?
Interior and exterior angles are supplementary. So subtract 90º, the measure of the interior angle,
from 180º.
180º 2 90º 5 90º
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14. What is the measure of an exterior angle if it is adjacent to an interior angle of a polygon that
measures 120º?
15. What is the measure of an exterior angle if it is adjacent to an interior angle of a polygon that
measures 108º?
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16. What is the measure of an exterior angle if it is adjacent to an interior angle of a polygon that
measures 135º?
17. What is the measure of an exterior angle if it is adjacent to an interior angle of a polygon that
measures 115º?
© Carnegie Learning
18. What is the measure of an exterior angle if it is adjacent to an interior angle of a polygon that
measures 124º?
Given the regular polygon, calculate the measure of each of its exterior angles.
19. What is the measure of each exterior angle of a square?
360 5 90º
____
4
20. What is the measure of each exterior angle of a regular pentagon?
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21. What is the measure of each exterior angle of a regular hexagon?
22. What is the measure of each exterior angle of a regular octagon?
23. What is the measure of each exterior angle of a regular decagon?
24. What is the measure of each exterior angle of a regular 12-gon?
10
Calculate the number of sides of the regular polygon given the measure of each exterior angle.
25. 45°
360 5 45
____
n
26. 90°
45n 5 360
n58
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27. 36°
28. 60°
29. 40°
30. 30°
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Quadrilateral Family
Categorizing Quadrilaterals Based on Their Properties
Problem Set
List all of the quadrilaterals that have the given characteristic.
1. all sides congruent
2. diagonals congruent
square and rhombus
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3. no parallel sides
4. diagonals bisect each other
5. two pairs of parallel sides
6. all angles congruent
Identify all of the terms from the following list that apply to each figure: quadrilateral, parallelogram,
rectangle, square, trapezoid, rhombus, kite.
7.
8.
rhombus
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parallelogram
quadrilateral
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9.
10.
11.
12.
Name the type of quadrilateral that best describes each figure. Explain your answer.
Rectangle. The quadrilateral has two pairs of parallel sides and four right angles, but the four
sides are not all congruent.
14.
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13.
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15.
10
16.
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17.
18.
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Draw the part of the Venn diagram that is described.
19. Suppose that part of a Venn diagram has two circles. One circle represents all types of quadrilaterals
with four congruent sides. The other circle represents all types of quadrilaterals with four congruent
angles. Draw this part of the Venn diagram and label it with the appropriate types of quadrilaterals.
Four Congruent
Sides
Rhombus
Four Congruent
Angles
Square
Rectangle
10
© Carnegie Learning
20. Suppose that part of a Venn diagram has two circles. One circle represents all types of quadrilaterals
with two pairs of congruent sides (adjacent or opposite). The other circle represents all types of
quadrilaterals with at least one pair of parallel sides. Draw this part of the Venn diagram and label it
with the appropriate types of quadrilaterals.
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21. Suppose that part of a Venn diagram has two circles. One circle represents all types of quadrilaterals
with two pairs of parallel sides. The other circle represents all types of quadrilaterals with four
congruent sides. Draw this part of the Venn diagram and label it with the appropriate types of
quadrilaterals.
10
© Carnegie Learning
22. Suppose that part of a Venn diagram has two circles. One circle represents all types of quadrilaterals
with four right angles. The other circle represents all types of quadrilaterals with two pairs of parallel
sides. Draw this part of the Venn diagram and label it with the appropriate types of quadrilaterals.
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23. Suppose that a Venn diagram has two circles. One circle represents all types of quadrilaterals with
four congruent sides. The other circle represents all types of quadrilaterals with congruent diagonals.
Draw this part of the Venn diagram and label it with the appropriate types of quadrilaterals.
10
© Carnegie Learning
24. Suppose that a Venn diagram has two circles. One circle represents all types of quadrilaterals with
diagonals that bisect the vertex angles. The other circle represents all types of quadrilaterals with
perpendicular diagonals. Draw this part of the Venn diagram and label it with the appropriate types of
quadrilaterals.
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Tell whether the statement is true or false. If false, explain why.
25. A trapezoid is also a parallelogram.
False. Parallelograms have two pairs of parallel sides. Trapezoids only have one pair.
26. A square is also a rhombus.
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27. Diagonals of a rectangle are perpendicular.
28. A parallelogram has exactly one pair of opposite angles congruent.
29. A square has diagonals that are perpendicular and congruent.
30. All quadrilaterals have supplementary consecutive angles.
List the steps to tell how you would construct the quadrilateral with the given information.
___
© Carnegie Learning
31. Rectangle HIJK given only diagonal HJ .
___
___
1. Duplicate HJ and bisect it to determine the midpoint.
___
2. Duplicate HJ again, labeling it IK , and bisecting it to determine the midpoint.
___
___
3. Draw segments HJ and IK so that their midpoints intersect.
4. Connect the endpoints of the segments to form rectangle HIJK.
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32. Square ABCD given only diagonal AC .
___
33. Kite RTSU given only diagonal RS .
10
___
34. Rhombus MNOP given only diagonal MO .
___
___
36. Isosceles trapezoid BCDE given only diagonal BD .
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35. Parallelogram JKLM given only diagonal JL .
LESSON 10.7
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Name That Quadrilateral!
Classifying Quadrilaterals on the Coordinate Plane
Problem Set
Calculate the distance between the two points.
1. A(22, 5) B(3, 2)
2. C(23, 21) D(4, 0)
___________________
AB 5 √ (22 2 3) 1 (5 2 2)
2
2
__________
5 √ (25)2 1 32
10
_______
5 √ 25 1 9
___
© Carnegie Learning
5 √ 34
3. R(5, 1) S(26, 4)
4. T(21, 0) W(25, 21)
5. M(0, 0) N(5, 1)
6. P(0, 28) Q(2, 6)
___
___
Calculate the slope of AB and CD . Then state if the segments are parallel, perpendicular or neither.
Explain your reasoning.
7. A(24, 2), B(4, 4), C(0, 0), D(4, 1)
___
Slope of AB
422
m 5 ________
4 2 (24)
2
5 __
8
1
5 __
4
___
Slope of CD
120
m 5 ______
420
1
__
5
4
The slopes of the segments are the same. The segments are parallel.
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8. A(21, 23), B(1, 2), C(21, 7), D(5, 1)
9. A(22, 1), B(0, 24), C(25, 24), D(0, 22)
10
11. A(1, 2), B(21, 26), C(0, 25), D(2, 3)
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10. A(22, 21), B(10, 2), C(3, 6), D(5, 22)
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12. A(23, 0), B(1, 2), C(1, 2), D(3, 22)
Determine the equation of the line with the given slope and passing through the given point.
10
2 passing through (9, 1)
13. m 5 __
3
2 (x 2 9)
(y 2 1) 5 __
3
__
y 2 1 5 2x 2 6
3
__
y 5 2x 2 5
3
1
14. m 5 2__ passing through (24, 2)
4
© Carnegie Learning
15. m 5 0 passing through (3, 5)
3 passing through (28, 2)
16. m 5 __
5
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17. m 5 25 passing through (6, 23)
6 passing through (0, 210)
18. m 5 __
5
10
Determine the coordinates of point D, the solution to the system of linear equations.
3
19. y 5 2__ x 1 8 and y 5 2x 1 10
2
3
3
y 5 2__x 1 8
2__x 1 8 5 2x 1 10
2
2
3
2__x 5 2x 1 2
2
1
2__x 5 2
2
x 5 24
3
y 5 2__(24) 1 8
2
y5618
y 5 14
The coordinates of point D are (24, 14).
3
7
21. y 5 4x 2 2 and y 5 2__x 1 __
2
2
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20. y 5 __x 2 2 and y 5 22x 1 5
2
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3
22. y 5 22x 1 5 and y 5 __x 2 2
2
10
5x 1 2 and y 5 x 2 4
23. y 5 __
3
© Carnegie Learning
1
3 x 1 ___
10
4 and y 5 __
24. y 5 2__x 1 __
5
5
7
7
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Use the given information to determine if quadrilateral ABCD can best be described as a trapezoid,
a rhombus, a rectangle, a square, or none of these. Explain your reasoning.
___
___
___
___
25. Side lengths: AB 5 √ 20 , BC 5 √45 , CD 5 √ 20 , DA 5 √ 45
___
___
1
Slope of BC is __
2
___
1
Slope of DA is __
2
Slope of AB is 22
___
Slope of CD is 22
The slopes of the line segments have a negative reciprocal relationship. This means the line
segments are perpendicular which means the angles must be right angles. Also, the opposite
sides have the same slope, so I know the opposite sides are parallel. Finally, opposite sides are
congruent. Quadrilateral ABCD can best be described as a rectangle.
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___
___
___
___
___
___
___
___
27. Side lengths: AB 5 √ 13 , BC 5 √ 17 , CD 5 √ 52 , DA 5 √ 10
___
___
1
2
Slope of BC is 2__
Slope of AB is __
4
3
___
___
2
Slope of DA is 23
Slope of CD is __
3
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26. Side lengths: AB 5 √ 13 , BC 5 √13 , CD 5 √ 13 , DA 5 √ 13
___
___
3
Slope of AB is 2__
Slope of BC is 1
2
___
___
3
Slope of CD is 2__
Slope of DA is 1
2
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__
___
__
___
28. Side lengths: AB 5 √ 8 , BC 5 √32 , CD 5 √ 8 , DA 5 √32
___
___
Slope of AB is 21
Slope of BC is 1
___
___
Slope of CD is 21
Slope of DA is 1
10
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___
___
___
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29. Side lengths: AB 5 √ 14 , BC 5 √14 , CD 5 √ 14 , DA 5 √ 14
___
___
1
Slope of BC is 28
Slope of AB is __
8
___
___
1
Slope of DA is 28
Slope of CD is __
8
30. Side lengths: AB 5 √ 17 , BC 5 √ 26 , CD 5 √ 50 , DA 5 √ 34
___
___
1
Slope of AB is 4
Slope of BC is __
5
___
___
1
4
__
__
Slope of CD is
Slope of DA is
7
5
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