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Mastering the California Mathematics Standards, Geometry
Printed in the United States of America.
1 2 3 4 5 6 7 8 9 10 047 15 14 13 12 11 10 09 08 07
Contents
California Mathematics Standards, Geometry . . . . . . . . . . . . . . . . . . . . . . . . Av
Diagnostic Test—Student Recording Sheet . . . . . . . . . . . . . . . . . . . . . . . . . Avii
Diagnostic Test—Student Answer Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . A1
Diagnostic Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A3
Practice by Standard
Logic and Geometric Proofs (G1.0−G7.0) . . . . . . . . . . . . . . . . . . . . . . . A21
Volume and Area Formulas (G8.0−G11.0) . . . . . . . . . . . . . . . . . . . . . . . A33
Angle Relationships, Constructions, and Lines (G12.0–G17.0) . . . . . . . . . . . . A39
Trigonometry (G18.0−G22.0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A47
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Standards Assessment—Student Answer Sheet . . . . . . . . . . . . . . . . . . . . . . . A57
Standards Assessment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A59
Mastering the California Mathematics Standards, Geometry
Aiii
California Mathematics Standards
Geometry
= Key standards
1.0 Students demonstrate understanding by identifying and giving examples of
undefined terms, axioms, theorems, and inductive and deductive reasoning.
2.0 Students write geometric proofs, including proofs by contradiction.
3.0 Students construct and judge the validity of a logical argument and give
counterexamples to disprove a statement.
4.0 Students prove basic theorems involving congruence and similarity.
5.0 Students prove that triangles are congruent or similar, and they are able to
use the concept of corresponding parts of congruent triangles.
6.0 Students know and are able to use the triangle inequality theorem.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
7.0 Students prove and use theorems involving the properties of parallel lines
cut by a transversal, the properties of quadrilaterals, and the properties
of circles.
8.0 Students know, derive, and solve problems involving perimeter,
circumference, area, volume, lateral area, and surface area of common
geometric figures.
9.0 Students compute the volumes and surface areas of prisms, pyramids,
cylinders, cones, and spheres; and students commit to memory the
formulas for prisms, pyramids, and cylinders.
10.0 Students compute areas of polygons, including rectangles, scalene
triangles, equilateral triangles, rhombi, parallelograms, and trapezoids.
11.0 Students determine how changes in dimensions affect the perimeter, area,
and volume of common geometric figures and solids.
12.0 Students find and use measures of sides and of interior and exterior angles
of triangles and polygons to classify figures and solve problems.
13.0 Students prove relationships between angles in polygons by using
properties of complementary, supplementary, vertical, and exterior angles.
14.0 Students prove the Pythagorean theorem.
15.0 Students use the Pythagorean theorem to determine distance and find
missing lengths of sides of right triangles.
Mastering the California Mathematics Standards, Geometry
Av
California Mathematics Standards
Geometry (continued)
16.0 Students perform basic constructions with a straightedge and compass,
such as angle bisectors, perpendicular bisectors, and the line parallel to a
given line through a point off the line.
17.0 Students prove theorems by using coordinate geometry, including the
midpoint of a line segment, the distance formula, and various forms of
equations of lines and circles.
18.0 Students know the definitions of the basic trigonometric functions
defined by the angles of a right triangle. They also know and are able to
use elementary relationships between them. For example,
sin (x)
cos (x)
tan (x) = _, (sin (x))2 + (cos (x))2 = 1.
19.0 Students use trigonometric functions to solve for an unknown length of a
side of a right triangle, given an angle and a length of a side.
20.0 Students know and are able to use angle and side relationships in
problems with special right triangles, such as 30°, 60°, and 90° triangles
and 45°, 45°, and 90° triangles.
22.0 Students know the effect of rigid motions on figures in the coordinate
plane and space, including rotations, translations, and reflections.
Avi Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
21.0 Students prove and solve problems regarding relationships among chords,
secants, tangents, inscribed angles, and inscribed and circumscribed
polygons of circles.
Diagnostic Test
Student Recording Sheet
Color in the bubble for each question that you answered correctly on the Diagnostic Test. For each
question you did not answer correctly, your teacher may ask you to do the exercises on the practice
sheet prescribed.
Standard
Assessed
Practice Page
1
G3.0
A24
2
G11.0
3
Standard
Assessed
Practice Page
26
G8.0
A33
A38
27
G1.0
A21
G10.0
A36
28
G12.0
A39
4
G3.0
A24
29
G16.0
A43
5
G22.0
A54
30
G22.0
A54
6
G16.0
A43
31
G14.0
A42
7
G1.0
A21
32
G12.0
A39
8
G19.0
A49
33
G21.0
A52
9
G17.0
A45
34
G9.0
A35
10
G8.0
A33
35
G20.0
A51
11
G18.0
A47
36
G21.0
A52
12
G7.0
A30
37
G17.0
A45
13
G4.0
A26
38
G2.0
A22
14
G5.0
A28
39
G13.0
A41
15
G12.0
A39
40
G15.0
A42
16
G2.0
A22
41
G4.0
A26
17
G7.0
A30
42
G7.0
A30
18
G22.0
A54
43
G21.0
A52
19
G4.0
A26
44
G16.0
A43
20
G7.0
A30
45
G8.0
A33
21
G19.0
A49
46
G2.0
A22
22
G6.0
A29
47
G18.0
A47
23
G18.0
A47
48
G10.0
A36
24
G10.0
A36
49
G3.0
A24
25
G3.0
A24
50
G5.0
A25
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Question
Question
continued on next page
Mastering the California Mathematics Standards, Geometry Avii
Name
Date
Diagnostic Test
Student Recording Sheet
(continued)
Practice Page
66
G9.0
A35
A41
67
G3.0
A24
G16.0
A43
68
G10.0
A36
54
G4.0
A26
69
G7.0
A30
55
G17.0
A45
70
G12.0
A39
56
G8.0
A33
71
G17.0
A45
57
G10.0
A36
72
G18.0
A47
58
G4.0
A26
73
G16.0
A43
59
G21.0
A52
74
G4.0
A26
60
G15.0
A42
75
G8.0
A33
61
G7.0
A30
76
G11.0
A38
62
G12.0
A39
77
G12.0
A39
63
G21.0
A52
78
G18.0
A47
64
G9.0
A35
79
G21.0
A52
65
G12.0
A39
Practice Page
51
G19.0
A49
52
G13.0
53
Question
Total Number of Questions Correct
Count how many questions you answered correctly.
Find your score in the table below and circle your level.
Far Below
Below Basic
Basic
Proficient
Advanced
0–20
21–43
44–55
56–67
68–79
Aviii
Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Standard
Assessed
Standard
Assessed
Question
Diagnostic Test
Student Answer Sheet
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Record your answers by coloring in the appropriate bubble for the best answer to
each question.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
Mastering the California Mathematics Standards, Geometry A1
Name
Date
Diagnostic Test
1
“Any two right triangles are similar.”
3
What is the area, in square units, of the
trapezoid below?
Which of the following provides a
counterexample to the statement
above?
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A a triangle with side lengths 3, 4, and 5
and another triangle with side lengths
9, 12, and 15
B a triangle with side lengths 3, 4, and 5
and another triangle with side lengths
6, 8, and 11
C a triangle with side lengths 3, 4, and 5
and another triangle with side lengths
5, 12, and 13
D a triangle with side lengths 6, 10, and
14 and another triangle with side
lengths 12, 20, and 28
2
A rectangle has an area of 12 cm 2.
What would the area be if both the
length and the width were doubled?
F
G
H
J
12 cm 2
24 cm 2
36 cm 2
48 cm 2
A
B
C
D
4
15
18
24
40
“The diagonals of a quadrilateral must
bisect each other.”
Which of the following could provide
a counterexample to the above
statement?
F
G
H
J
Mastering the California Mathematics Standards, Geometry
A3
Name
Date
Diagnostic Test
5
(continued)
Quadrilateral ABCD is to be translated
to quadrilateral A'B'C'D' by the
following rule.
(x, y)
7
A a statement proved from given
assumptions
B a statement assumed to be true
without proof
C a statement that is never true
D a statement that is almost always true
(x - 2, y + 5)
y
4
3#
2
1
−4−3−2
"
O
$
%
1 2 3 4x
−2
−3
−4
8
What will be the coordinates of D'?
A
B
C
D
Which of the following statements best
describes what an axiom is?
In the figure below, m∠X = 42°, and
XZ = 35. Which equation could be used
to find a in XYZ?
;
(-2, 5)
(5, 2)
(-1, 6)
(1, 5)
35
6
Aaron is using a straightedge and a
compass to do the construction below.
9
G a = 35 tan 42°
H a = 35 sin 42°
J a = 35 cos 42°
F constructing a line parallel to ℓ
through point P
G bisecting ℓ
H constructing the angle between ℓ and
point P
J constructing a line perpendicular to ℓ
through point P
42‚
35
F a=_
sin 42°
1
Which best describes the construction
that Aaron is doing?
a
9
Which type of triangle is formed
by the points X(3, 4), Y(2, 1), and
Z(4, 1)?
A
B
C
D
isosceles
equilateral
scalene
right
A4 Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
:
Name
Date
Diagnostic Test
10
(continued)
Find the perimeter of the rhombus
pictured below.
13
4
−−
−−
Suppose that AB is twice as long as DE
−−
−−
and that AC is twice as long as DF .
Which of the following would be
sufficient to prove that ABC and
DEF are similar?
#
3
"
F 24
G 20
11
H 14
J 5
&
%
6
If cos x = _
, then what are sin x
A
B
C
D
10
and tan x?
8
8
, and tan x = _
A sin x = _
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
$
10
6
8
6
B sin x = _
, and tan x = _
10
8
8
8
C sin x = _
, and tan x = _
10
6
10
10
D sin x = _
, and tan x = _
8
6
14
'
∠BCA ∠EFD
−−
−−
BC is twice as long as EF .
∠ABC ∠EDF
−−
−−
BC is twice as long as DF .
In the figure below, which of the
following would not be sufficient to
prove that ABC EDC?
"
12
1
In the figure below, ℓ and m are parallel
lines. Which of the following statements
must be true?
2
#
3
$
%
4
5
6
1
2
F
G
H
J
3
m
&
F
G
H
J
−− −−
AC CE
∠2 ∠5
−−
−−
AB is parallel to DE .
∠1 ∠2
∠1 and ∠3 are congruent.
∠1 and ∠3 are complementary.
∠2 and ∠3 are congruent.
∠2 and ∠3 are complementary.
Mastering the California Mathematics Standards, Geometry
A5
Name
Date
Diagnostic Test
15
Two angles of a triangle have measures
35° and 85°. Which of the following
could not be a measure of an exterior
angle of the triangle?
A
B
C
D
16
(continued)
17
In the figure below, find m∠a.
80‚
a
95°
120°
135°
145°
60‚
50‚
A
B
C
D
−−
In the figure below, AD bisects ∠BAC,
−− −−
AB AC.
10°
100°
170°
190°
"
18
%
$
5
4
3
2
1
Amanda wants to prove that
ABD ACD.
Statement
−− −−
1. AB AC
−−
2. AD bisects ∠BAC
3. ∠BAD ∠CAD
−− −−
4. AD AD
5. ABD ACD
−2
−3
−4
F
G
H
J
(-3, 1)
(-3, -1)
(-1, -3)
(-1, 3)
What reason can be given to justify the
last step of the proof?
F
G
H
J
-
,
−3−2−1O + 1 2 3 4 5 x
Reason
1. Given
2. Given
3. Definition
of Angle
Bisector
4. Reflexive
Property
5. ?
y
ASA
SSA
SSS
SAS
A6 Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
#
If triangle JKL were rotated 90°
counterclockwise about the origin,
what would be the coordinates of the
new vertex K'?
Name
Date
Diagnostic Test
19
Quadrilateral ABCD is a
parallelogram. Which of the following
would be sufficient to prove that
ABE CBE?
#
(continued)
21
$
&
The figure below shows a 10-foot rope
stretched from the window of a house
down to the ground. The rope makes a
27° angle with the ground. How high
off the ground is the window?
window
10 ft
"
A
B
C
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
20
?
%
−− −−
AD CD
∠CAD ∠ACB
−− −−
AE CE
ABE is an isosceles triangle.
27‚
sin 27‚≈ 0.45
cos 27‚≈ 0.89
tan 27‚≈ 0.51
A
B
C
D
In the figure below, what is the arc
length from A to B?
"
120‚
4.5 ft
5.1 ft
8.9 ft
22.2 ft
#
3 cm
22
F π cm
G 2π cm
A triangle has side lengths 7 meters
and 4 meters. According to the triangle
inequality theorem, which of the
following is not a possible value for
the length of the third side of the
triangle?
H 3π cm
J 6π cm
F
G
H
J
4m
6m
10 m
12 m
Mastering the California Mathematics Standards, Geometry
A7
Name
Date
Diagnostic Test
23
In the figure below, sin A = 0.6. What
−−
is the length of BC ?
(continued)
25
$
Which of the following could provide
a counterexample to the above
statement?
25
#
A
B
C
D
A
B
C
D
"
15
20
25
41.7
26
24
“A triangle cannot have an
obtuse angle.”
What is the area of the equilateral
triangle shown below?
scalene triangle
right triangle
equilateral triangle
acute triangle
Find the lateral area of a cylinder
that has a diameter of 6 inches and a
height of 4 inches.
24π sq in.
36π sq in.
42π sq in.
48π sq in.
4 in.
3 sq in.
F 8 √
G 8 sq in.
27
In the figure below, lines ℓ and m are
parallel. Which of the following must
be true?
H 4 √
3 sq in.
1
3
J 2 √
3 sq in.
6
5
7
A
B
C
D
8
2
4
m
∠1 and ∠4 are supplementary.
∠3 ∠6
∠2 and ∠7 are supplementary.
∠4 ∠7
A8 Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
F
G
H
J
Name
Date
Diagnostic Test
28
A particular regular polygon has
interior angles 120°. How many sides
does the polygon have?
F
G
H
J
29
30
3
5
6
10
The vertices of DEF are D(3, 6),
E(5, 1), and F(0, 0). If the triangle
were reflected across the x-axis,
what would be the coordinates of
the triangle D'E'F'?
F
G
H
J
Which of the following should
be the first step toward bisecting
angle ∠BAC?
A
(continued)
31
D '(3, -6), E '(5, -1), F '(0, 0)
D '(-3, 6), E '(-5, 1), F '(0, 0)
D '(-3, -6), E '(-5, -1), F '(0, 0)
D '(-6, 3), E '(5, -1), F '(0, 0)
The figure below is from a proof of the
Pythagorean theorem.
a
b
c
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
B
C
Which statement would not be
used in the proof of the Pythagorean
theorem?
A The four right triangles are congruent.
D
B The area of the inner square is equal
to half the area of the larger square.
C The area of the larger square is
equal to the sum of the areas of
the smaller square and the four
congruent triangles.
1
D The area of a triangle is _
ab.
2
Mastering the California Mathematics Standards, Geometry
A9
Name
Date
Diagnostic Test
32
(continued)
34
What is m∠x?
Find the surface area of the cylinder
shown below.
8 cm
45‚
55‚
F
G
H
J
33
x
10 cm
80°
90°
100°
120°
A square is circumscribed about
a circle. What is the ratio of the area
of the square to the area of the
circle?
35
96π cm 2
112π cm 2
160π cm 2
208π cm 2
In the triangle below, what is the
value of x?
π
B _
x
2
1
C _
4
4
D _
π
3
2
90‚
x
16
4
A
B
C
D
A10
32
16 √
2
8
8 √
2
Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
π
A _
4
F
G
H
J
Name
Date
Diagnostic Test
36
(continued)
−−
−−
In the circle below, EF and GH are
chords intersecting at K.
38
(
&
3
,
12
10
Jose wants to prove that no
quadrilateral can have exactly three
right angles. He begins by drawing the
quadrilateral below, assuming that
∠1, ∠2, and ∠3 are right angles, and
that ∠4 is not a right angle. Which
theorem or fact can Jose use to reach
a contradiction?
)
1
2
3
4
'
If EK = 3, FK = 10, and HK = 12, then
what is GK?
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
F
G
H
J
37
1
2.5
4.5
6
F A square has four right angles.
G The sum of the measures of the angles
of a quadrilateral is 360°.
H In any quadrilateral, opposite angles
are equal.
J He can use the Pythagorean theorem.
The figure below shows ABC.
y
$
"
39
What is m∠ x?
50‚
#
Which statement would prove that
ABC is an isosceles triangle?
A
B
C
D
−−
−−
(slope AC )(slope CB ) = 1
AB = AC
−−
−−
(slope AC )(slope CB ) = -1
AB ≤ BC + CA
120‚
A
B
C
D
x
40°
60°
70°
110°
Mastering the California Mathematics Standards, Geometry
A11
Name
Date
Diagnostic Test
40
(continued)
A right triangle has hypotenuse length 8
and side length 5. What is the length of
the remaining side?
F
G
H
J
42
In the figure pictured below, lines ℓ and
m are parallel. Which pair of angles are
supplementary?
3
√
13
√
39
√
89
1
3
5
7
41
Which of the following would not be
sufficient to prove that ABC and
DEC are similar?
F
G
H
J
"
%
#
&
$
6
m
8
∠1 and ∠6
∠2 and ∠3
∠3 and ∠7
∠4 and ∠5
−−
In the figure below, ST is tangent to
−−
circle R at point S. RS is a radius of the
circle.
A DCE is a right triangle.
5
B AD = DC
2
67‚
C ∠CAB ∠CDE
4
CA
CB _
CB
D _
=
=_
CE
CE
CD
3
What is m∠QST?
A
B
C
D
23°
46°
67°
113°
A12 Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
43
2
4
Name
Date
Diagnostic Test
44
(continued)
Moesha is using a straightedge and a
compass to perform the construction
shown below.
46
In the figure below, m∠1 + m∠4 = 150°.
1
k
3
2
4
m
1
Assume that lines ℓ and m are parallel.
We know that ∠1 and ∠2 are
supplementary. Since ∠2 ∠4,
we conclude that ∠1 and ∠4 are
supplementary. That is, m∠1 + m∠4 =
180°, which contradicts m∠1 + m∠4 =
150°. This contradiction allows us to
conclude which of the following?
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Which best describes the construction
Moesha is doing?
F
G
H
J
45
a line through P congruent to line ℓ
a line through P parallel to line k
a line through P perpendicular to line ℓ
a line through P parallel to line ℓ
F
G
H
J
Jim wants to carpet his living room,
which is shaped like the right triangle
shown below.
47
45‚
∠1 and ∠4 are congruent.
∠1 and ∠2 are not supplementary.
Lines ℓ and m are not parallel.
∠3 and ∠4 are supplementary.
Approximately how many meters long
is the lake?
sin 35‚≈ 0.57
cos 35‚≈ 0.82
tan 35‚≈ 0.70
41 m
12 ft
35‚
If carpet costs
foot, how
CAG$2
DT per
038A square
8 9 3
much will it cost Jim to carpet his
living room?
A $36
B $72
C $144
D $288
A 28.7 m
B 33.6 m
C 58.6 m
D 71.9 m
Mastering the California Mathematics Standards, Geometry
A13
Name
Date
Diagnostic Test
48
Compute the area, in square units,
of the parallelogram shown below.
(continued)
50
8
2
F 8
G 9
49
In the figure below, which of the
following would be sufficient to prove
that ABD ACD?
30‚
"
H 8 √
3
J 16
1 4
“A quadrilateral must have at least one
right angle.”
#
Which of the following provides a
counterexample of the statement
above?
F
G
H
J
A
3 5
6
%
$
−−
D is the midpoint of BC.
ABD is a right triangle.
∠1 ∠5
m∠1 + m∠2 = m∠4 + m∠6
Triangle MNO is shown below.
0
B
12
52‚
/
C
Which equation should be used to find
−−−
the length of MN ?
MN
A tan 52° = _
D
.
12
MN
B sin 52° = _
12
12
C tan 52° = _
MN
12
D sin 52° = _
MN
A14 Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
51
2
Name
Date
Diagnostic Test
52
(continued)
54
What is m∠ ABC?
#
"
64‚
124‚
"
F
G
H
J
Which of the following would be
sufficient to prove that ABC and
DEF are similar?
1
$
%
8°
26°
56°
98°
#
2
3
&
53
What is the first step toward
constructing the perpendicular bisector
of the line segment AB?
F
G
H
J
4
$
5
6
'
−− −−
AB DE
∠1 ∠6
−− −−
BC EF
∠1 ∠4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
&
"
%
$
#
55
Figure LMNO is a parallelogram.
'
−−
A Draw EF.
B From point C, draw an arc that
−−
intersects AB and points E and F.
C From point A, draw an arc that
−−
intersects AB at point D.
D From point A, draw an arc that
−−
intersects AB at point C.
5
4
3
2
1
−1 O
y
-
0
.
/
1 2 3 4 5 6 7x
−2
−3
What are the coordinates of the
midpoint of the diagonal?
A (6, 4)
1
B _
,2
(2 )
5
C (_
,2
2 )
D (3, 2)
Mastering the California Mathematics Standards, Geometry
A15
Name
Date
Diagnostic Test
56
(continued)
The parallelogram below has an area of
160 cm2. Find the perimeter.
58
Which of the following is sufficient to
prove that ABD and CBD are
congruent?
8 cm
10 cm
F
G
H
J
DN
F
G
H
J
57
Given: ABD and CBD share side
−− −− −−
BD, AD CD.
20 cm
24 cm
52 cm
60 cm
59
What is the area of the rhombus shown
below?
∠ABD ∠CBD
∠ADB ∠CDB
−− −−
BD AD
∠DAB ∠BDC
⎯ is tangent to
In the figure below, UT
⎯⎯
circle O at point T, and secant line UW
intersects the circle at points V and W.
TV measures 50°, and VW measures 130°.
5
0
4 cm
7
130‚
8
60‚
A
B
C
D
2
2 √
3 cm
√
4 3 cm 2
8 √
3 cm 2
16 √
3 cm 2
What is m∠TUV?
A
B
C
D
65°
80°
90°
115°
A16 Mastering the California Mathematics Standards, Geometry
6
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
50‚
Name
Date
Diagnostic Test
60
A new highway is being built to route
traffic around a residential area. A
figure of the old and new roads is
shown below.
(continued)
62
The measure of an exterior angle of a
regular polygon is 72°. Which type of
polygon is it?
F
G
H
J
/FXDPOTUSVDUJPO
12 mi
a hexagon
a pentagon
a quadrilateral
a triangle
5 mi
0METFDUJPOPGIJHIXBZ
63
How many extra miles will cars now
have to drive?
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
F
G
H
J
61
A circle is circumscribed around
a square. What is the ratio of the
area of the circle to the area of the
square?
1
A _
4 mi
6 mi
13 mi
17 mi
2
π
B _
4
π
C _
2
2
D _
π
In the figure below, ABCD is a
parallelogram. Which of the following
two line segments must have the same
length?
#
64
What is the volume of the square
pyramid shown below?
$
6 cm
&
"
A
B
C
D
−−
−−
AB and BC
−−
−−
AC and BD
−−
−−
AE and ED
−−
−−
BE and ED
5 cm
%
F
G
H
J
10 cm 3
30 cm 3
50 cm 3
150 cm 3
Mastering the California Mathematics Standards, Geometry
A17
Name
Date
Diagnostic Test
65
−− −−
In the figure below, AB || CD.
"
x‚+ 20‚
x‚+ 60‚
(continued)
68
What is the area of the trapezoid
shown below?
%
14 in.
&
#
13 in.
$
12 in.
20 in.
What is the value of x°?
A
B
C
D
66
Find the surface area of a rectangular
prism with dimensions 3 feet by 5 feet
by 8 feet.
69
168 in 2
294 in 2
318.5 in 2
490 in 2
In the figure below, ℓ || m, and
m∠3 = 52°. Find m∠2 + m∠5.
79 ft 2
120 ft 2
128 ft 2
158 ft 2
1
3
5
7
67
“Every rhombus has an acute angle.”
Which of the following provides a
counterexample of the statement
above?
A
B
C
D
a rectangle
a square
a parallelogram
a trapezoid
A
B
C
D
104°
128°
180°
256°
A18 Mastering the California Mathematics Standards, Geometry
6
8
2
4
m
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
F
G
H
J
F
G
H
J
50°
70°
90°
95°
Name
Date
Diagnostic Test
70
(continued)
72
What is m∠x?
_
In the figure below, if tan x = 24 , then
7
what are sin x and cos x?
x
x
80‚
F
G
H
J
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
71
25
24
F sin x = _
, and cos x = _
140‚
7
7
25
7
G sin x = _
, and cos x = _
24
24
7
24
H sin x = _
, and cos x = _
25
25
7
24
J sin x = _
, and cos x = _
25
25
60°
100°
120°
140°
Figure PQRS is a rhombus.
y
73
3 (3, 10)
4 (1, 6)
Which of the following best describes
the construction illustrated below?
2 (5, 6)
1 (3, 2)
x
O
What are the coordinates of the
−−
midpoint of side RQ?
A
B
C
D
(4, 8)
(4, 4)
(3, 6)
(3, 2)
A
B
C
D
bisecting an angle
constructing an equilateral triangle
constructing a parallel line
constructing a right triangle
Mastering the California Mathematics Standards, Geometry
A19
Name
Date
Diagnostic Test
74
(continued)
Which of the following would not be
sufficient to prove that the triangles are
congruent?
77
Which type of polygon has the sum of
its interior angles equal to the sum of
its exterior angles?
A
B
C
D
#
$
a triangle
a quadrilateral
a pentagon
a hexagon
"
78
%
F
G
H
J
In the figure below, sin T = 0.55.
&
6
−− −−
AD BC
−− −−
BC DE
∠DAE ∠BAC
∠ABC ∠ADE
11
5
7
−−
What is the length of TV?
A swimming pool is shaped like a half
sphere. The pool has a diameter of
30 feet. What is its volume?
A
B
C
D
76
450π ft 3
2,250π ft 3
4,500π ft 3
18,000π ft 3
If the radius of a cylinder were tripled,
how would the volume change?
F The volume would not change.
G The volume would triple.
H The volume would be six times
as large.
J The volume would be nine times
as large.
F
G
H
J
79
6.1
20
22.8
36.4
−−
−−−
Chords KL and MN intersect at point O
in the interior of a circle. KO = 9,
LO = 4, MO = 3. Find NO.
A
B
C
D
6.8
8
10
12
A20 Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
75
Name
Date
Practice by Standard
Geometry 1.0
Students demonstrate understanding by
identifying and giving examples of undefined
terms, axioms, theorems, and inductive and
deductive reasoning.
G1.0
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1
4
axiom
counterexample
inductive reasoning
deductive reasoning
Triangle ABC is a right triangle.
#
c
a
$
b
"
Which of the following conclusions does
not have to be true?
F
G
H
J
Which of the following best describes
inductive reasoning?
F accepting the meaning of a term
without definition
G using logic to draw conclusions based
on accepted statements
H inferring a general truth by examining
a number of specific examples
J defining mathematical terms to
correspond with physical objects
The statement “Given any two points,
there is only one line that passes
through them” is an example of which
of the following?
A
B
C
D
Which of the following is not an example
of deductive reasoning?
A Everyone Rickey knows owns a cell
phone. Therefore, everyone owns a
cell phone.
B Mammals are warm blooded. Dogs
are mammals. Therefore, dogs are
warm blooded.
C Fish breathe through their gills. Sharks
are fish. Therefore, sharks breathe
through gills.
D Every U.S. senator must be a citizen of
the United States. Barbara Boxer is a
U.S. senator. Therefore, Barbara Boxer
is a citizen of the United States.
2
3
5
a2 + b2 = c2
m∠A + m∠B = 90°
−− −−
AC BC
−− −−
AC ⊥ BC
Which statement would be best supported
by deductive reasoning?
A The number of automobiles made in
the United States is greater this year
than last year.
B Mexico is the most populous Spanishspeaking country in the world.
C Residents of Sacramento, California,
are residents of the United States.
D Fewer people own personal computers
than own cell phones.
Mastering the California Mathematics Standards, Geometry
A21
Name
Date
Practice by Standard
Geometry 2.0
Students write geometric proofs,
including proofs by contradiction.
G2.0
1
Which postulate or theorem supports
the statement that line ℓ and point P lie
in the same plane?
Which of the following is a basic
assumption that is accepted
without proof ?
A hypothesis
B postulate
2
4
1
C theorem
D conclusion
F A line and a point not on the line are
contained in exactly one plane.
G Two lines that intersect are contained
in exactly one plane.
H If two points of a line are in a given
plane, then the line is in the plane.
J A plane contains a minimum of three
non-collinear points.
Which of the following is not sufficient
to prove that a quadrilateral is a
parallelogram?
5
Given: Lines m and n are perpendicular
to line ℓ.
m
3
To prove the following theorem by
contradiction, Twan assumed that there
are two lines, m and n, that intersect at
two points:
If two lines intersect, they intersect at
exactly one point.
Which postulate or theorem did they
use to reach a contradiction?
A A line contains at least two points.
B Through any two points there is
exactly one line.
C A line segment has exactly one midpoint.
D Every angle, except a straight angle,
has exactly one bisector.
n
Which is the first step in a proof that
lines m and n are parallel?
A assuming line m is not parallel to line n
B assuming line m is parallel to line n
C assuming line m is perpendicular to
line ℓ
D assuming line m is not perpendicular
to line ℓ
A22 Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
F showing that both pairs of opposite
sides are congruent
G showing that both pairs of opposite
angles are congruent
H showing that the diagonals bisect
each other
J showing that the sum of the measures
of the interior angles is 360°
Name
Date
Practice by Standard
Geometry 2.0 (continued)
6
Which of the following cannot be used
as a reason in a proof?
F definition
G property
7
9
H postulate
J conjecture
A
B
C
D
Mai is proving the following theorem
by contradiction.
A triangle has at most one right angle.
10
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
She started by assuming that XYZ
has two right angles. Which postulate
or theorem will help Mai reach
a contradiction?
2
If AB = CD, then which property
justifies the statement that
AB + BC = CD + BC ?
F
G
H
J
11
Which reason best justifies the
statement that points P, Q, and R lie in
the same plane?
hypothesis
conditional
postulate
contradiction
"
A The sum of the measures of the angles
of a triangle is 180°.
B A triangle has, at most, one obtuse angle.
C Every angle, except for a straight
angle, has exactly one bisector.
D All right angles are congruent.
8
In an indirect proof, which of the
following is a result of assuming that
the statement to be proved is false?
$
#
%
Transitive Property of Equality
Reflexive Property of Equality
Addition Property of Equality
Subtraction Property of Equality
Given: Points A and B lie in plane W.
"
w
#
$
Which theorem or postulate justifies the
conclusion that line ℓ lies in plane W ?
3
1
F Two intersecting lines are contained in
exactly one plane.
G If two points of a line are in a given
plane, then the line is in the plane.
H Three points that are not on the same
line form the vertices of a triangle.
J Three points that are not on the same
line are contained in exactly one plane.
A If two points of a line are in a given
plane, the line is in the plane.
B Three points that are not on the same
line are contained in exactly one plane.
C If two planes intersect, they intersect at
exactly one line.
D Two intersecting lines are contained in
exactly one plane.
Mastering the California Mathematics Standards, Geometry
A23
Name
Date
Practice by Standard
Geometry 3.0
Students construct and judge the
validity of a logical argument and give
counterexamples to disprove a statement.
G3.0
1
hypothesis
conclusion
conditional
converse
Which of the following can be used
to prove that a conditional statement
is false?
A counterexample
B converse
In the statement “If a figure has three
sides, it is a triangle,” the phrase “…it is
a triangle” is which of the following?
A
B
C
D
2
3
4
C conclusion
D hypothesis
Which figure can serve as a
counterexample to the conjecture
below?
If one pair of opposite sides of a
quadrilateral is parallel, and the
other pair is congruent, then the
quadrilateral is a parallelogram.
Which figure is a counterexample to
the statement below?
F square
G rectangle
For any quadrilateral, the lengths of its
diagonals are equal.
H rhombus
J trapezoid
5
Suppose a conditional statement is true.
Which of the following is true about its
converse?
A
B
C
D
G
It is true
It is false
It is possibly true
It is neither true nor false
H
6
J
Which statement is the converse of the
statement “If a polygon has eight sides,
then it is an octagon”?
F An octagon has eight sides.
G An eight-sided polygon is an octagon.
H If a polygon is an octagon, then it has
eight sides.
J If a polygon is an octagon, then it has
eight angles.
A24 Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
F
Name
Date
Practice by Standard
Geometry 3.0 (continued)
7
Given: ABCD is a parallelogram with
−−
−−
diagonals AC and BD. Which of the
following must be true?
A
B
C
D
8
−− −−
AC ⊥ BD
−− −−
AC BD
−− −−
AC BD
−−
−−
AC bisects BD.
Write the following statement as
a conditional statement: “All fish
can swim.”
A If an animal is a fish, then it can swim.
B If an animal can swim, then it is a fish.
C If an animal cannot swim, then it is
not a fish.
D If an animal is not a fish, then it
cannot swim.
Given the statements below, which
conclusion is valid?
All birds have feathers. A penguin is
a bird.
F
G
H
J
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
11
12
In the figure below, line m is parallel to
line n. Which of the following does not
have to be true?
All penguins have feathers.
All birds are penguins.
All penguins can fly.
All birds lay eggs.
m
n
9
If the conclusion is false in a valid
argument, then which of the following
must be false?
A inverse
B converse
10
F
G
H
J
C argument
D hypothesis
Which of the following is the
inverse of the statement “A square
is a rectangle”?
F If a figure is not a square, then it is not
a rectangle.
G If a figure is not a rectangle, then it is
not a square.
H A rectangle is a square.
J Some rectangles are squares.
13
1
2
3
5
4
∠1 ∠2
∠1 ∠3
∠1 ∠4
∠1 ∠5
How many counterexamples are needed
to disprove the conjecture “Two lines
in a plane always intersect at exactly
one point”?
A
B
C
D
0
1
2
many more than 2
Mastering the California Mathematics Standards, Geometry
A25
Name
Date
Practice by Standard
Geometry 4.0
Students prove basic theorems involving
congruence and similarity.
G4.0
1
3
Given: ∠B and ∠D are right angles.
#
$
"
%
Which reason justifies the statement
m∠B = m∠D?
A
B
C
D
−−
In parallelogram ABCD, diagonals AC
−−
and BD intersect at E. Which of the
following statements does not have to
be true?
A
B
C
D
4
definition of rectangle
definition of hypotenuse
equality of right angles
the sum of the measures of the angles
of a 180° triangle
∠AEB ∠DEC
∠AED ∠BEC
∠BCE ∠DAE
∠ABD ∠BCD
Jasmine wants to prove that MNP OPN in the parallelogram MNOP.
.
/
2
1
1
0
2
Devon wants to prove that
ABD CBD.
F If two parallel lines are intersected by
a transversal, then alternate interior
angles are congruent.
G If two parallel lines are intersected by
a transversal, then corresponding
angles are supplementary.
H If a quadrilateral is a parallelogram,
then its opposite sides are congruent.
J If a quadrilateral is a parallelogram,
then its opposite angles are congruent.
#
"
%
$
One step in Devon’s proof is the
−− −−
statement BD BD. Which reason
justifies that statement?
5
F definition of midpoint
G congruency of corresponding parts of
congruent triangles
H Substitution Property
J Reflexive Property
Which triangles must be congruent?
A two similar right triangles
B two obtuse triangles with
congruent bases
C two equilateral triangles with
congruent bases
D two similar isosceles triangles
A26 Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Which of the following supports
Jasmine’s assertion that ∠1 ∠2?
Name
Date
Practice by Standard
Geometry 4.0 (continued)
6
Which of the following would be enough
to prove CDT∼RST?
9
In which of the following triangles are
corresponding angles congruent and
corresponding sides proportional?
4
A
B
C
D
%
3
F
G
H
J
$
5
∠SRT ∠STR
∠SRT ∠DCT
−− −−
RC CT
−− −−
SD DT
10
corresponding
congruent
scalene
similar
−− −−
In the quadrilateral ABCD, AB CD,
−− −−
and AC BD.
#
7
In the triangles below, ∠ABC ∠DEF.
#
"
&
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
$
"
$
%
Which postulate can be used to prove
ABD DCA?
'
F
G
H
J
Which of the following is sufficient to
prove that the triangles are similar?
A
B
C
D
−− −−
AB DE
−− −−
AB BC
∠BAC ∠EDF
∠ABC ∠DEF
%
11
SAS
ASA
SSS
AAS
−− −−
−− −−
Given: AB WY, and AC XY.
"
#
9
8
In the quadrilateral ABCD, diagonals
−−
−−
AC and BD bisect each other. Which
statement does not have to be true?
F
G
H
J
−− −−
AB CD
ABD CDB
ABCD is a rectangle.
ABCD is a parallelogram.
$
8
:
Which is enough to prove that the
triangles are congruent?
A
B
C
D
−− −−−
CB WX
−− −−
CB XY
∠CAB ∠WXY
∠ABC ∠YWX
Mastering the California Mathematics Standards, Geometry
A27
Name
Date
Practice by Standard
Geometry 5.0
3
Students prove that triangles are
congruent or similar, and they are able to
use the concept of corresponding parts of
congruent triangles.
G5.0
1
The triangles below are similar.
Find the value of x.
#
−−
−−
Given: AC and DE intersect at B,
−− −−
AB BC, and ∠1 ∠2.
Y
&
"
&
"
1
A 9
B 10
2
#
%
$
Which can be used to prove
ABD CBE?
A
B
C
D
4
ASA
HL
SAS
SSS
−− −− −− −− −− −−
AB XY, BC YZ, AC XZ
−− −− −− −− −− −−
AB XY, AC YZ, BC XZ
−− −− −− −− −− −−
AB YZ, BC XY, AC XZ
−− −− −− −− −− −−
AB XZ, BC XY, AC YZ
%
−− −−
AB DE
−− −−
AC DF
−− −−
AC BC
−− −−
AB EF
;
5
Two triangles are congruent if each side
of one triangle is congruent to which of
the following of the second triangle?
A
B
C
D
base
hypotenuse
corresponding side
corresponding angle
A28 Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
F
G
H
J
"
:
&
Which is enough to prove that
ABC DEF?
9
F
G
H
J
'
#
$
'
C 16
D 25
$
The right triangles shown below are
congruent. Which are the
corresponding sides?
#
%
Given: ABC and DEF are right
−− −−
triangles, and AC DF.
"
2
$
Name
Date
Practice by Standard
Geometry 6.0
Students know and are able to use the
triangle inequality theorem.
G6.0
1
$
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
D
F
G
H
J
What is the largest possible
whole number base of an isosceles
triangle with equal sides each
7.3 inches?
7 in.
10 in.
14 in.
15 in.
C
#
"
Which of the following triangles could
not be drawn?
A
B
C
D
B
the 2 ft branch
the 4 ft branch
the 7 ft branch
the 12 ft branch
F an equilateral triangle whose sides
are each 5 inches long
G a scalene triangle with one 6 inch side,
one 8 inch side, and one 10 inch side
H an isosceles triangle with two 5 inch
sides and one 12 inch side
J an isosceles triangle with two sides
5 inches and the remaining side
2 inches
3
Given the triangle sketched below, which
of the following must be true?
Maurice is hiking through a forest.
He wants to make a triangular fire pit.
He has a 3-foot branch and an 8-foot
branch. He finds four more branches
with lengths listed below. Which will
allow him to complete his triangular
fire pit?
A
B
C
D
2
4
5
Which of the following sets of line
segment lengths could be used to create
a triangle?
A
B
C
D
6
b+c>a
a2 + b2 = c2
m∠A + m∠B = m∠C
a>b
3 in., 4 in., 8 in.
5 in., 10 in., 12 in.
7 in., 11 in., 20 in.
19 in., 82 in., 120 in.
The triangle inequality theorem
references which fact?
F The angles in all triangles that are not
equilateral do not have equal measures.
G The sum of the lengths of any two
sides of a triangle must be greater than
the length of the third side.
H Two congruent triangles must have all
sides equal.
J The sum of the measures of the acute
angles in a right triangle is equal to the
measure of the largest angle.
Mastering the California Mathematics Standards, Geometry
A29
Name
Date
Practice by Standard
Geometry 7.0
Students prove and use theorems using
the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the
properties of circles.
G7.0
1
3
A
B
C
D
In the figure below, parallel lines b and
c are cut by transversal a.
B
C
D
4
A circle is drawn on a coordinate
plane. It has its center at (-3, 5).
The point (-3, 0) is on the circle.
Which of the following points is also
on the circle?
F
G
H
J
(0, 5)
(0, -3)
(-3, 10)
(-5, -3)
5
Given parallelogram ABCD, which
expression represents m∠B?
A
B
C
D
6
All squares are rectangles.
All rectangles are parallelograms.
All parallelograms are trapezoids.
All trapezoids are quadrilaterals.
m∠A + m∠C + m∠D
90° + m∠A
180° - m∠C
m∠A - m∠C
Which of the following pairs of
angles would not necessarily
be supplementary?
F
G
H
J
angles that would form a straight line
adjacent angles in a parallelogram
any pair of angles in a rectangle
alternate exterior angles
A30 Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
180° - m∠1 = m∠2
m∠1 ≠ m∠2
m∠1 ≥ m∠2
m∠1 + m∠2 = 90°
Quadrilateral ABCD is a square.
Quadrilateral ABCD is a rhombus.
Quadrilateral ABCD is a rectangle.
Quadrilateral ABCD is a
parallelogram.
Which of the following statements is
not true?
F
G
H
J
Which of the following statements must
be true?
A
B
C
D
Figure ABCD is a quadrilateral. If
the lengths of opposite sides are
congruent, which of the following
must be true?
Name
Date
Practice by Standard
Geometry 7.0 (continued)
7
Circle A has diameter d meters, radius
r meters, and circumference c meters.
What is a result of dividing the
circumference by the diameter?
A 2r
B π
11
A rhombus
B square
C 2
D 2π
12
8
What minimal set of conditions
would have to be true to prove
that parallelogram ABCD is
a rectangle?
"
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
F
G
H
J
9
m
3
7
4
8
n
11 12
15 16
p
if m∠1 + m∠15 = 180°, then which of
the following need not be true?
F np
G ∠1 ∠11
Which of the following is true for
all quadrilaterals?
Which of the following quadrilaterals
has the most lines of symmetry?
2
6
9 10
13 14
−− −−
AB = CD
−− −−
AC = BD
m∠A + m∠B = 180°
m∠A = m∠C
F rhombus
G square
Parallel lines l and m are cut by
transversals n and p
1
5
$
A The diagonals bisect each other.
B The sum of the interior angles is equal
to the sum of the exterior angles.
C The lengths of at least two sides are equal.
D The diagonals bisect the angles of the
quadrilateral.
10
C rectangle
D kite
#
%
Which of the following quadrilaterals
can have diagonals that do not intersect
to form 90° angles?
13
H m∠3 = m∠16
J n⊥m
If the diagonals of a quadrilateral are
perpendicular bisectors, then…
A …all interior angles of the
quadrilateral must be right angles.
B …the quadrilateral is an isosceles
trapezoid.
C …all interior angles are bisected by
the diagonals.
D …the quadrilateral must be a square.
H rectangle
J kite
Mastering the California Mathematics Standards, Geometry
A31
Name
Date
Practice by Standard
Geometry 7.0 (continued)
14
In parallelogram ABCD, which two
angles are congruent?
"
17
In the figure below, line m is parallel to
line n. Find the measure of ∠BCA.
#
$
40‚
&
%
F
G
H
J
15
35°
70°
105°
140°
"
A
B
C
D
18
#
40°
50°
90°
140°
−−
In circle X, the midpoints of AB
−−
and CD are 5 inches from the center.
Which of the following statements is
true?
#
$
"
X
Which of the following can be drawn
without any congruent sides?
F
G
H
J
n
130‚
∠DAE and ∠EAB
∠CED and ∠CEB
∠ABC and ∠CDA
∠ADC and ∠DAB
%
rhombus
parallelogram
kite
trapezoid
−−
F The length of AB is equal to the length
−−
of CD.
−−
G AB must be a diameter.
−−
−−
H AB and CD are perpendicular
bisectors.
is equal to
J The measure of AB
the measure of BC.
A32 Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
16
$
M and N are points on circle O.
The measure of central angle MON
is 70°. What is the measure of MN ?
A
B
C
D
m
Name
Date
Practice by Standard
Geometry 8.0
Students know, derive, and solve
problems involving perimeter, circumference,
area, volume, lateral area, and surface area of
geometric figures.
G8.0
1
4
A cylinder has radius 2 inches and
height 8 inches
A student knows that the area of a
parallelogram is found by multiplying
the base by the height. Drawing the
diagonal for the parallelogram is one
way to illustrate which formula?
A
B
C
D
2 in.
area of a triangle
area of a rectangle
perimeter of a parallelogram
area of a trapezoid
8 in.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
If you needed to paint the entire
cylinder, with the exception of the two
bases, what area would you paint?
A
B
C
D
5
F
G
H
J
10π sq in.
16π sq in.
32π sq in.
64π sq in.
6
2
Circle A has radius 3 cm. Circle B
has diameter 8 cm. What is the sum
of their areas?
F
G
H
J
Circle A has area 81π square
inches. Find the circumference
of circle A.
A
B
C
D
9π in.
18π in.
81π in.
162π in.
72 cm
22 cm
11 cm
8 cm
A truck tire has a diameter of 3 feet.
How far will the truck travel in 20
rotations?
A
B
C
D
11π cm2
24π cm2
25π cm2
73π cm2
7
3
A rectangle has an area 24 cm2 and
length 3 cm. What is its perimeter?
30π ft
60π ft
120π ft
180π ft
A triangle has a base 12 inches long
and an area of 36 square inches. Find
the length of the altitude.
A
B
C
D
3 in.
4 in.
5 in.
6 in.
Mastering the California Mathematics Standards, Geometry
A33
Name
Date
Practice by Standard
Geometry 8.0 (continued)
8
Find the area of trapezoid ABCD.
"
5 in.
%
A
B
C
D
4 in.
4.5 in.
9 in.
11
#
6 in.
$
F
G
H
J
24 sq in.
28.5 sq in.
29.25 sq in.
40.5 sq in.
12
9
A prism has volume 90 cm3. It has
a square base whose area is 9 cm2.
What is its surface area?
10
64 ft2
108 ft2
144 ft2
504 ft2
A school put in a new football field.
The field has a running track around
its perimeter. The dimensions are
shown in the figure below.
360 ft
138 cm2
198 cm2
270 cm2
810 cm2
120 ft
If the groundskeeper could mow
400 square feet per minute, how long
would it take her to mow the entire
field, to the nearest minute?
Which of the following techniques can
be used to find the volume of any right
prism or cylinder?
A find the area of each side and multiply
by the height
B multiply the length and the width and
the height
C double the area of each side and add
the results together
D find the area of the base and multiply
by the height
A
B
C
D
13
100 minutes
108 minutes
136 minutes
185 minutes
Refer to the figure in question 12.
A runner wants to jog around the
perimeter of the field. How far will
the runner go in one lap?
F 720 ft
G 960 ft
A34 Mastering the California Mathematics Standards, Geometry
H 1005 ft
J 1097 ft
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
F
G
H
J
A carpenter needs 42 feet of crown
molding to finish the perimeter of a
rectangular room. One side of the room
is 12 feet long. How much carpet will
he need to finish the room?
Name
Date
Practice by Standard
Geometry 9.0
Students compute the volumes and
surface areas of prisms, pyramids, cylinders,
cones, and spheres; students commit to
memory the formulas for prisms, pyramids
and cylinders.
G9.0
1
4
A fruit juice company sells its juice
in containers that are 4.5 cm long,
2.5 cm wide, and 6 cm tall. How much
material is needed to cover the outside
of one container?
F
G
H
J
Which of the following volume
formulas is incorrect?
1
A volume of a pyramid = _
Bh
3
53.25 cm2
67.5 cm2
106.5 cm2
135 cm2
B volume of a cube = s3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
D volume of rectangular prism = ℓ2h
A beach ball has a diameter of
24 inches. What is its approximate
surface area, in square inches?
(Surface Area = 4πr 2)
A
B
C
D
F
G
H
J
3
5
A pyramid has a square base with each
side measuring 6 meters. The distance
from the center of the base to the top
of the pyramid is 9 meters, What is the
volume of the pyramid?
C volume of a cylinder = πr2h
3617 sq in.
1809 sq in.
1152 sq in.
96 sq in.
6
A company sells seasoning in cans
that are 14 cm tall. The cans have
radius 4 cm. How much seasoning
will the cylinder hold, to the nearest
cubic centimeter?
A
B
C
D
18 m3
54 m3
108 m3
324 m3
Which of the following surface area
formulas is incorrect?
F surface area of a cube = 6s2
G surface area of a rectangular
prism = 2ℓwh
H surface area of a
cylinder = 2πr2 + 2πrh
J surface area of a regular square
pyramid = s2 + 2sℓ
56 cm3
224 cm3
704 cm3
2463 cm3
Mastering the California Mathematics Standards, Geometry
A35
Name
Date
Practice by Standard
Geometry 10.0
Students compute areas of polygons,
including rectangles, scalene triangles,
equilateral triangles, rhombi, parallelograms
and trapezoids.
G10.0
1
4
F a scalene triangle with perimeter
54 in., base 20 in., and height 10 in.
G a rhombus with side 13 in., short
diagonal 10 in., and long
diagonal 24 in.
H a trapezoid with short base 12 in., long
base 16 in., and height 8 in.
J a parallelogram with short side 12 in.,
long side (base) 13 in., and height 9 in.
The triangle shown on the coordinate
plane below has vertices at (-3, 5),
(-8, -4) and (5, 5). What is its area,
in square units?
6
5
4
3
2
1
−8−7−6−5−4−3−2−1O
y
1 2 3 4 5 6x
−2
−3
−4
Which of the following figures will have
the greatest area?
5
Which of the following sketches does
not illustrate the altitude (height) of
a triangle?
A
C 36
D 72
h
B
2
F 30 sq in.
G 43 sq in.
3
h
One side of an equilateral triangle is
10 inches long. Find the area, to the
nearest square inch.
H 50 sq in.
J 100 sq in.
Juanita is going to put a new floor
in her kitchen. Her kitchen is 15 feet
long and 18 feet wide. She wants to
use tiles that are 9 inches square. How
many tiles will she need to cover the
kitchen floor?
A 480
B 270
C
h
D
C 136
D 66
A36 Mastering the California Mathematics Standards, Geometry
h
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A 24
B 32
Name
Date
Practice by Standard
Geometry 10.0 (continued)
6
Figure ABCD is a rhombus. Find
its area.
9
#
Rectangle ABCD has a length of 24 cm
and a width of 16 cm. Find the area of
the inscribed rhombus MNOP.
"
5 in.
3 in.
$
4 in.
A
B
C
D
%
12 sq in.
20 sq in.
24 sq in.
60 sq in.
10
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
7
8
#
/
1
"
%
F
G
H
J
.
0
$
80 cm2
96 cm2
192 cm2
384 cm2
Isosceles trapezoid EFGH has area
80 cm2, height 20 cm, and legs 22 cm.
One of the bases is 5 cm. How long is
the remaining base?
Carlos has developed software that
allows the user to instantly find the
lengths of all sides of any polygon.
Which of the following figures needs
more information before its area can
be calculated?
A
B
C
D
F
G
H
J
3 cm
4 cm
20 cm
24 cm
If the base of parallelogram MNOP
is 1 inch less than twice its height,
which expression represents the area
of the parallelogram?
F (x)(2x - 1)
G 2(2x - 1)
H (2x - 1)2
(x)(2x - 1)
J _
2
11
right triangle
rectangle
square
rhombus
A kite has one diagonal 12 inches long
and another diagonal 8 inches long.
Which figure has the same area as the
kite?
A rectangle with length 12 in. and width
8 in.
B triangle with base 12 in. and height 8 in.
C parallelogram with base 12 in. and
height 8 in.
D square with side 10 in.
Mastering the California Mathematics Standards, Geometry
A37
Name
Date
Practice by Standard
Geometry 11.0 (continued)
Students determine how changes
in dimensions affect the perimeter, area, and
volume of common geometric figures and solids.
G11.0
1
x
A The volume doubles.
B The volume triples.
C The volume becomes four
times greater.
D The volume becomes eight
times greater.
Zach filled the boxes shown below
with packing peanuts.
20 in.
15 in.
6
10 in.
10 in.
10 in.
10 in.
How much more will fit in the
taller box?
A 500 in3
B 100 in3
The side lengths of a cube are
doubled. What happens to the
volume of the cube?
The heights of two similar rectangular
prisms are in a ratio of 9 to 4. What is
the ratio of the volumes of these two
rectangular prisms?
F 3 to 2
G 9 to 4
C 1500 in3
D 2000 in3
A38 Mastering the California Mathematics Standards, Geometry
H 81 to 16
J 729 to 64
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5
4 to 3
16 to 9
32 to 18
256 to 81
2x
F The circumference of the larger circle
is twice that of the smaller circle.
G The circumference of the larger circle
is four times that of the smaller circle.
H The area of the larger circle is twice
that of the smaller circle.
J The area of the larger circle is eight
times that of the smaller circle.
60 cm2
90 cm2
120 cm2
300 cm2
The ratio of the two perimeters of
two similar rectangles is 16 to 9.
What is the ratio of the areas of
these two rectangles?
F
G
H
J
3
Which is a true statement about the
circles shown below?
A parallelogram has an area of 30 cm2.
If both the base and the height of the
parallelogram were doubled, what
would be the new area?
A
B
C
D
2
4
Name
Date
Practice by Standard
Geometry 12.0
Students find and use measures of
sides and of interior and exterior angles of
triangles and polygons to classify figures and
solve problems.
G12.0
1
4
F
G
H
J
What is the value of x?
2
88‚
A
B
C
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
x‚
31‚
1
3
5
A regular hexagon is shown below.
(x + 50)‚
3
What is the value of x?
A
B
C
D
4
(x - 25)‚
(x + 35)‚
6
3
3
4
5
6
61
92
119
149
−− −−
In the figure below, RS || TU. What is
the value of x?
F
G
H
J
If the measure of an exterior angle of a
regular polygon is 72°, how many sides
does the polygon have?
5
60
65
85
90
What is the value of x?
69‚
158‚
Two exterior angles of a triangle
measure 153° and 105°. Which
could not be an interior angle
measure of the triangle?
A 27°
B 75°
6
40
70
120
130
F
G
H
J
x‚
22
47
69
111
C 78°
D 102°
Mastering the California Mathematics Standards, Geometry
A39
Name
Date
Practice by Standard
Geometry 12.0 (continued)
7
The sum of the interior angles of
a polygon is two times the sum of
its exterior angles. What type of
polygon is it?
A
B
C
D
8
10
Two angles of a triangle measure 84°
and 35°. Which of the following could
not be a measure of an exterior angle
of the triangle?
F
G
H
J
triangle
quadrilateral
hexagon
octagon
What is m∠C in the quadrilateral
shown below?
11
96°
119°
131°
145°
What is the value of x?
x‚
#
x‚
$
116‚
(x + 45)‚
"
9
50‚
A 26
B 54
%
65°
100°
135°
145°
12
A regular pentagon is shown below.
(x - 31)‚
What is the value of x?
A
B
C
D
41
72
77
103
If the measure of an interior angle
of a regular pentagon is (x + 26)°,
what is the value of x?
F 46
G 72
13
C 64
D 154
H 82
J 108
Two exterior angles of a quadrilateral
measure 112° and 38°. Which could
be the measures of the other two
exterior angles?
A 90°, 100°
B 100°, 100°
A40 Mastering the California Mathematics Standards, Geometry
C 100°, 110°
D 150°, 150°
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
F
G
H
J
65‚
Name
Date
Practice by Standard
Geometry 13.0
Students prove relationships between
angles in polygons by using properties of
complementary, supplementary, vertical, and
exterior angles.
G13.0
1
3
A quadrilateral has two pairs of
supplementary angles. Two of the
angles are 135° and 55°. What are
the other two angles?
A
B
C
D
What is m∠1?
150‚
4
45°, 45°
90°, 90°
35°, 145°
45°, 125°
What is m∠ACB?
1
%
114‚
37‚
#
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
$
A
B
C
D
2
30°
66°
84°
96°
&
"
F
G
H
J
A quadrilateral and its diagonals are
shown below.
87‚
40‚
17°
30°
53°
87°
45°
61°
82°
119°
110‚
3
5
What is m∠3?
62‚
What is m∠3?
F
G
H
J
98‚
120‚
3
A
B
C
D
28°
30°
32°
58°
Mastering the California Mathematics Standards, Geometry
A41
Name
Date
Practice by Standard
Geometry 14.0 and 15.0
Students prove the Pythagorean theorem.
Students use the Pythagorean theorem
to determine distance and find missing lengths
of sides of right triangles.
G14.0
4
G15.0
A new bike path is being built to go
through a forest. The figure below
shows the old path and the new path.
OLD BIKE PATH
1
A right triangle’s hypotenuse has
length 7. If one leg has length 4,
what is the length of the other leg?
A
B
C
D
2
NEW BIKE PATH
8 miles
15 miles
About how many miles longer is
the new bike path?
3
√
33
√
65
11
F
G
H
J
6 mi
7 mi
17 mi
23 mi
Find the value of x in the right triangle
below.
5
a
27
c
F 5
G 7 √
5
3
x
H √
1213
J 49
The legs of a right triangle have
lengths 7 and 24. What is the
length of the hypotenuse?
A
B
C
D
17
√
527
25
31
b
Which statement could be used in the
proof of the Pythagorean theorem?
A The area of each triangle
1
equals _
ac.
2
B The area of the smaller square
is equal to half the area of the larger
square.
C The area of the smaller square
equals bc.
D All of the right triangles around
the smaller square are congruent.
A42 Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
22
A figure from a proof of the Pythagorean
theorem is pictured below.
Name
Date
Practice by Standard
Geometry 16.0
Students perform basic constructions
with a straightedge and compass, such as angle
bisectors, perpendicular bisectors, and the
line parallel to a given line through a point off
the line.
G16.0
1
3
Tanesha is constructing a line parallel
to line ℓ through point P. Which of the
following should be her first step?
A
1
Fayad is using a straightedge and
a compass to do the construction
shown below.
B
1
"
#
C
1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Which best describes Fayad’s
construction?
A
B
C
D
D
bisecting an angle
bisecting a segment
−−
making a line parallel to AB
−−
making a line congruent to AD
4
2
Anna wants to use a straightedge and a
compass to construct an angle
congruent to ∠R shown below.
3
What is the first step she
should take?
F Use a straightedge to draw a ray.
G Use a protractor to measure ∠R.
H Adjust the compass so that it is
the width of the largest part of ∠R.
J From the vertex of ∠R, draw an
arc through one side of the angle.
1
Jacob plans to use a straightedge
and a compass to construct a line
that is perpendicular to line ℓ and
passes through point J, which is a
point not on ℓ. What is the first
step he should take?
F From point J, draw an arc that intersects
line ℓ in two different places.
G From point J, draw an arc above J and
an arc below J.
H Draw a line through point J
intersecting line ℓ.
J Draw a line through point J parallel
to line ℓ.
Mastering the California Mathematics Standards, Geometry
A43
Name
Date
Practice by Standard
Geometry 16.0 (continued)
5
Emily is using a straightedge and
a compass to do the construction
shown below.
7
Carlos plans to use a straightedge and
compass to construct a perpendicular
−−
bisector of AC in ABC shown below.
#
1
"
2
3
Which shows the construction?
Which best describes Emily’s
construction?
A
B
C
D
A
−−
a line through P parallel to QR
−−
a line through P intersecting QR
−−
a bisector of QR
a bisector of ∠Q.
What is the first step in constructing
a line perpendicular to line m through
point D?
#
"
B
$
#
"
$
"
%
#
C
#
$
m
F Draw line CD
.
G From point A, draw an arc through
point B.
H From point D, draw equal arcs that
intersect at A and B.
J From points A and B, draw equal
arcs that intersect at C.
"
D
$
#
"
A44 Mastering the California Mathematics Standards, Geometry
$
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6
$
Name
Date
Practice by Standard
Geometry 17.0
Students prove theorems by using
coordinate geometry, including the midpoint
of a line segment, the distance formula, and
various forms of equations of lines and circles.
G17.0
1
4
Given that quadrilateral RSTU is a
parallelogram, which is necessary in
order to conclude that RSTU is a
rectangle?
y
What are the coordinates of the point
of intersection of the diagonals
of JKLM?
S
T
O R
U
y
K (j, k)
O
A
J (j, 0)
x
L (m, k)
M (m, 0)
−−
−−
F (slope SU )(slope RT ) = 1
−−
−−
G (slope SU )(slope RT ) = -1
H distance from R to T = distance
from R to U
J distance from R to T = distance
from S to U
x
(_m2 , _2k )
(_2j , _2k )
j+m k
C ( _, _
2
2)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
B
(
5
The figure below shows rectangle ABCD.
)
y
j+m j+k
D _, _
2
2
B
A
2
What type of triangle is formed by
the points P(1, 6), Q(-2, 3), and
R(8, -1)?
F right
G acute
3
H isosceles
J equilateral
What type of figure is formed by
the points F(-2, 1), G(0, 5), H(6, 5),
and J(4, 1)?
A square
B rectangle
O
C
D
x
Which is a true statement?
A
B
C
D
−−
−−
(slope AB )(slope BC ) = -1
−−
−−
(slope AB )(slope BC ) = 1
−−
−−
slope AB = slope BC
−−
−−
slope AB = 2(slope BC)
C trapezoid
D parallelogram
Mastering the California Mathematics Standards, Geometry
A45
Name
Date
Practice by Standard
Geometry 17.0 (continued)
6
The figure below shows FGH.
y
O F
9
G
A
B
C
D
Hx
Which statement would prove that
FGH is an isosceles triangle?
10
−−
−−−
F (slope FG )(slope GH ) = 1
−−
−−−
G (slope FG )(slope GH ) = -1
H distance from F to G = distance
from G to H
J distance from F to G = -(distance
from G to H)
8
F
G
H
J
(6, 4)
(4, 6)
(2, 3)
(3, 2)
P
N
Q
O
M
x
Which statement would prove that
MNPQ is a rhombus?
−−
−−−
F (slope MP )(slope NQ ) = 1
−−
−−−
G (slope MP )(slope NQ ) = –1
H distance from N to Q = distance from
M to P
1
J distance from N to Q = _
(distance
2
from M to P)
right
scalene
isosceles
equilateral
The diameter of a circle has endpoints
at (1, -1) and (5, 5). What are
the coordinates of the center of
the circle?
The figure below shows
parallelogram MNPQ.
y
What type of triangle is formed by
the points J(-3, 5), K(1, 10), and
L(4, 0)?
A
B
C
D
square
triangle
kite
trapezoid
11
What type of figure is formed by
the points W(-1, 6), X(5, 6), Y(2, 3),
and Z(-1, 3)?
A square
B rhombus
C trapezoid
D rectangle
A46 Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
7
A figure is formed by the points A(0, 0),
B(a, 0), C(a, a), and D(0, a). What type
of figure is formed?
Name
Date
Practice by Standard
Geometry 18.0
Students know the definitions of
the basic trigonometric functions defined
by the angles of a right triangle. They also
know and are able to use elementary
relationships between them. For example,
G18.0
tan (x) =
1
sin (x)
_
, (sin (x))
cos (x)
2
3
In the figure below, the flagpole has
height h. In the triangle, tan x = 1.5.
How many feet tall is the flagpole?
+ (cos (x)) 2 = 1.
I
In the figure below, sin B = 0.8.
#
Y
GU
"
A 16 ft
B 25.5 ft
$
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
−−
What is the length of AB ?
A
B
C
D
2
9.6
12
12.8
15
4
17
what is cos x?
Y
G
H
J
10
6 , and tan x = _
6
F sin x = _
10
8
6
8
_
_
G sin x = , and tan x =
10
6
10
6
H sin x = _, and tan x = _
8
6
10
10
_
_
J sin x = , and tan x =
8
6
_
F
_
In a right triangle, cos x = 8 . What
are sin x and tan x?
In the figure below, if sin x = 8 , then
8
_
15
15
_
17
15
_
8
17
_
15
C 36 ft
D 48 ft
5
_
In a right triangle, tan x = 35 .
12
What is sin x?
A
12
_
35
12
B _
37
35
_
C
37
37
D _
35
Mastering the California Mathematics Standards, Geometry
A47
Name
Date
Practice by Standard
Geometry 18.0 (continued)
6
_
In a right triangle, cos x = 7 .
25
Which correctly shows the triangle?
8
In the triangle below, tan x ≈ 0.47.
Approximately how far is the cat from
the girl?
F
Y
GU
G
Y
Y
E
F
G
H
J
H
Y
J
Y
9
11.9 ft
10.6 ft
4.7 ft
2.4 ft
_
In a right triangle, cos x = 24
a , and
_
sin x = 7a . What is tan x?
7
7
7
B _
25
In the figure below, tan A = 1.5.
"
10
24
C _
7
7
D_
24
_
In the figure below, if tan x = 48 , then
14
what is sin x?
$
#
Y
−−
What is the length of AC ?
A
B
C
D
24
36
40
54
14
F _
48
14
G _
50
50
H _
48
48
J _
50
A48 Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
25
A _
Name
Date
Practice by Standard
Geometry 19.0
Students use trigonometric functions to
solve for an unknown length of a side of a right
triangle, given an angle and a length of a side.
G19.0
1
3
In the triangle below, which equation
should be used to find the length of the
hypotenuse?
Triangle RST is shown below.
35‚
5
56‚
3
C
4
Which equation should be used to find
−−
the length of RS ?
A b = 24 sin 35°
B b = 24 cos 35°
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
RS
A sin 56° = _
31
31
B sin 56° = _
RS
RS
C cos 56° = _
31
31
D cos 56° = _
RS
2
24
C b=_
sin 35°
24
D b=_
cos 35°
4
In the figure below, m∠K = 41°, and
MK = 18. Which equation could be
used to find x in KLM?
The figure below shows a 10-foot ladder
leaning against a wall. The ladder
makes a 62° angle with the ground.
Which is closest to how far up the
ladder reaches on the wall?
,
41‚
10 ft
62‚
.
Y
sin 62‚≈ 0.88
cos 62‚≈ 0.47
tan 62‚≈ 1.88
-
F x = 18 sin 41°
H x = 18 tan 41°
G x = 18 cos 41°
18
J x=_
sin 41°
F 4.7 ft
G 6.2 ft
H 8.8 ft
J 18.8 ft
Mastering the California Mathematics Standards, Geometry
A49
Name
Date
Practice by Standard
Geometry 19.0
5
In a right triangle, one angle has
measure 26°. The side opposite that
angle is 9 cm long. Which is closest
to the length of the hypotenuse?
8
In the figure below, m∠Q = 17°, and
NP = 23. Which equation could be used
to find the value of x in NPQ?
1
sin 26° ≈ 0.44
cos 26° ≈ 0.90
tan 26° ≈ 0.49
A 2.9 cm
B 10.0 cm
6
2
17‚
Y
/
C 18.4 cm
D 20.5 cm
23
F x=_
sin 17°
23
G x=_
cos 17°
tan 17°
H x=_
23
cos
17°
J x=_
23
Which equation could be used to find
a in the right triangle below?
9
27‚
55
F sin 27° = _
a
55
G cos 27° = _
a
a
H cos 27° = _
GU
55
55
J tan 27° = _
a
32‚
7
In a right triangle, one angle has
measure 50° and hypotenuse 6 inches.
Which equation could be used to find x,
the side opposite the 50° angle?
x
A sin 50° = _
6
6
B sin 50° = _
x
sin 32‚≈ 0.53
cos 32‚≈ 0.85
tan 32‚≈ 0.62
A
B
C
D
10.6
12.4
23.6
32.0
x
C cos 50° = _
6
6
D cos 50° = _
x
A50 Mastering the California Mathematics Standards, Geometry
Y
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
B
The figure below shows a 20-foot water
slide. The slide makes a 32° angle with
the ground. Which is closest to the
length of the ladder?
Name
Date
Practice by Standard
Geometry 20.0
Students know and are able to use
angle and side relationships in problems with
special right triangles, such as 30°, 60°, and 90°
triangles and 45°, 45°, and 90° triangles.
G20.0
1
3
What is the value of x in the
triangle below?
'
What is the value of x in the
triangle below?
#
)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A
B
C
D
2
A
B
C
D
"
$
Y
4
4 √
2
8 √
2
16
4
C
5
Y
(
10
10 √
2
20 √
3
40
The hypotenuse of a 45°, 45°, 90°
triangle is 26 √
2 inches. What is
the length of each of the other
two sides?
F
G
H
J
In the right triangle below, a = 12.
What is the value of c?
D
Y
13 in.
13 √
2 in.
√
13 3 in.
26 in.
If b = 4 √
3 in the right triangle below,
then what is the value of c?
C
30‚
60‚
B
D
F
G
H
J
6
6 √
3
12 √
3
24
A
B
C
D
B
8
12
8 √
3
16
Mastering the California Mathematics Standards, Geometry
A51
Name
Date
Practice by Standard
Geometry 21.0
Students prove and solve problems
regarding relationships among chords, secants,
tangents, inscribed angles, and inscribed and
circumscribed polygons of circles.
G21.0
1
3
In the figure below, AD
is tangent
to circle M at point D, AC
intersects
circle M at points B and C,
m
BD = 64°, and m
BC = 96°.
−−
−−
In the circle below, HJ and KL are
chords intersecting at M.
$
96‚
,
.
#
64‚
)
"
+
.
%
What is m∠DAB?
-
A
B
C
D
If HM = 6, JM = 6, and LM = 9, then
−−
what is the length of KM ?
C 12
D 36
4
2
BG
is tangent at point B to a circle
−−
whose center is C. BD is a diameter.
In the figure below, secants QS
and RT
intersect at point P, m
QR = 63°, and
m
TS = 81°.
3
"
63‚
%
4
1
81‚
(
$
2
32‚
5
#
What is m∠SPT?
What is m∠ABG?
F 40°
G 58°
H 90°
J 116°
F
G
H
J
18°
36°
72°
144°
A52 Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A 3
B 4
68°
100°
136°
200°
Name
Date
Practice by Standard
Geometry 21.0
5
In the figure below, FGH is
inscribed in circle A, m
FH = 156°, and
m
GH = 116°.
7
(
A square is circumscribed about a
circle. What is the ratio of the
perimeter of the square to the
circumference of the circle?
A
'
116‚
"
_8
2
C _
π
4
D_
π
1
4
B _
1
156‚
)
8
What is m∠FHG?
A 88°
Triangle ABC is circumscribed about the
circle. In the figure, AD = 6, DB = 4,
and the perimeter of ABC is 40.
#
B 64°
C 44°
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
D 32°
"
6
&
%
−−
What is the length of FC ?
In the figure below, ABC is inscribed
in circle T and m
AB = 50°.
#
50‚
F 10
G 15
H 20
J 30
$
5
"
9
−−
In the figure below, CD is tangent to
circle B at point C.
$
What is m∠BAC?
F
G
H
J
$
'
25°
40°
55°
65°
#
'
%
−−
What is the length of FD ?
A 7
B 8
C 9
D 17
Mastering the California Mathematics Standards, Geometry
A53
Name
Date
Practice by Standard
Geometry 22.0
Students know the effect of rigid
motions on figures in the coordinate plane
and space, including rotations, translations,
and reflections.
G22.0
1
3
4
3
)
2
1
If triangle EFG is rotated 180 degrees
about the origin, what would be the
coordinates of G´?
5
4
3
2
1
O
−5−4−3−2
−4−3−2
y
'
1 2 3 4 5x
A
B
C
D
C (–2, 4)
D (–2, –4)
4
/
.
5
4
3
2
-
1
−5−4−3−2
O
y
−2
−3
F (–5, –1)
G (5, 1)
,
(x, y)
(x + 1, y + 2)
(x, y)
(x – 1, y + 2)
(x, y)
(x + 2, y – 1)
(x, y)
(x – 2, y + 1)
Square PQRS below is to be translated
to square P´Q´R´S´ by the following
motion rule.
(x, y)
(x + 2, y – 6)
1
1 2 3x
1 2 3 4x
O
Which motion rule describes the
translation?
(
&
If trapezoid LMNP is reflected across
the y-axis, what would be the
coordinates of L´?
+
5
2 4
3
2
3
1
4
−4−3−2
O
y
1 2 3 4x
−2
−3
H (1, 5)
J (–1, 5)
What will be the coordinates of
vertex P´?
F (–2, –3)
G (–3, –1)
A54 Mastering the California Mathematics Standards, Geometry
H (1, –1)
J (–10, 5)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A (–4, –2)
B (–4, 2)
y
−2
−3
−4
−2
−3
−4
−5
2
Triangle HJK below is translated so
that the coordinates of the new vertices
are H´(–2, 4), J´(1, 4), and K´(2, 0).
Name
Date
Practice by Standard
Geometry 22.0
5
The vertices of parallelogram ABCD
are A(–3, 0), B(–1, 3), C(–1, –2), and
D(–3, –5). If the figure is translated
4 units to the right and 2 units up, what
are the coordinates of vertex B´?
8
Triangle TUV has vertices T(–2, 1),
U(2, 4), and V(0, –1). Which shows
TUV translated 3 units down and
1 unit to the left?
F
A
B
C
D
6
(–3, –1)
(–5, 1)
(1, 7)
(3, 5)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5h
+
F
G
H
J
3
2
1
O
−2
−3
−4
−5
G
1 2 3 4x
5
4
3
2
1
5h
y
y 6h
1 2 3 4x
−4−3−2 7hO
−2
−3
1 2 3 4x
H
(5, 3)
(3, 0)
(3, 5)
(3, –5)
4
3
2
1
5h
y
6h
1 2 3 4x
O
−4−3−2
The vertices of ABC are A(0, 6),
B(2, 1), and C(–3, 4). If the figure is
reflected across the x-axis to create
WXY, what would be the coordinates
of the vertices of WXY?
A
B
C
D
6h
−2
−3
−4
7h
−5
−2
−3
−4
7h
J
7
y
O
−4−3−2
If triangle JKL is rotated 180 degrees
about the origin, what are the
coordinates of J´?
,
−4−3−2
3
2
1
5h
−5−4−3−2
7h
6h 4
3
2
1
O
y
1 2 3x
−2
−3
−4
W(–6, 0), X(2, 1), Y(–3, –4)
W(–3, –4), X(2, 1), Y(0, –6)
W(0, –6), X(2, –1), Y(–3, –4)
W(0, 6), X(–2, 1), Y(3, –4)
Mastering the California Mathematics Standards, Geometry
A55
Standards Assessment
Student Answer Sheet
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Record your answers by coloring in the appropriate bubble for the best answer to
each question.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
Mastering the California Mathematics Standards, Geometry A57
Name
Date
Standards Assessment
1
_
Use the triangle below. If cos x = 3 ,
5
then what is tan x?
4
A truck tire has a diameter of 3 feet.
How many feet will the tire roll in 5
revolutions along a smooth surface?
F
G
H
J
x
9.4 ft
15 ft
47.1 ft
360 ft
1
A _
2
3
B _
4
5
C _
6
4
D _
3
5
Find the length of the missing leg in the
right triangle shown below.
3
7
x
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
Timo’s aquarium is 12 inches long and
8 inches wide. The water is 6 inches
deep. How much water is in the
aquarium?
F
G
H
J
26 in 3
432 in 3
576 in 3
1152 in 3
A
B
C
D
6
2.5
6.3
36
40
What is the area of the shaded region
of the figure shown below?
15 m
3
Which type of quadrilateral is formed
by the points P(2, 3), Q(-3, 3),
R(-3, -5), and I(2, -5)?
A
B
C
D
rectangle
square
trapezoid
kite
4m
10 m
F
G
H
J
4m
16 m 2
110 m 2
134 m 2
150 m 2
Mastering the California Mathematics Standards, Geometry
A59
Name
Date
Standards Assessment
7
10
Find the measure of angle A.
%
(continued)
A baseball has diameter 3 inches. What
is the approximate surface area of a
baseball? (Surface Area = 4πr 2)
#
120‚
"
30‚
$
3 in.
&
A
B
C
D
8
150°
120°
90°
45°
Which of the following pairs of figures
must be similar?
11
9 sq in.
10 sq in.
28.3 sq in.
113.1 sq in.
Quadrilateral QUAD is translated
4 units to the left and 3 units up.
two right triangles
two squares
two rectangles
two parallelograms
4
3
2
2
1
−4−3−2% O
9
−2
−3
−4
y
6
1 2 3 4x
"
The sum of the exterior angles of a
particular polygon is twice the sum
of the polygon’s interior angles.
What type of polygon is it?
What are the coordinates of
vertex A?
A
B
C
D
A
B
C
D
triangle
quadrilateral
pentagon
hexagon
(1, -2)
(-2, 1)
(-1, 2)
(2, -1)
A60 Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
F
G
H
J
F
G
H
J
Name
Date
Standards Assessment
12
A square is inscribed in a circle. What
is the ratio of the area of the square to
the area of the circle?
2
F _
π
15
Fill in the missing reason in the
proof shown.
Given: ABCD is a square.
−− −−
Prove: AC BD.
π
G _
2
2
H _
1
1
J _
2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
13
14
(continued)
"
#
%
$
If PQRS is a quadrilateral whose sides
are all congruent, which statement must
be true?
Statement
−− −−
1. AB DC
−− −−
2. BC BC
A
B
C
D
3. ∠B ∠C
3. They are both
right angles.
4. ABC DCB
−− −−
5. AC BD
4. ?
PQRS is a square.
PQRS is a rhombus.
PQRS is a trapezoid.
PQRS is kite.
“A triangle can have only one
right angle.”
Xing is trying to prove the theorem by
contradiction. She assumes that a triangle
has two right angles. What will she use
next to reach a contradiction?
F The sum of the angles in a rectangle
is 360°.
G The sum of the angles in a triangle
is 180°.
H A square has four 90° angles.
J A rectangle has four 90° angles.
A
B
C
D
Reason
1. ABCD is a square.
2. Reflexive Property
5. Corresponding
parts of congruent
triangles are
congruent.
SSS
SAS
AAS
ASA
Mastering the California Mathematics Standards, Geometry
A61
Name
Date
Standards Assessment
16
A soup can is shaped like a cylinder. It
has height 5 inches and radius 2 inches.
What is the volume of the can?
18
(continued)
A right circular cone has a radius of
4 inches and a height of 10 inches.
2 in.
5 in.
10 in.
4 in.
F
G
H
J
17
31.4 in 3
62.8 in 3
157.1 in 3
251.3 in 3
What is the volume of the cone?
F
G
H
J
19
−− −−
−− −−
Given: BC DC and AC EC.
"
x
%
15.5
$
12
#
A
B
C
D
2
3
4
5
&
What theorem or postulate can be used
to prove ABC EDC?
A
B
C
D
AAA
SSS
SAS
AAS
A62 Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
In the figure below, x is a whole
number. What is the smallest
possible value for x?
167.5 in 3
251.2 in 3
502.4 in 3
670.2 in 3
Name
Date
Standards Assessment
20
If ABCD is a square, then what is the
area of ABO?
%
18
23
$
(continued)
A ladder is leaning on the Freeman
House in Hollywood. The ladder is
12 feet long and makes a 28° angle
with the wall of the house. About
how far up the wall does the
ladder reach?
0
"
F
G
H
J
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
21
#
9
81
162
324
A
B
C
D
If triangle TRI were plotted on a
coordinate grid, which statement
would prove that TRI is a
right triangle?
−−
A The slope of TR times the slope of
−−
RI is -1.
−−
B The slope of TR times the slope of
−−
RI is 1.
−−
C The slope of TR equals the slope
−−
of RI.
−−
D The slope of TR is greater than that
−−
of RI.
24
5.6 ft
10.6 ft
13.6 ft
22.6 ft
If the sum of the measures of the
interior angles of a regular polygon is
900°, how many sides does the polygon
have?
F
G
H
J
sin 28° ≈ 0.47
cos 28° ≈ 0.88
tan 28° ≈ 0.53
Which values for x and y make LMNO
a parallelogram?
-
16
.
x+4
9
0
22
28‚
12 ft
F
G
H
J
3x + y
/
x = 1 and y = 1
x = 5 and y = 1
x = 5 and y = 4
x = 6 and y = 3
5
6
7
8
Mastering the California Mathematics Standards, Geometry
A63
Name
Date
Standards Assessment
25
About how tall is the flagpole?
27
(continued)
Jorge is doing a construction with
a compass and a straightedge. The
construction is shown below.
CALIFORNIA REPUBLIC
3
4
30‚
15 ft
A
B
C
D
26
7.5 ft
8.7 ft
13.1 ft
15 ft
sin 30° = 0.5
cos 30° ≈ 0.87
tan 30° ≈ 0.58
1
A bisecting an angle
B drawing a perpendicular line through
a point not on the line
C drawing a parallel line through a point
not on the line
D drawing an equilateral triangle
#
60‚
$
28
What is the measure of ∠ABC?
−− −−
In triangles XYZ and ABC, XY AB,
−− −−
−− −−
YZ BC , and XZ AC.
:
#
9
;
"
$
Which can be used to prove that triangle
XYZ is congruent to triangle ABC?
F
G
H
J
SSA
SAS
SSS
AAA
A64 Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
"
15°
30°
60°
120°
2
Which best describes the construction
Jorge is doing?
−−
−−
In circle O, AB is a diameter, and CB is
a chord.
F
G
H
J
+
Name
Date
Standards Assessment
29
Which of the following best describes
inductive reasoning?
31
(continued)
In the pentagon shown below, what
is m∠A + m∠D?
A proving a statement
B using logic to draw conclusions based
on accepted general statements
C inferring a general truth by examining
a number of specific examples
D doing a paragraph proof
"
&
120‚
140‚
75‚
%
30
−−
What is the length of the diagonal QA
if QUAD is a rectangle?
2
A
B
C
D
6
6
#
$
25°
205°
385°
535°
0
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
%
F
G
H
J
2
3 √
3 √
3
6
12
"
32
If triangle TRI is reflected across the
y-axis, what will be the coordinates
of T?
3
−4−3−2
4
3
2
1
O
−2
−3
−4
F
G
H
J
y
5
1 2 3 4x
*
(-3, 2)
(3, -2)
(-2, 3)
(-3, -2)
Mastering the California Mathematics Standards, Geometry
A65
Name
Date
Standards Assessment
33
−−
Given: KITE is a kite with diagonals KT
−−
and IE.
36
34
F
G
−− −−
RC ET
−−
−−
RC is perpendicular to ET.
−− −−
RX EX
−− −−
XC XT
H
An isosceles trapezoid is shown below.
5
J
3
9
1
"
Which pair of triangles can be shown
to be congruent, in order to prove
∠T ∠R?
A
B
C
D
ATP and PRA
TXP and RXA
TXR and PXA
TRP and TRA
37
A triangle has angles of 35° and 95°.
Which could possibly be the measure
of an exterior angle of the triangle?
A
B
C
D
20°
90°
130°
180°
A66 Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
35
The diagonals are perpendicular.
The diagonals bisect each other.
The diagonals are congruent.
The diagonals are perpendicular
bisectors.
−−
In rectangle RECT, diagonals RC and
−−
ET intersect at point X. Which of
the following statements need not be
true?
F
G
H
J
Which is a counterexample to the
following statement?
If a quadrilateral has four congruent
sides, then it is a square.
Which of the following must be true?
A
B
C
D
(continued)
Name
Date
Standards Assessment
38
Which of the following is not used
in constructions?
F
G
H
J
39
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
O
8
x
x
What is the area, in square units, of the
triangle shown below?
A
B
C
D
In the isosceles triangle shown below,
what is the value of x?
protractor
compass
straightedge
pencil
8
7
6
5
4
3
2
1
40
41
(continued)
A
B
C
D
y
42
1 2 3 4 5 6 7 8x
6
12
18
36
√
2
4 √
2
√
16 2
8
In triangle ABC, which expression
should be used to find the length
−−
of AB?
"
16
Which can be used as a counterexample
to the following statement?
#
36‚
$
F sin 36°
If an animal is a dog, then it is a
German shepherd.
F
G
H
J
a cat
a pig
a German shepherd
a Jack Russell terrier
sin 36°
G _
36
16
H _
sin 36°
J 16(sin 36°)
Mastering the California Mathematics Standards, Geometry
A67
Name
Date
Standards Assessment
43
−− −−
In the figure below, DE AC. Which
could be used to prove that ABC is
similar to DBE?
46
(continued)
Which picture is a counterexample to
the following statement?
A line and a circle in a plane intersect
either at zero points or at one point.
#
F
%
"
A
B
C
D
$
ASA
SSA
AA
SSS
G
The vertices of triangle PQR are P(2, 6),
Q(4, -3), and R(-1, -3). What will be
the vertices of triangle PQR if triangle
PQR is reflected across the y-axis?
H
F
G
H
J
45
P(2, -6), Q(4, 3), R(-1, 3)
P(-2, -6), Q(-4, 3), R(1, 3)
P(-2, 6), Q(-4, -3), R(1, -3)
P(6, 2), Q(-3, 4), R(-3, -1)
Which trigonometric identity is true?
A
B
C
D
J
sin 2 x = 1
sin 2 x = cos 2 x
sin 2 x + cos 2 x = 1
sin 2 x - cos 2 x = 1
A68 Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
44
&
Name
Date
Standards Assessment
47
What is the straight-line distance from
the corner of Irvine Center Drive and
Culver Drive to the corner of Sand
Canyon Avenue and Trabuco Drive?
49
(continued)
−−
−−−
In the circle shown, LM and PQ are
chords of the circle and intersect at
point O.
1
rD
r.
0
lve
Tra
b
.
Cu
uc
d.
rD
2.5 mile
2
Sa
nd
r.
ny
nte
Ca
Ce
on
Irv
ine
oR
1.5 mile
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
If LO = 4 and MO = 5, find PO
if QO = 10.
A
B
C
D
48
A
B
C
D
2 mi
2.9 mi
3 mi
4 mi
If cos A = 0.4, then what is the length
−−
of BA?
50
2
3
8
12
What is the area of the figure
shown below?
"
6 in.
10
#
F
G
H
J
0.4
4
8
25
14 in.
$
F
G
H
J
84.0 sq in.
102.8 sq in.
112.3 sq in.
197.1 sq in.
Mastering the California Mathematics Standards, Geometry
A69
Name
Date
Standards Assessment
51
Janine is bisecting an angle. Which of
the following should be her first step?
52
A
What is the relationship between two
consecutive angles in a parallelogram?
F
G
H
J
%
"
53
B
(continued)
They are right angles.
They are congruent.
They are complementary.
They are supplementary.
−−
AB is to circle O at B, and OA = 5.
Find the value of x in the figure
shown below.
%
"
#
x
0
C
4
5
#
A
B
C
D
$
2
3
4
5
D
#
54
"
$
Which of the following would be
sufficient to prove that ABC
and DEF are similar?
F
G
H
J
∠A ∠D and ∠B ∠E
∠A ∠D
AB = DE and BC = EF
m∠A + m∠B + m∠C =
m∠D + m∠E + m∠F
A70 Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
"
"
Name
Date
Standards Assessment
55
−− −−
In the figure below, AB DE, and
−− −−
CB FE. What information is sufficient
to prove that ABC DEF?
#
57
(continued)
What is m∠BAC?
%
&
$
"
35‚
"
$
%
&
'
# 125‚
A
B
C
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
56
∠A ∠D
∠C ∠F
−− −−
AC DF
−− −−
AB BC
A
B
C
D
Quadrilateral PQRS is a parallelogram.
Which statement must be true about
angles 1 and 3?
1
4
F
G
H
J
58
35°
55°
90°
125°
Figure PQRS is a rectangle.
y
2
1
3
3
∠1 and ∠3 are both right angles.
m∠1 + m∠3 = 90°
m∠1 + m∠3 = 180°
m∠1 = m∠3
Q (0, p)
R (s, p)
P (0, 0)
S (s, 0)
x
What are the coordinates of the point
at which the diagonals intersect?
F (0, 0)
G (s, p)
p
H _s , _
(2 2)
J (2, 2)
Mastering the California Mathematics Standards, Geometry
A71
Name
Date
Standards Assessment
59
In the figure below, m∠4 = m∠ 8.
t
61
(continued)
What is the first step in constructing a
perpendicular to a line through a given
point on the line?
A
2
3
6
m
7
1
"
4
1
#
5
8
B
"
1
#
Which of the following conclusions is
not necessarily true?
A
B
C
D
Line ℓ is parallel to line m.
m∠2 = m∠3
m∠2 = m∠8
m∠2 + m∠7 = 180°
C
"
1
#
60
The ratio of the sides of two squares is
3 to 2. What is the ratio of the areas of
the two squares?
F
G
H
J
√
3 to √2
3 to 2
9 to 4
27 to 8
A72 Mastering the California Mathematics Standards, Geometry
"
1
#
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
D
Name
Date
Standards Assessment
62
What is the area of square C in
square units?
7
6
5
4
3
2
1
−3 −2
64
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
63
O
(2x + 5)‚
$
(x + 9)‚
1 2 3 4 5 6 7x
F
G
H
J
9
12
16
25
65
What is the measure of y?
"
A
B
C
D
75‚
2
4
5
9
Modoc County in California is shaped
like a rectangle. If the length of the
rectangle is 100 miles, and the width
is 60 miles, what is the area of the
county?
A
B
C
D
50‚
$
The figure shown is a parallelogram
with angle measures as shown. What
is the value of x?
y
−2
−3
−4
F
G
H
J
(continued)
600 mi 2
6,000 mi 2
60,000 mi 2
600,000 mi 2
y
#
55°
100°
125°
180°
Mastering the California Mathematics Standards, Geometry
A73
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Except as
permitted under the United States Copyright Act, no part of this publication may be
reproduced or distributed in any form or by any means, or stored in a database or
retrieval system, without prior permission of the publisher.
Send all inquiries to:
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ISBN: 978-0-07-879537-4
MHID: 0-07-879537-0
Mastering the California Mathematics Standards, Geometry
(Standards Practice and Periodic Assessments)
Printed in the United States of America.
1 2 3 4 5 6 7 8 9 10 047 15 14 13 12 11 10 09 08 07
Contents
California Mathematics Standards, Geometry . . . . . . . . . . . . . . . . . . . Bv
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Practice by Chapter
Chapter 1
Tools of Geometry . . . . . . . . . . . . . . . . . . . . . . . . B1
Chapter 2
Reasoning and Proof . . . . . . . . . . . . . . . . . . . . . . B3
Chapter 3
Parallel and Perpendicular Lines . . . . . . . . . . . . . . . . . B5
Chapter 4
Congruent Triangles . . . . . . . . . . . . . . . . . . . . . . . B7
Chapter 5
Relationships in Triangles . . . . . . . . . . . . . . . . . . . . B9
Chapter 6
Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . B11
Chapter 7
Proportions and Similarity . . . . . . . . . . . . . . . . . . . B13
Chapter 8
Right Triangles and Trigonometry . . . . . . . . . . . . . . . B15
Chapter 9
Transformations . . . . . . . . . . . . . . . . . . . . . . . . B17
Chapter 10
Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B19
Chapter 11
Areas of Polygons and Circles . . . . . . . . . . . . . . . . . B21
Chapter 12
Extending Surface Area . . . . . . . . . . . . . . . . . . . . B23
Chapter 13
Extending Volume . . . . . . . . . . . . . . . . . . . . . . . B25
Periodic Assessment 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B27
Periodic Assessment 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B32
Periodic Assessment 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B37
Periodic Assessment 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B42
Periodic Assessment—Student Answer Sheets . . . . . . . . . . . . . . . . . . B47
Mastering the California Mathematics Standards, Geometry
Biii
California Mathematics Standards
Geometry
= denotes Key standards
1.0 Students demonstrate understanding by identifying and giving examples of
undefined terms, axioms, theorems, and inductive and deductive reasoning.
2.0 Students write geometric proofs, including proofs by contradiction.
3.0 Students construct and judge the validity of a logical argument and give
counterexamples to disprove a statement.
4.0 Students prove basic theorems involving congruence and similarity.
5.0
Students prove that triangles are congruent or similar, and they are able to
use the concept of corresponding parts of congruent triangles.
6.0
Students know and are able to use the triangle inequality theorem.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
7.0 Students prove and use theorems involving the properties of parallel lines
cut by a transversal, the properties of quadrilaterals, and the properties of
circles.
8.0 Students know, derive, and solve problems involving perimeter,
circumference, area, volume, lateral area, and surface area of common
geometric figures.
9.0
Students compute the volumes and surface areas of prisms, pyramids,
cylinders, cones, and spheres; and students commit to memory the
formulas for prisms, pyramids, and cylinders.
10.0 Students compute areas of polygons, including rectangles, scalene
triangles, equilateral triangles, rhombi, parallelograms, and trapezoids.
11.0 Students determine how changes in dimensions affect the perimeter, area,
and volume of common geometric figures and solids.
12.0 Students find and use measures of sides and of interior and exterior angles
of triangles and polygons to classify figures and solve problems.
13.0 Students prove relationships between angles in polygons by using
properties of complementary, supplementary, vertical, and exterior angles.
14.0 Students prove the Pythagorean theorem.
15.0 Students use the Pythagorean theorem to determine distance and find
missing lengths of sides of right triangles.
16.0 Students perform basic constructions with a straightedge and compass,
such as angle bisectors, perpendicular bisectors, and the line parallel to a
given line through a point off the line.
Mastering the California Mathematics Standards, Geometry
Bv
California Mathematics Standards
Geometry (continued)
17.0 Students prove theorems by using coordinate geometry, including the
midpoint of a line segment, the distance formula, and various forms of
equations of lines and circles.
18.0 Students know the definitions of the basic trigonometric functions
defined by the angles of a right triangle. They also know and are able to
use elementary relationships between them. For example,
sin (x)
cos (x)
tan (x) = _, (sin (x))2 + (cos (x))2 = 1.
19.0 Students use trigonometric functions to solve for an unknown length of
a side of a right triangle, given an angle and a length of a side.
20.0 Students know and are able to use angle and side relationships in
problems with special right triangles, such as 30°, 60°, and 90° triangles
and 45°, 45°, and 90° triangles.
21.0 Students prove and solve problems regarding relationships among chords,
secants, tangents, inscribed angles, and inscribed and circumscribed
polygons of circles.
Bvi Mastering the California Mathematics Standards, Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
22.0 Students know the effect of rigid motions on figures in the coordinate
plane and space, including rotations, translations, and reflections.
Name
Date
Practice by Chapter
Chapter 1 Tools of Geometry
1
A washer is a thin, circular metal plate
with a circular hole in the middle, as
shown below.
If the circumference of the washer is
8π cm, and the area of the hole in the
middle is 4π cm 2, then what is the area
of the metal part of the washer, in
square centimeters?
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A
B
C
D
2
A rectangle drawn on a coordinate grid
has three of its vertices at points (2, 6),
(2, 1), and (10, 1). What is the area of
the rectangle, in square units?
A
B
C
D
4
2
10π cm
12π cm 2
14π cm 2
16π cm 2
What results from the following steps?
Draw a line. Draw two arcs equidistant
from a point on the line. From the
points at which these arcs intersect the
line, draw two intersecting arcs with
equal radii on each side of the line.
Connect the points at which the arcs
intersect to form a line.
F
G
H
J
3
A cylindrical container with a volume
of 45π meters and a height of 5 meters
is rolled on its side for 72π meters.
How many revolutions did the
cylindrical container make?
F
G
H
J
5
20
40
48
54
6
12
18
24
What is the area of the triangle in
square units?
4
3
(−1, 2)
2
1
a line perpendicular to the original line
a line parallel to the original line
a plane perpendicular to the line
a plane parallel to the line
O
−2
−3
−4
A
B
C
D
y
(4, 2)
1 2 3 4 5 6 7x
(7, −1)
7.5
12
15
24
California Mathematics, Geometry Standards Practice
B1
Name
Date
Practice by Chapter
Chapter 1 (continued)
6
A cone’s slant height is 10 inches, and
8
the height of the cone is 8 inches. What
is the volume of the cone, in cubic
_
inches? (V = 1 πr 2h)
3
8 in.
Eric purchased a pair of shoes. He
placed them in a box that measured
15 inches by 8 inches by 5 inches. He
wrapped the shoe box with wrapping
paper. What is the minimum amount of
wrapping paper, in square inches, that
Eric will need to completely cover the
shoe box?
10 in.
F 150 sq in.
G 235 sq in.
F 24π in 3
G 72π in 3
7
H 96π in 3
J 120π in 3
A scalene triangle is inscribed in a
rectangle, as shown below.
A 2m
B 4m
What construction is described by the
following steps? From the vertex of an
angle, draw an arc that intersects both
sides of the angle. From the points at
which the arc intersects the legs, draw
two intersecting arcs with equal radii.
Connect the point at which the arcs
intersect to the vertex of the angle to
form a line.
A
B
C
D
C 8m
D 12 m
B2 California Mathematics, Geometry Standards Practice
a vertex of the angle
a copy of the angle
a bisector of the angle
a trisector of the angle
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
If the height of the scalene triangle is
12 meters, and the area of the rectangle
is 48 square meters, what is the length
of the base of the scalene triangle?
9
H 470 sq in.
J 600 sq in.
Name
Date
Practice by Chapter
Chapter 2 Reasoning and Proof
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1
Which of the following is an example of
inductive reasoning?
A Parallel lines never intersect. Line a
and line b are parallel lines. Therefore,
line a and line b never intersect.
B In a random sample of line segments,
line a never intersects with line b.
Therefore, line a and line b never
intersect.
C If line a and line b are parallel lines,
then they never intersect. Line a and
line b intersect. Therefore, line a and
line b are not parallel lines.
D If line a and line b are parallel lines,
then they never intersect. If two lines
never intersect, then they don’t form an
angle with each other. Therefore, if line
a and line b are parallel lines, then they
don’t form an angle with each other.
2
Two triangles have congruent
corresponding exterior angles. What
additional information could be used to
prove that the triangles are congruent?
F Both triangles are equilateral triangles.
G Both triangles are isosceles triangles.
H The triangles have a pair of congruent
corresponding interior angles.
J The triangles have a pair of congruent
corresponding sides.
3
Which of the following can serve as a
counterexample to the assertion that
two lines that are not parallel always
intersect?
A skew lines
B parallel lines
4
C intersecting lines
D perpendicular lines
Which of the following statements must
be true in regards to the three triangles
shown below?
9
$
'
5
35‚
35‚
6
6
&
5
:
6
5
%
"
#
F
G
H
J
5
70‚
;
ABC DEF
ABC XYZ
DEF XYZ
ABC DEF XYZ
Which of the following is an example of
deductive reasoning?
A The corresponding angles of two
regular pentagons are congruent.
Therefore, the two regular pentagons
are congruent.
B All of the sides of a regular pentagon
are congruent.
C All of the angles of a regular pentagon
are congruent.
D If two regular pentagons have the same
perimeter, then they are congruent. Two
regular pentagons have the same
perimeter. Therefore, they are congruent.
California Mathematics, Geometry Standards Practice
B3
Name
Date
Practice by Chapter
Chapter 2 (continued)
6
Which of the following can serve as a
counterexample to the assertion that
two perpendicular lines in the same
plane never intersect?
F
G
H
J
7
the origin of a coordinate grid
the x-axis of a coordinate grid
the y-axis of a coordinate grid
the four quadrants of a coordinate grid
10
The hypotenuses of two right triangles
are congruent. Is this enough
information to prove that the two right
triangles are congruent?
Which of the following facts proves
AC = BC?
%
"
F
G
H
J
$
#
AC = CD
CB = CD
∠ACD ∠BCD
−−
−−
CD is a perpendicular bisector of AB.
F Yes—the triangles must have a pair of
corresponding sides and an included
angle that are congruent.
G Yes—the triangles must have three
corresponding sides that are congruent.
H Yes—the triangles must have a pair of
corresponding angles and an included
side that are congruent.
J No—this is not enough information to
prove that the two right triangles are
congruent.
B4 California Mathematics, Geometry Standards Practice
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
triangle
quadrilateral
pentagon
hexagon
Which of the following statements is
true?
A Deductive reasoning often involves the
making of a general argument based
on observed examples.
B Inductive reasoning often involves the
making of arguments based on more
general arguments that are known to
be true.
C Deductive reasoning is logically valid.
D An argument that is formed by
inductive reasoning will always
be correct.
Which of the following polygons serves
as a counterexample to the assertion
that the sum of the interior angles of
any polygon is always greater than or
equal to 360°?
A
B
C
D
8
9
Name
Date
Practice by Chapter
Chapter 3 Parallel and Perpendicular Lines
1
In the figure below, lines a and b are
parallel and are cut by transversal c.
4
In the figure below, what term is used
to describe angles 1 and 2?
c
c
1
2
1
a
b
2
F
G
H
J
What is the sum of angles 1 and 2?
A 90°
B 180°
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
C 360°
D cannot be determined
What is the first step in constructing a
line parallel to a given line through a
point not on the line with a straightedge
and a compass?
5
6
Which of the following statements is
true when constructing a line parallel
to a given line through a point off the
line with a straightedge and a compass?
A
B
C
D
No arcs need to be drawn.
Only one arc needs to be drawn.
Two arcs need to be drawn.
More than two arcs need to be drawn.
b
alternate interior angles
alternate exterior angles
consecutive interior angles
vertically opposite angles
Which of the following can serve as a
counterexample to the assertion that
alternate interior angles are never
supplementary?
A a line that is parallel to two parallel lines
B a transversal that forms a 45° angle
with two parallel lines
C a transversal that is perpendicular to
two parallel lines
D a transversal that forms a 180° angle
with two parallel lines
F Draw a transversal of the line through
the point off the line.
G Draw a perpendicular bisector of the
line through the point off the line
H Draw an arc from the point off the line
that intersects the line
J Draw an arc from a point on the line
that intersects the point off the line
3
a
Which is a counterexample to the
following statement?
If line a and line b are parallel, and line c intersects
line a, then line c will also intersect line b.
F Line c is in the same plane as line a,
but it is not in the same plane as line b.
G Line c is in the same plane as line b,
but it is not in the same plane as line a.
H Line c is in the same plane as line a
and line b.
J Line c is not in the same plane as
line a or line b.
California Mathematics, Geometry Standards Practice
B5
Name
Date
Practice by Chapter
Chapter 3 (continued)
7
In the figure below, what term is used
to describe angles 1 and 2?
9
c
1
2
A
B
C
D
8
a
corresponding angles
alternate interior angles
alternate exterior angles
consecutive interior angles
to construct equal length segments
to construct alternate interior angles
to bisect the given line
to bisect an angle
10
When constructing a line parallel to a
given line through a point not on the
line with a straightedge and a compass,
after drawing the first arc, will you
have to adjust your compass before
drawing the second arc?
F Yes—I will be drawing the second arc
from the same point.
G Yes—I will be drawing the second arc
from a different point.
H No—I will be drawing the second arc
from the same point.
J No—I will be drawing the second arc
from a different point.
B6 California Mathematics, Geometry Standards Practice
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
F
G
H
J
A a traversal that intersects two parallel
lines produces vertically opposite
angles, each measuring 35°
B a traversal that intersects two parallel
lines produces vertically opposite
angles, each measuring 45°
C a traversal that intersects two parallel
lines produces vertically opposite
angles, each measuring 55°
D a traversal that intersects two lines that
are not parallel
b
When constructing a line parallel to a
given line through a point not on the
line, why should you construct a line
that intersects the given line?
Which of the following can serve as a
counterexample to the assertion that
consecutive interior angles are always
supplementary?
Name
Date
Practice by Chapter
Chapter 4 Congruent Triangles
1
What type of triangle is formed from
the points A(-4, 2), B(-3, -4), and
C(-2, 2)?
A
B
C
D
2
4
F
G
H
J
right
equilateral
isosceles
scalene
5
What is m∠DCE?
"
#
42‚
136‚
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
%
&
90°
92°
94°
96°
6
SAS
SSS
ASA
AAA
The measurements of the three interior
angles of a triangle are x, 2x - 9, and
3x + 9. What type of triangle is it?
A
B
C
D
$
F
G
H
J
Which of the following cannot be used
to prove the congruence of two
triangles?
acute
right
obtuse
cannot be determined
−−
In the figure below, BD is
−−
perpendicular to and bisects AC.
#
3
The diagonals of parallelogram VWXY
intersect at point Z. Which of the
following must be true?
A
B
C
D
VZW XZY
VZW VXW
XYZ XWZ
XYZ VYW
"
%
$
What additional information is needed
to prove that ABD CBD?
F
G
H
J
−− −−
AB BC
−− −−
AB BD
∠ABD ∠CBD
No additional information is needed
California Mathematics, Geometry Standards Practice
B7
Name
Date
Practice by Chapter
Chapter 4 (continued)
7
A right triangle is plotted on a
coordinate grid, and the hypotenuse of
the right triangle has endpoints (8, 9)
and (36, -12). Which of the following
could be the perimeter of the triangle?
A
B
C
D
8
10
F Yes—and the other two interior angles
of the isosceles triangle measure 15°.
G Yes—and the other two interior angles
of the isosceles triangle measure 25°.
H Yes—and the other two interior angles
of the isosceles triangle measure 65°.
J No—this is not enough information
to determine the measurements of
the other two interior angles of the
isosceles triangle.
21 units
28 units
35 units
84 units
ABC and DEF are isosceles
−− −− −− −−
triangles. AB BC, DE EF, and
−− −−
AB DE.
'
#
11
$
%
ABC shown below is an equilateral
triangle. What is m∠DAC?
&
"
Which would not be sufficient to prove
that ABC DEF?
F
G
H
J
9
∠B ∠E
∠A ∠D
−− −−
AC DF
∠A ∠B
XYZ is an equilateral triangle with
one vertex at the origin of a coordinate
grid and another vertex at point
(12, -5). What is the perimeter of
XYZ?
A
B
C
D
12 units
13 units
36 units
39 units
#
A 10°
B 20°
12
70‚
%
$
C 60°
D 110°
The hypotenuse of a right triangle has
a length of 100 inches. One of the
interior angles of the right triangle
measures 45°. What is the area of the
right triangle?
F
G
H
J
B8 California Mathematics, Geometry Standards Practice
1,250 sq in.
2,500 sq in.
5,000 sq in.
10,000 sq in.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
"
If you know that one of the interior
angles of an isosceles triangle
measures 130°, can you determine the
measurements of the other two
interior angles?
Name
Date
Practice by Chapter
Chapter 5 Relationships in Triangles
1
If two of the exterior angles of a
triangle measure 123° and 147° then
which of the following statements is
true?
A
B
C
D
4
F 16 √3
G 32 √
3
The triangle is a right triangle.
The triangle is an acute triangle.
The triangle is an obtuse triangle.
The triangle is an equilateral triangle.
5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
What is the first step toward
constructing a line perpendicular to a
given line through a point off the line
with a straightedge and a compass?
F Draw an arc from the point off the line
that intersects the line at one point.
G Draw an arc from the point off the line
that intersects the line at two points.
H Draw an arc from a point on the line
that intersects the point off the line.
J Draw an arc from a point on the line
that intersects the point off the line and
another point on the line.
3
A trapezoid has congruent diagonals.
Which of the following must be true of
the trapezoid?
A
B
C
D
It has at least one pair of congruent sides.
It has at least two pairs of congruent sides.
Its diagonals bisect each other.
Its diagonals are parallel to each other.
A right triangle has an exterior angle
measuring 150° and a hypotenuse
measuring 16 inches. What is the area
of the right triangle?
H 48 √
3
J 64 √
3
Joy wants to prove by contradiction
that the longest side of a right triangle
is its hypotenuse. She begins by
assuming that the hypotenuse of the
right triangle measures 5 units, and
that one of its other sides measures
6 units. Which theorem will Joy use to
reach a contradiction?
A If two sides of a triangle are
congruent, then the angles opposite
them are congruent.
B If two complementary angles are
congruent, then the angles each
measure 45°.
C If the measures of the interior angles
of a triangle are added together, the
sum equals 180°.
D She will use the Pythagorean theorem.
6
Based on the figure below, which of the
following statements must be true?
4
6
x
F x<2
G x=2
H x≤2
J x>2
California Mathematics, Geometry Standards Practice
B9
Name
Date
Practice by Chapter
Chapter 5 (continued)
7
If one of the exterior angles of a
triangle measures less than 90°, then
which of the following statements
is true?
A
B
C
D
8
The triangle is a right triangle.
The triangle is an acute triangle.
The triangle is an obtuse triangle.
The triangle is an equilateral triangle.
8
x
9
A x < 17
B x = 17
C x > 17
D x = √
145
11
Haruki wants to prove the theorem that
the sum of the exterior angles of a
triangle does not equal 180°. He begins
by assuming that each of the three
exterior angles of an equilateral
triangle measures 60°. Which theorem
will Haruki definitely not use to reach a
contradiction?
A If two angles are adjacent interior and
exterior angles of a polygon, then the
angles are supplementary angles.
B If two angles are supplementary
angles, then the sum of their measures
is 180°.
C If the measures of the interior angles
of a triangle are added together, the
sum equals 180°.
D He will not use the Pythagorean
theorem.
B10 California Mathematics, Geometry Standards Practice
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Based on the figure below, which of the
following statements must be true?
If you were constructing a line
perpendicular to a given line through a
point off the line with a straightedge
and a compass, which of the following
would be true?
F The second and third arcs should not
intersect.
G The second and third arcs should
intersect only at the point off the line.
H The second and third arcs should
intersect only at a point on the given line.
J The second and third arcs should
intersect at a point off the line and at
another point off the line.
An obtuse triangle has an exterior
angle measuring 150°. Which of
the following statements must
be true?
F The obtuse triangle must have another
exterior angle greater than 120°.
G The obtuse triangle must have another
exterior angle greater than 130°.
H The obtuse triangle must have another
exterior angle greater than 140°.
J The obtuse triangle must have another
exterior angle greater than 150°.
9
10
Name
Date
Practice by Chapter
Chapter 6 Quadrilaterals
1
What is the measure of an exterior
angle of the regular polygon shown
below?
4
A parallelogram is graphed on a
coordinate grid, and three of its
vertices are (-4, -2), (-3, 1), and
(1, 1), respectively. What are the
possible coordinates of the fourth
vertex?
F (-2, 0)
G (0, -2)
A
B
C
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
36°
45°
135°
144°
5
What are the values of x and y in the
rectangle shown below?
H (2, 0)
J (0, 2)
To construct a line perpendicular to a
given line through a point not on the
line with a straightedge and a compass,
how many arcs should you strike in
total?
A 1
B 2
C 3
D4
15
6
5x - y
7
F All squares are quadrilaterals, and all
rhombuses are squares.
G All quadrilaterals are squares, and all
squares are rhombuses.
H All rhombuses are squares, and all
squares are quadrilaterals.
J All squares are rhombuses, and all
rhombuses are quadrilaterals.
6x + y
F
G
H
J
3
x = 2, y = 2
x = 2, y = 3
x = 3, y = 2
x = 3, y = 3
The sum of the measures of the exterior
angles of a regular polygon equals 360°.
How many sides does the regular
polygon have?
A
B
C
D
4
6
8
cannot be determined
Which of the following is a true
statement?
7
An isosceles trapezoid has sides that
have lengths 18, 25, 25, and 58. What
is the area of the trapezoid, in square
units?
A 270
B 300
C 570
D 1140
California Mathematics, Geometry Standards Practice
B11
Name
Date
Practice by Chapter
Chapter 6 (continued)
8
Guillermo graphed a triangle on a
coordinate grid. The product of the
slopes of two of the sides equals -1.
Which of the following statements must
be true of the triangle?
11
F The triangle is an acute triangle.
G The triangle has an exterior angle of
less than 90°.
H The sum of the exterior angles of the
triangle equals 180°.
J The triangle is a right triangle.
9
16
18
20
22
If the perimeter of a rhombus is y units,
which of the following statements must
be true of the length of either of the
diagonals of the rhombus?
F The length of either of the diagonals
y
8
must be less than _ units.
G The length of either of the diagonals
y
4
must be less than _ units.
H The length of either of the diagonals
y
2
must be less than _ units.
12
A rectangle is graphed on a coordinate
grid. The slope of one of the sides of the
_
rectangle is 2 . What are the slopes of
3
the other three sides of the rectangle?
3
3
2
F -_
, -_
, and _
3
2
2
3 _
2
G -_
, 2 , and _
3
2 3
3
2
2
H _, _, and _
2 3
3
3 _
2
J _
, 3 , and _
2 2
3
J The length of either of the diagonals
must be greater than y units.
B12 California Mathematics, Geometry Standards Practice
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
10
A The measures of all of the exterior
angles of Jabari’s polygon are the same.
B The sum of the measures of all of the
exterior angles of Jabari’s polygon
equals the sum of the measures of all
of the exterior angles of Juanita’s
polygon.
C The measure of one of the exterior
angles of Juanita’s polygon is greater
than the measure of one of the exterior
angles of Jabari’s polygon.
D The measure of one of the exterior
angles of Juanita’s polygon plus the
measure of the adjacent interior angle
of the polygon equals 180°.
Each of the exterior angles of a regular
polygon measures 20°. How many sides
does the regular polygon have?
A
B
C
D
Jabari and Juanita have both
constructed regular polygons. Jabari’s
polygon has 14 sides. Juanita’s polygon
has 15 sides. Which of the following
statements is false?
Name
Date
Practice by Chapter
Chapter 7 Proportions and Similarity
1
Lydia wants to paint two square walls
in her house with paint that has a
spread rate of 350 square feet per
gallon. The perimeter of the first
square wall is 72 feet. The perimeter of
the second square wall is 76 feet. Which
wall can Lydia paint completely if she
has 1 gallon of paint?
A
B
C
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
the first square wall
the second square wall
either wall
neither wall
4
F They must have the same number
of sides.
G They must be similar.
H They must have the same perimeter.
J They must have corresponding interior
angles that are congruent.
5
Two right triangles each have an
exterior angle measuring 151°. Which
statement must be true?
F The two right triangles are congruent,
but not necessarily similar.
G The two right triangles are similar, but
not necessarily congruent.
H The two right triangles are both similar
and congruent.
J The two right triangles are neither
similar nor congruent.
Which of the following statements is
false about any two regular pentagons?
Two right triangles are similar. The area
of the first triangle is x 2 square units,
while the area of the second triangle is
4x 2 square units. If the hypotenuse of the
first triangle has a length of 14 units, then
what is the length of the hypotenuse of
the second triangle?
A 28 units
B 56 units
6
C 112 units
D 224 units
In the figure below, is ABC similar to
DEF?
"
y
3
'
2
# 1
−6−5−4−3−2
3
If the midpoints of two sides of a
triangle are connected by a line
segment, a smaller triangle is formed.
From which theorem or postulate does
it immediately follow that the two
triangles are similar?
A
B
C
D
AA
SSS
SAS
The triangles are not similar.
%
O &1 2 3 4 x
−2
−3
−4
$
−5
F No—the corresponding sides of the
two triangles are not congruent.
G No—the three sets of corresponding sides
of the triangles are not in proportion.
H Yes—the corresponding sides of the
two triangles are congruent.
J Yes—the three sets of corresponding
sides of the triangles are in proportion.
California Mathematics, Geometry Standards Practice
B13
Name
Date
Practice by Chapter
Chapter 7 (continued)
7
Which two triangles must be similar?
10
A two scalene triangles with the same
perimeter
B two scalene triangles with different
perimeters
C two isosceles triangles with the same
perimeter
D two equilateral triangles with different
perimeters
F pentagon
G hexagon
11
8
Which theorem or postulate can be
used to prove that the two triangles
shown below are similar?
24
24
F
G
H
J
9
12
AA
SSS
SAS
The triangles are not similar.
Quadrilateral ABCD is graphed on a
coordinate grid. The product of the
slopes of any two adjacent sides of the
quadrilateral equals -1. Also, AB =
BC = CD. Is the quadrilateral similar
to square WXYZ?
A no
B only if AB = WX, BC = XY, and
CD = YZ
C yes
D cannot be determined
12
H heptagon
J octagon
Two equilateral triangles have
perimeters with a ratio of 2 to 3. What
is the ratio of the areas of the two
equilateral triangles?
A √2 to √
3
C 4 to 9
B 2 to 3
D 16 to 81
−−
In the figure below, AB is parallel
−−
to CD. What additional information is
needed to prove that ABE is similar
to CDE?
&
$
"
F
G
H
J
B14 California Mathematics, Geometry Standards Practice
%
#
m∠ECD = m∠EDC
m∠EAB = m∠EBA
m∠CED = m∠ECD
No additional information is needed.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
12
A regular polygon with perimeter
40 units is graphed on a coordinate
grid. One vertex is at point (-4, 4), and
an adjacent vertex is at point (-8, 1).
What kind of regular polygon is
it?
Name
Date
Practice by Chapter
Chapter 8 Right Triangles and Trigonometry
1
−−
In right triangle JKL, KM is an
altitude. Which statement must be
true?
4
The figure below shows how areas of
triangles can be used to prove the
Pythagorean theorem.
,
a
b
b
c
c
+
.
JK
MK
=_
A _
c
a
JK
KL
C _
=_
JL
ML
JK
KM
B _
=_
JL
KL
2
-
a
KM
ML
JK
JL
D_
=_
KM
ML
c
b
b
a
Which is a true statement about the
figure?
Which is a true statement about RST
shown below?
F The area of the larger square is
a 2 + b 2.
4
G The area of the larger square is
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4(a + b).
13
H The total area of the four right
8
1
ab.
triangles is _
2
3
10
J The total area of the four right
5
triangles is 2ab.
F It is a right triangle. ∠R and ∠S are
acute angles, and ∠T is a right angle.
−−
G It is a right triangle. RT appears to be
−−
perpendicular to ST.
H It is not a right triangle.
J It is impossible to determine whether
the triangle is a right triangle or not.
3
_
In right triangle ABC, sin B = 16 .
34
What is tan B?
34
A _
16
34
B _
30
30
C _
16
16
D_
30
5
The graph of DEF is shown below.
−−
What is the length of EF?
7
6
5
4
3
2
1
O
y
&
'
%
1 2 3 4 5 6 7x
A 10 units
B 2 √
13 units
C 6 units
D 2 √
5 units
California Mathematics, Geometry Standards Practice
B15
Name
Date
Practice by Chapter
Chapter 8 (continued)
6
A right triangle has hypotenuse 8 cm
and leg 5 cm. What is the length of the
other leg?
F
G
H
J
7
9
3 cm
√
39 cm
√
89 cm
13 cm
6
d
18‚
A
B
C
D
10
sin 41° 0.66 sin 49° 0.75
cos 41° 0.75 cos 49° 0.66
tan 41° 0.87 tan 49° 1.15
518 ft
492 ft
168 ft
152 ft
F 7.9
G 9.1
_
In a right triangle, sin x = m
p , and
_
In right triangle LMN, m∠L = 49°,
m∠M = 41°, and MN = 12. What is the
−−
length of LN to the nearest tenth?
11
H 10.4
J 13.8
Find the value of x in the triangle
below.
"
m
F _
n
n
G _
m
x
p
H _
n
p
J _
m
$
A
B
C
D
B16 California Mathematics, Geometry Standards Practice
4
8
8 √
2
16 √
2
16
45‚
#
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
To the nearest foot, what is d, the
distance from the top of the lighthouse
to the boat?
cos x = np . What is tan x?
36 ft
24 √
3 ft
√
12 3 ft
18 ft
115 ft
45 ft
8
x
3 ft
30‚
The Pigeon Point Lighthouse in
Pescadero, California, is 115 feet tall.
A
B
C
D
A skateboard ramp is 6 √
3 feet high
and makes a 30° angle with the ground,
as shown in the figure below. What is x,
the length of the ramp?
Name
Date
Practice by Chapter
Chapter 9 Transformations
1
JKL is translated 3 units left
and 2 units up to create J K L.
What are the coordinates of the
vertices?
+
−4−3−2
4
3
2
1
y
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
The vertices of LMN are L(5, 6),
M(2, 0), and N(-8, 8). If the figure is
translated, and the new vertices are
L(1, 6), M(-2, 0), and N(-12, 8),
which rule describes the
transformation?
A
B
C
D
1 2 3 4x
O
−2
−3
−4 -
A
B
C
D
3
,
J(1, 4), K(6, 0), L(3, -1)
J(-5, 4), K(0, 0), L(-3, -1)
J(0, -1), K(5, –5), L(2, 0)
J(-5, 0), K(0, -4), L(-3, -5)
4
(x, y) → (x - 4, y)
(x, y) → (x, y - 4)
(x, y) → (x + 4, y)
(x, y) → (x, y + 4)
Triangle QRS is reduced in size to form
QRS. What is the ratio of the area
of QRS to the area of QRS?
7
6
5
4
3
2
1
Trapezoid ABCD is reflected across
the y-axis. What are the coordinates
of C?
4
3
2
1
−4 −3−2 $ O
#
"
(2, 1)
(-2, 1)
(2, -1)
(-1, -2)
3h
2 2h
O
y
2
F _
1
4
G _
1
1 2 3 4x
−2
−3
%
−4
5
F
G
H
J
3
y
4h
4
1 2 3 4 5 6 7x
1
H _
2
2
J _
3
Triangle JKL has vertices at J(0, 1),
K(2, 3), and L(4, 0). If the triangle is
rotated 180° about the origin, what will
be the coordinates of K?
A
B
C
D
(3, 2)
(-2, 3)
(-2, -3)
(-3, -2)
California Mathematics, Geometry Standards Practice
B17
Name
Date
Practice by Chapter
Chapter 9 (continued)
6
The state flag of California is shown on
the grid below. Suppose the flag were
enlarged so that the vertices of the new
flag were (0, 0), (0, 6), (9, 6), and (9, 0).
What is the ratio of the perimeter of
the original flag to that of the enlarged
flag?
7
6
5
4
3
2
1
O
8
Triangle DEF has vertices D(-1, -3),
E(3, 1), and F(3, -3). Which shows a
reflection of the triangle over the line
y = x?
F
y
&h
−2
−3
−4
CALIFORNIA REPUBLIC
1 2 3 4 5 6 7x
G
2
G _
'h
−2
−3
−4
H
6
4
3
2
1
−4−3−2 O
−2
−3
−4
%h
8
C (4, -3)
D (4, -4)
B18 California Mathematics, Geometry Standards Practice
1 2 3 4x
&h
7
1 2 3 4x
&h
−2
−3
−4
J
y
y
O
−4−3−2
%h
%h
4
3
2
1
'h
Quadrilateral TUVW is translated so
that the new vertices are T(-1, 0),
U(1, 3), and V(4, 2). What are the
coordinates of W?
1 2 3 4x
O
−4−3−2
4
3
2
1
y
'h
O
−2
−3
−4
1 2 3 4x
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3
3
H _
2
9
J _
4
y
4
3
2
1
−4−3−2
A (0, -3)
B (0, -4)
1 2 3 4x
O
−4−3−2
&h
5
'h
2
1
4
F _
9
7
y
4
%h 3
Name
Date
Practice by Chapter
Chapter 10 Circles
1
⎯⎯ is tangent to
In the figure below, WX
⎯ is tangent to
circle O at point W, XY
circle O at point Y, and m
WY = 140°.
3
What could be the first step in
constructing the perpendicular
−−
bisector of PQ?
3
8
2
1
140‚
9
0
:
4
−−
A From a point on PQ that is halfway
between P and Q, draw equal arcs that
intersect at P and Q.
−−
B Draw segment RS.
C From point P, draw an arc that extends
−−
more than halfway across PQ.
−−
D Draw point R above PQ.
What is m∠WXY?
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A
B
C
D
2
40°
70°
80°
140°
−−
AB is a diameter of the circle.
⎯ is tangent to the circle at point B.
BE
m
BCD = 110°.
4
"
Which of the following is always true
when a radius and a chord intersect at
a right angle?
F
G
H
J
$
The radius is bisected.
The chord is bisected.
The chord length equals the radius.
The chord intercepts a 90° arc.
%
&
#
5
−−
In the circle below, AB is a diameter.
What is m∠ABC?
F
G
H
J
35°
55°
70°
110°
"
12 m
#
What is the circumference of the circle?
(π ≈ 3.14)
A 37.68 m
B 75.36 m
C 113.04 m
D 452.16 m
California Mathematics, Geometry Standards Practice
B19
Name
Date
Practice by Chapter
Chapter 10 (continued)
6
∠ACB is inscribed in the circle below,
intercepting arc ADB.
8
"
%
$
Which of the following can be used
to prove that the equation of a circle
with center (h, k) and radius r is
r 2 = (x - h)2 + (y - k)2?
F
G
H
J
#
the formula for the area of a circle
the midpoint formula
the angle addition postulate
the distance formula
If mADB = 70°, then what is the
measure of ∠ACB?
F
G
H
J
−−
−−
LP and LQ are secants of the circle
below.
35°
70°
110°
140°
.
0
−−
−−
In the circle below, PS and QR are
chords intersecting at T. If QT = 8,
RT = 3, and PT = 4, then what is the
−−
length of ST?
2
1
2
If LM = 6, MP = 10, and LO = 8, what
−−
is the length of LQ?
1
A 4
B 6
5
C 10
D 12
3
4
A
B
C
D
5
6
7
8
10
In the figure below, if MN = 13 inches,
and MO = 12 inches, then what is the
−−
length of NO?
.
1
/
0
F 4 in.
G 5 in.
B20 California Mathematics, Geometry Standards Practice
H 6 in.
J 7 in.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
7
9
Name
Date
Practice by Chapter
Chapter 11 Areas of Polygons and Circles
1
Parallelogram WXYZ is shown below.
4
What is the area, in square units, of
rhombus PQRS?
9
8
5 cm
;
2
4 cm
6 cm
:
1
30‚
24
What is the area of the parallelogram?
A
B
C
D
2
3
18 cm 2
30 cm 2
40 cm 2
50 cm 2
4
What is the area, in square meters, of
the figure shown below?
5
30 m
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
F
G
H
J
12 m
96
144 √
3
288 √
3
576
What is the area, in square units, of the
parallelogram shown below?
18 m
20 m
F
G
H
J
3
220 m 2
300 m 2
360 m 2
450 m 2
What is the area, in square units,
of an equilateral triangle with side
length 12?
A
B
C
D
9 √
3
18 √
3
36 √
3
72 √
3
A
B
C
D
9
15
15 √
2
20
California Mathematics, Geometry Standards Practice
B21
Name
Date
Practice by Chapter
Chapter 11 (continued)
6
A circular spinner is shown below.
8
Which of the following shaded sectors
shows a region with an area of 48π
square inches?
F
100‚
60‚
12 in.
If the spinner is spun one time, what is
the probability that it will land on the
shaded area?
G
90‚
5
F _
12 in.
18
1
G _
3
1
H _
4
5
J _
9
H
120‚
12 in.
What is the area, in square units, of the
triangle shown below?
J
y
12 in.
240‚
(0, 3)
(−4, 0)
O
(4, 0)
x
9
A
B
C
D
6
10
12
24
What is the area of a rhombus that has
diagonal lengths of 18 cm and 10 cm?
A
B
C
D
B22 California Mathematics, Geometry Standards Practice
56 cm 2
90 cm 2
112 cm 2
180 cm 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
7
Name
Date
Practice by Chapter
Chapter 12 Extending Surface Area
1
Helena is painting the top and sides
of the cabinet shown below. What is
the area of the surface that she
is painting?
4
F
G
H
J
5 ft
2 ft
A
B
C
D
A rectangular prism is 4 inches long,
2 inches wide, and 9 inches high. What
is the surface area of the prism?
124 in 2
72 in 2
62 in 2
45 in 2
3 ft
5
30 ft 2
33 ft 2
56 ft 2
62 ft 2
The curved surface of the swimming
pool shown below is being cleaned.
5 ft
4 ft
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
A cylinder is 25 cm tall and has a
radius of 10 cm. What is the surface
area of the cylinder?
F
G
H
J
3
A
B
C
D
200π cm 2
300π cm 2
500π cm 2
700π cm 2
What is the surface area of the right
triangular prism shown below?
14 cm
8 cm
6 cm
A
B
C
D
What is the area of the surface being
cleaned?
6
40π ft 2
90π ft 2
100π ft 2
140π ft 2
A right circular cone has a diameter of
12 units and a slant height of 9 units.
What is the surface area of the cone, in
square units? (SA = πr 2 + πr)
F
G
H
J
54π
90π
108π
252π
164 cm 2
188 cm 2
336 cm 2
384 cm 2
California Mathematics, Geometry Standards Practice
B23
Name
Date
Practice by Chapter
Chapter 12 (continued)
7
The Walter Pyramid sports arena in
Long Beach, California, has a square
base that is 115 yards on each side.
10
86 yd
64 yd
F
G
H
J
115 yd
The entire surface of the pyramid is
blue. What is the area of the blue
surface? (Hint: Find the lateral area.)
A
B
C
D
In a model of the solar system, Earth
has a diameter of 15 inches, and Mars
has a diameter of 8 inches. How much
greater is the surface area of Earth?
(SA = 4πr 2)
A sphere has a radius of 6 units. What
is the surface area of the sphere, in
square units, rounded to the nearest
tenth? (SA = 4πr 2)
F
G
H
J
9
11
1307 sq in.
1508 sq in.
3016 sq in.
3820 sq in.
15 in.
8 in.
75.4
452.4
904.8
2714.3
Earth
A
B
C
D
A right circular cone has a height of
11 cm and a radius of 3 cm. What is the
lateral area of the cone? (LA = πrℓ)
A
B
C
D
3π √
33 cm 2
33π cm 2
3π √
130 cm 2
390π cm 2
12
Mars
161π sq in.
225π sq in.
289π sq in.
644π sq in.
A pyramid has a square base that has
area 81 cm 2. The slant height of the
pyramid is 7 cm. What is the surface
area of the pyramid?
F 126 cm 2
G 207 cm 2
B24 California Mathematics, Geometry Standards Practice
H 364.5 cm 2
J 567 cm 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
8
14,720 yd 2
19,780 yd 2
27,945 yd 2
33,005 yd 2
Michael uses a cylindrical container to
store his art supplies. The diameter of
the container is 16 inches, and the
height is 22 inches. He wants to cover
the whole surface except the lid with
contact paper. Which is the best
estimate for the amount of contact
paper that he will need?
Name
Date
Practice by Chapter
Chapter 13 Extending Volume
1
Trey placed a rectangular box in an
empty cylindrical can, as shown below.
4
24 in.
30 in.
F
G
H
J
6 in.
10 in.
8 in.
What is the approximate volume of the
empty space in the can? (π ≈ 3.14)
A 3,840 in 3
B 13,085 in 3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
5
C 16,800 in 3
D 53,779 in 3
Cylinder A and cylinder B have equal
heights, but the diameter of cylinder B
3
the ratio of the volume of cylinder A to
the volume of cylinder B?
3
H 9 to 25
J 25 to 9
6
Dolly has substituted different values
for one of the dimensions of a cone into
the formula for the volume of a cone.
She has plotted the resulting volumes
on a coordinate grid. If the data points
she has plotted form a straight line,
then which dimension of the cone has
she changed?
A
B
C
D
the height
the radius
the diameter
cannot be determined
8
8π
12π
72π
A cylindrical container and a coneshaped container are filled with water.
If both containers have equal diameters
and equal heights, how much more
water can the cylindrical container
hold than the cone-shaped
container?
A
B
C
D
5
the diameter of cylinder A. What is
is _
F 3 to 5
G 5 to 3
The distance between points X and
Y on the surface of a sphere is
12π inches. If the sphere is translated
8 inches up, what is the distance
between points X and Y, in inches?
two times as much
three times as much
four times as much
five times as much
A pyramid and a prism have bases
with equal areas, but the height of
the pyramid is nine times the height of
the prism. Which of the following
statements is true?
F The pyramid and the prism have
equal volumes.
G The pyramid has volume three times
that of the prism.
H The prism has volume three times that
of the pyramid.
J The pyramid has volume nine times
that of the prism.
California Mathematics, Geometry Standards Practice
B25
Name
Date
Practice by Chapter
Chapter 13 (continued)
7
A spherical balloon has decreased in
1
of what it
size so that its volume is _
10
64
was before the decrease. What can be
said about the diameter of the balloon?
A rectangular prism that is similar to
the one shown below has height 40 cm.
What is the volume of the rectangular
prism?
4
πr 3)
(V = _
3
16 cm
1
A It is _
of what it was before.
64
1
B It is _
of what it was before.
16
1
C It is _
of what it was before.
8
1
D It is _
of what it was before.
4
8
F
G
H
J
9
F
G
H
J
11
A cylinder has radius equal to its
height. Which would result in a greater
increase in volume, doubling the radius
or doubling the height?
A
B
C
D
(3, 4)
(4, 3)
(-3, 4)
(-4, 3)
Jimmy is trying to shoot three balls of
different sizes through a hoop with a
diameter of 18 inches. The balls have
volumes of 850π, 900π, and 950π cubic
inches. How many of the balls will fit
4
through the hoop? (V = _
πr 3)
3
A
B
C
D
6,144 cm 3
15,360 cm 3
64,000 cm 3
96,000 cm 3
0
1
2
3
B26 California Mathematics, Geometry Standards Practice
doubling either
doubling the height
doubling the radius
It cannot be determined.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Rectangle ABCD is centered about the
origin. Vertex A has coordinates (3, 4).
Rectangle ABCD is a result of a 90°
counterclockwise of rectangle ABCD
about the origin. What are the
coordinates of A?
12 cm
32 cm
Name
Date
Periodic Assessment 1
1
In ABC and DEF, ∠A ∠D,
−−
−−
∠C ∠F, and BC EF. Which rule
proves that ABC DEF?
#
3
Let ABC be an equilateral triangle.
−−
Suppose that AD bisects ∠A. Which of
the following must be true?
#
&
%
"
A
B
C
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
$
%
'
"
ASA
AAS
SSS
SAS
A
B
C
D
Dave is folding cardboard cutouts into
pizza boxes. What is the volume of the
pizza box formed from the cardboard
cutout shown below?
4
$
−− −−
AD ⊥ BC
−− −−
AD AB
−− −−
AD BC
−−
−−
AD bisects AB.
The figure below is a trapezoid with
three congruent right triangles inside.
6
2 in.
4
15 in.
15 in.
F
G
H
J
60 in3
225 in3
227 in3
450 in3
10
What is the area, in square units, of the
shaded portion?
F
G
H
J
10
20
24
32
California Mathematics, Geometry Standards Practice
B27
Name
Date
Periodic Assessment 1
5
Ashish needs to calculate the volume
of a basketball. The diameter of the
basketball is 9 inches. What is the
volume of the basketball?
4 3
(Volume = _
πr )
3
A
B
C
D
7
(continued)
Mary is drawing a line that passes
through point P and is parallel to line ℓ.
Which of the following should be her
first step?
A
201 in3
381.5 in3
1017.4 in3
3052.1 in3
1
6
Which of the following describes two
parallel lines in a plane?
F
G
H
J
B
1
The lines are similar.
The lines are congruent.
The lines intersect only once.
The lines never intersect.
1
D
1
B28 California Mathematics, Geometry Standards Practice
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
C
Name
Date
Periodic Assessment 1
8
Which figure is a counterexample to the
conjecture below?
11
A quadilateral with four equal sides is
always a square.
F
G
H
J
9
Given: MNOP is a parallelogram with
−−−
−−
diagonals MO and NP. Which of the
following is always true?
A
B
C
D
irregular quadrilateral
square
rhombus
cube
12
(continued)
−−− −−
MO NP
−−− −−
MO NP
−−− −−
MO ⊥ NP
−−−
−−
MO bisects NP.
What is the first step toward
constructing the angle bisector of
angle A?
What values of x and y make
quadrilateral PQRS a parallelogram?
%
2
12‚
18‚
4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
#
1
A
B
C
D
10
2x
(
)
3 ‚
3y
(
)
4 ‚
"
5
F From points B and C, draw equal arcs
that intersect at D.
G Draw ray AD
.
H From point A, draw an arc that
intersects the sides of the angle at
points B and C.
J Draw a line segment that connects
points B and C.
x = 12, y = 18
x = 18, y = 12
x = 27, y = 16
x = 16, y = 27
Which of the following best describes
inductive reasoning?
F inferring a general truth by examining
a number of specific examples
G defining mathematical terms to
correspond to physical objects
H using logic to draw conclusions based
on accepted statements
J accepting the meaning of a term
without its definition
$
13
Which of the following is true of any
two points, A and B, in a plane?
A
B
C
D
A line exists that contains A and B.
Point A is greater than B.
Point A bisects B.
Point A is perpendicular to B.
California Mathematics, Geometry Standards Practice
B29
Name
Date
Periodic Assessment 1
14
What is the area, in square units, of the
scalene triangle shown below?
16
(continued)
Use the proof to answer the question
below.
−− −−
Given: AB DE and ∠A ∠C
Prove: ABC ∼ DEC
3.5
7
#
5
3
&
F
G
H
J
15
5.25
15
24.5
35
"
A
B
C
D
1590 sq in.
1802 sq in.
1809 sq in.
7206 sq in.
Statement
Reason
1. ∠A ∠D
Transversal Postulate
2. ∠C ∠C
Reflexive Property
3. ∠D ∠C
Transitive Property
4. ABC ∼ DEF
F
G
H
J
%JBNFUFSGU
%JBNFUFSJO
$
17
?
SAS
AA
SSS
AAS
Which figure can serve as a
counterexample to the conjecture
below?
If both pairs of opposite sides of a
quadrilateral are parallel, then the
quadrilateral is a square.
A
B
C
D
B30 California Mathematics, Geometry Standards Practice
trapezoid
equilateral triangle
rectangle
irregular quadrilateral
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Geena just bought a circular patio
table that has diameter 4 feet. There
is a hole in the middle of the table
for an umbrella. What is the area of
the shaded portion of the table, in
square inches?
%
Name
Date
Periodic Assessment 1
18
STUV is a rhombus whose angles are
all congruent. Therefore, STUV is a
square. What kind of reasoning is this?
F
G
H
J
21
(continued)
Given: ABE and CDF are isosceles
right triangles. Prove that the trapezoid
ABCD is isosceles.
inductive
deductive
proof by contradiction
reductio ad absurdum
"
19
#
$
&
'
%
“All right triangles are isosceles.”
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Which of the following best describes a
counterexample to the assertion above?
A
B
C
D
20
equilateral triangle
scalene right triangle
obtuse triangle
isosceles trapezoid
What is the volume of a shed with the
following dimensions.
The floor is 10 feet by 8 feet, two sides
are 8 feet by 7 feet, and the other two
sides are 10 feet by 7 feet.
F
G
H
J
412 ft3
560 ft3
640 ft3
700 ft3
California Mathematics, Geometry Standards Practice
B31
Name
Date
Periodic Assessment 2
1
3
What is m∠x?
x‚
110‚
These are parallelograms. What is
m∠GDE?
160‚
"
160‚
A
B
C
D
2
105‚
(
20°
50°
70°
110°
%
'
"
#
4
135‚
%
30°
60°
75°
105°
−− −−
In the figure below, AC DF.
#
5b
(
)
7 ‚
&
$
F
G
H
J
a = 12, b = 21
a = 28, b = 49
a = 36, b = 63
a = 44, b = 77
"
$
%
Which additional information
would be sufficient to prove that
ABC DEF?
F
G
H
J
B32 California Mathematics, Geometry Standards Practice
−− −−
AB AC
−− −−
AB DF
−− −−
AB DE
−− −−
AB EF
'
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5a
( )‚
4
$
&
A
B
C
D
What values of a and b make
quadrilateral ABCD an isosceles
trapezoid?
#
105‚
Name
Date
Periodic Assessment 2
5
Which type of quadrilateral is
quadrilateral ABCD?
y
8
7
6
5
(
)
4 A 1, 3
3
2
1
O
A
B
C
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6
7
(continued)
ABCD is a trapezoid with congruent
−−
−−
diagonals AC and BD. The diagonals
intersect at point F. Which of the
following would be sufficient to prove
that ABC BAD?
B (4, 5)
#
"
C (7, 3)
'
D (4, 1)
1 2 3 4 5 6 7 8x
%
A
B
C
D
square
rhombus
irregular quadrilateral
trapezoid
8
What is the value of x?
x‚
$
−− −−
AB AC
−− −−
AF DF
−− −−
AD BC
−− −−
AC CD
A triangle is formed by the points
A(-2, -3), B(-2, 5), and C(6, -3).
What are the coordinates of the
−−
midpoint of BC?
F (1, 2)
35‚
F
G
H
J
50‚
1
G 1, -_
(
35°
55°
95°
85°
2
)
H (2, 3)
J (2, 1)
9
Two angles of a triangle measure 90°
and 45°, respectively. What type of
triangle is it?
A
B
C
D
equilateral
scalene
isosceles
obtuse
California Mathematics, Geometry Standards Practice
B33
Name
Date
Periodic Assessment 2
10
In the quadrilateral shown below, what
is m∠a + m∠b?
12
(continued)
What are the values of x and y if the
−−
−−
lengths of AB and BC are 1.5 times the
−−
length of AC?
78‚
61‚
#
a
b
F
G
H
J
2x
3
102°
119°
221°
278°
"
11
3y
4
$
1
F x = 12, y = 10_
3
A parallelogram is formed by the
points A(-4, 6), B(2, 6), C(4, 3), and
D(-2, 3). What is the length of
−−
diagonal BD?
D (−2, 3)
−4
A
B
C
D
3
5
7
10
−2
7
6
5
4
3
2
1
O
y
H x = 16, y = 18
2
J x = 30, y = 26_
B (2, 6)
C (4, 3)
1 2 3 4x
3
13
Given: ABC is an equilateral
triangle. D is a point such that
AD
bisects ∠A.
Which of the following is not
necessarily true?
A
B
C
D
B34 California Mathematics, Geometry Standards Practice
AD
bisects BC
.
−− −−
AD AB
−−
AD
⊥ BC
−− −−
AB AC
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A (−4, 6)
G x = 18, y = 16
Name
Date
Periodic Assessment 2
14
16
What is m∠z?
(continued)
What is m∠x?
#
#
x
z
145‚
35‚
"
"
F
G
H
J
60‚
$
10°
30°
50°
70°
17
−− −−
In the figure below, AC DF, and
∠C ∠F.
15
#
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
F
G
H
J
110‚
If n is a whole number, what is the
smallest possible value for n in the
triangle below?
n
4
10
'
%
Which additional information
would be sufficient to prove that
ABC DEF?
A
B
C
D
35°
70°
110°
145°
&
$
"
$
A
B
C
D
3
4
6
7
−− −−
BC EF
−− −−
AB DE
−− −−
BC DF
−− −−
AB AC
California Mathematics, Geometry Standards Practice
B35
Name
Date
Periodic Assessment 2
18
21
The figure below shows ABC.
y#
(continued)
Using a straightedge and a compass,
construct the perpendicular bisector
of line segment AB. Explain your
process.
$ x
"
Which statement would not help prove
that ABC is isosceles?
−−
−−
F (slope AB) = -(slope BC)
G distance from A to B = distance from
B to C
−−
−−
H slope AB = slope BC
J m∠A = m∠C
19
#
"
40‚
%
A 40°
B 80°
20
$
x‚
C 100°
D 140°
If BDEF is a parallelogram, and C, B,
and D are collinear, what is m∠CBF?
"
%
60‚
#
60‚
&
$
80‚
'
F 60°
G 80°
H 100°
J 120°
B36 California Mathematics, Geometry Standards Practice
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
In the parallelogram below, what is the
value of x?
Name
Date
Periodic Assessment 3
1
The sun is at a height of 25° relative to
the horizon. How long is the shadow of
a tree that is 30 feet tall?
3
If point D is the center of the circle
below, what is m∠A?
#
"
35‚
%
$
30 ft
25‚
A 35°
B 55°
sin 25° ≈ 0.42
cos 25° ≈ 0.91
tan 25° ≈ 0.47
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A
B
C
D
2
4
12.6 ft
33.0 ft
63.8 ft
71.4 ft
C 70°
D 90°
What are the coordinates of B if
trapezoid ABCD is reflected across
the y-axis?
"
%
Suppose that ABC is an equilateral
triangle. What are the values of a
and b?
#
4
3
2
$ 1
−4−3−2
O
y
1 2 3 4x
−2
−3
−4
#
2a
3
"
F
G
H
J
F
G
H
J
18
3b
5
a = 12, b = 10.8
a = 18, b = 18
a = 27, b = 30
a = 30, b = 27
(-2, -4)
(2, -4)
(4, -2)
(2, 4)
$
5
A right triangle has one leg with length
7 and another leg with length 8. What
is the length of the hypotenuse of the
triangle?
A √
15
B √
113
C 4 √7
D 7 √
8
California Mathematics, Geometry Standards Practice
B37
Name
Date
Periodic Assessment 3
6
The figure below shows trapezoid
ABCD.
8
7
6
5
4
3
2
1
9
#
15
"
−−
−−
slope AB = slope CD
−−
−−
slope AB = -(slope CD)
−−
−−
(slope AB) · (slope CD) = 1
−−
−−
(slope AB) · (slope CD) = -1
10
20 revolutions
39 revolutions
40 revolutions
80 revolutions
16
A 0.6
C 0.8
B 0.75
4
D_
3
314 ft
628 ft
942 ft
1256 ft
If a = 2 √2 in the right triangle below,
then what is the value of c?
45‚
a
:
c
#
b
8
"
10
F 8
G 10
$
9
16
H 12.8
J 20
;
$
A circus has several large tents whose
bases are circular. The radii of two of
the tents are 200 feet and 300 feet.
What is the approximate difference in
their circumferences?
F
G
H
J
11
%
9
−−
What must be the length of XY if ABC
is similar to XYZ?
8
20
A 2
B 2 √
2
B38 California Mathematics, Geometry Standards Practice
C 4
D8
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A basketball has diameter 9 inches.
How many revolutions will the
basketball make if it rolls 94 feet?
A
B
C
D
12
$
#
What statement would prove that
ABCD is an isosceles trapezoid?
7
What is sin A?
y
O " 1 2 3 4 5 6%7 8 x
F
G
H
J
(continued)
Name
Date
Periodic Assessment 3
12
−−
If AB is a diameter, what are the
coordinates of the center of the circle?
−4−3−2
F
G
H
J
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
13
y
7
6
5
4
3
2
1
A (−4, 2)
14
B (2, 2)
15
Suppose that ABC is isosceles and
−−
−−
that AB is twice as long as AC. What
are the values of a and b?
#
(x, y)
4
# 3
2
1
−2
O
−2
−3
−4
a -3
b +2
4
3
(x - 2, y + 3)
"
$
3
y
A a = 7, b = 9
B a = 9, b = 7
C a = 16, b = 27
D a = 27, b = 16
1 2 3 4x
$
16
−− −−
In the figure below, AC || DE. Which
theorem or postulate can be used to
prove that ABC is similar to DBE?
What will be the coordinates of
point B?
A
B
C
D
H 47,925 ft2
J 60,025 ft2
1 2 3 4x
O
ABC is to be translated to ABC
by the following motion rule.
−4
The Smiths’ backyard is 110 feet by
165 feet. The Yings’ backyard is
adjacent to the that of the Smiths. The
total dimensions of the two backyards
are 245 feet by 165 feet. What is the area
of the Yings’ backyard?
F 22,275 ft2
G 32,800 ft2
(1, 2)
(-3, 2)
(3, 2)
(-1, 2)
"
(continued)
#
(-3, 5)
(2, 0)
(1, 5)
(2, 0)
%
"
F SAS
G SSS
&
$
H SSA
J AA
California Mathematics, Geometry Standards Practice
B39
Name
Date
Periodic Assessment 3
17
What are the coordinates of C if
rectangle ABCD is rotated 90°
clockwise about the origin?
19
(continued)
If ABC is isosceles, what is the
value of x?
#
4
3
2
1
−5−4−3−2
#
"
A
B
C
D
18
$
O
y
2x
3
10
"
1 2 3x
−2
−3
−4
A
B
C
D
%
(-4, 2)
(-2, 2)
(-4, -2)
(4, -2)
20
$
16
9
15
24
39
In the triangles below, ∠A ∠D. What
would prove that ABC is similar to
DEF?
What is the measure of leg b?
&
9
$
"
b
F
G
H
J
16
23.3
32
34
AB
DF
F _
=_
DE
AC
AB
AC
G _=_
DE
DF
BC
AB
H _
=_
DE
EF
AB
DE
J _
=_
BC
DF
B40 California Mathematics, Geometry Standards Practice
%
'
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
#
25
Name
Date
Periodic Assessment 3
21
(continued)
Prove the Pythagorean theorem using
the figure below.
Given: a square is inscribed within a
larger square.
a
b
b
c
c
c
b
c
a
a
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
b
California Mathematics, Geometry Standards Practice
B41
Name
Date
Periodic Assessment 4
1
What is the area, in square units, of
a parallelogram with vertices A(1, 0),
B(3, 6), C(7, 6), and D(5, 0)?
A 10
B 20
4
In the figure below, AB
is tangent
to circle P at point A, BD
intersects
= 40°,
circle P at points C and D, mAC
and mCD = 140°.
C 24
D 28
%
140‚
2
What is the volume of the silo shown
below?
1
40‚
$
30 ft
#
"
What is m∠ABC?
F
G
H
J
150 ft
3
H 135,000 ft 3
J 423,900 ft 3
5
What is the lateral area of the
rectangular prism shown below?
What is the area, in square units, of
isosceles ABC inscribed within the
circle below? Point D is the center of
the circle, and the radius is 5 units.
7 ft
8 ft
9 ft
#
%
"
A
B
C
D
$
A 15
B 25
C 50
D 100
B42 California Mathematics, Geometry Standards Practice
128 ft 2
135 ft 2
238 ft 2
504 ft 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
F 33,750 ft 3
G 105,975 ft 3
35°
40°
70°
80°
Name
Date
Periodic Assessment 4
6
What is the area, in square units, of the
trapezoid shown below?
7
6
5
4
3
2
1
#
"
F
G
H
J
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
7
What is the surface area of the prism
shown below?
y
5 ft
$
5 ft
%
5 ft
5 ft
4.3 ft
1 2 3 4x
O
−4−3−2
8
(continued)
F
G
H
J
18
21
36
42
What is the total area, in square units,
of the figure shown below?
9
5 ft
75 ft2
96.5 ft2
150 ft2
193 ft2
−−
−−
In the circle below, AB and CD are
chords that intersect at E.
%
"
8
5
3
&
$
5
6
A 6
B 12
C 24
D 30
#
4
If CE = 13, DE = 21, and BE = 19,
then what is AE?
A
B
C
D
11.8
14.4
15
30.7
California Mathematics, Geometry Standards Practice
B43
Name
Date
Periodic Assessment 4
10
What is the volume of the cone
1 2
shown below? (Volume = _
πr b)
3
(π ≈ 3.14)
13
15 cm
348 cm3
475 cm3
1425 cm3
1900 cm3
14
11
What is the volume of a shoe box
that is 5 inches tall, 9 inches long, and
6 inches wide?
What is the total surface area, in square
centimeters, of the pyramid below?
24
cm
C 99 in3
D 270 in3
25 cm
25 cm
2
12
is a tangent line, AB
is a secant line,
and m AB = 140°.
140‚
#
15
What is the area of equilateral
triangle ABC inscribed in circle M?
Circle M has radius 6 inches.
%
"
H 1200 cm 2
J 1825 cm 2
F 600 cm
G 625 cm 2
−−
In circle R below, AC is a diameter, DC
#
$
120‚
3
120‚
.
"
$
What is m∠BDC?
F 40°
G 45°
H 70°
J 90°
120‚
A 36 sq in.
B 27 √
3 sq in.
B44 California Mathematics, Geometry Standards Practice
C 54 sq in.
D 54 √
3 sq in.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A 75 in3
B 84 in3
The vertices of quadrilateral ABCD are
A(-2, 1), B(-2, 5), C(3, 5), and D(3, 1).
If ABCD is translated 6 units down and
5 units to the right to create DEFG,
what are the coordinates of the vertices
of DEFG?
A A(3, -5), B(3, -1), C(8, -1), D(8, -5)
B A(-7, -5), B(-7, -1), C(-2, -1),
D(–2, –5)
C A(3, 7), B(3, 7), C(8, 11), D(8, 11)
D A(-7, 7), B(-7, 11), C(-2, 11),
D(–2, 7)
11 cm
F
G
H
J
(continued)
Name
Date
Periodic Assessment 4
16
What is the lateral area of a cylinder
that has radius 8 cm and height 15 cm?
(π ≈ 3.14)
F
G
H
J
17
19
13 cm
−−
AB is the diameter of circle C.
A
B
C
D
"
$
#
What is m∠DAB?
A 30°
B 45°
530.7 sq in.
2122.6 sq in.
4245.3 sq in.
9203 sq in.
30‚
20
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
What is the surface area, in square
inches, of the sphere shown below?
(SA = 4πr 2) (π ≈ 3.14)
120.0 cm2
240.0 cm2
401.9 cm2
753.6 cm2
%
18
(continued)
What is the total surface area of the
cone below shown? (SA = πr 2 + πrℓ)
(π ≈ 3.14)
C 60°
D 90°
12 cm
What is the volume of the prism shown
below?
5 cm
5 cm
8 cm
11 cm
F
G
H
J
F
G
H
J
204.1 cm2
266.9 cm2
282.6 cm2
722.2 cm2
cm3
95
128 cm3
220 cm3
440 cm3
California Mathematics, Geometry Standards Practice
B45
Name
Date
Periodic Assessment 4
21
(continued)
Margo is shopping for a new
backpacking tent.
8 ft
4 ft
5 ft
a. What is the volume of the tent?
b. By how much would the volume
change if you increased the length
by 10%?
d. By how much would the volume
change if you increased the length,
width, and height all by 10%?
B46
California Mathematics, Geometry Standards Practice
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
c. By how much would the volume
change if you increased both the
width and the height by 10%?
Name
Date
Periodic Assessment 1
Student Answer Sheet
Record your answers by coloring in the appropriate bubble for the best answer to
each question.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Write your answer for Question 21 in the space below. Show all your work or reasoning.
California Mathematics, Geometry Standards Practice
B47
Name
Date
Periodic Assessment 2
Student Answer Sheet
Record your answers by coloring in the appropriate bubble for the best answer to
each question.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Write your answer for Question 21 in the space below. Show all your work or reasoning.
California Mathematics, Geometry Standards Practice
B49
Name
Date
Periodic Assessment 3
Student Answer Sheet
Record your answers by coloring in the appropriate bubble for the best answer to
each question.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Write your answer for Question 21 in the space below. Show all your work or reasoning.
California Mathematics, Geometry Standards Practice
B51
Name
Date
Periodic Assessment 4
Student Answer Sheet
Record your answers by coloring in the appropriate bubble for the best answer to
each question.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Write your answer for Question 21 in the space below. Show all your work or reasoning.
California Mathematics, Geometry Standards Practice
B53