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Edward B. Burger
David J. Chard
Earlene J. Hall
Paul A. Kennedy
Steven J. Leinwand
Freddie L. Renfro
Dale G. Seymour
Bert K. Waits
Geometry
Contents in Brief
CHAPTER 1
Foundations for Geometry
CHAPTER 2
Geometric Reasoning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CHAPTER 3
Parallel and Perpendicular Lines . . . . . . . . . . . . . . . . . 142
CHAPTER 4
Triangle Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
CHAPTER 5
Properties and Attributes of Triangles . . . . . . . . . . 296
CHAPTER 6
Polygons and Quadrilaterals
CHAPTER 7
Similarity
CHAPTER 8
CHAPTER 9
Right Triangles and Trigonometry . . . . . . . . . . . . . . . 514
Extending Perimeter, Circumference,
and Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584
C H A P T E R 10
Spatial Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650
C H A P T E R 11
Circles
C H A P T E R 12
Extending Transformational Geometry . . . . . . . . . 820
......................... 2
70
. . . . . . . . . . . . . . . . . . . . . 376
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742
Student Handbook
Extra Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S4
Problem Solving Handbook . . . . . . . . . . . . . . . . . . . . . . . . . . . S40
Skills Bank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S50
Postulates, Theorems, and Corollaries . . . . . . . . . . . . . . . . S82
Selected Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S88
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S115
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S161
Symbols and Formulas . . . . . . . . . . . . . . . . . . . Inside Back Cover
Copyright © 2007 by Holt, Rinehart and Winston
All rights reserved. No part of this publication may be reproduced or
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HOLT and the “Owl Design” are trademarks licensed to Holt, Rinehart and
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Printed in the United States of America
If you have received these materials as examination copies free of charge, Holt,
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Resale of examination copies is strictly prohibited.
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this publication, or any portion of it, into electronic format.
ISBN 0-03-035828-0
Cover photo: The Stata Center at
MIT, Boston, Massachusetts, USA.
© Scott Gilchrist/Masterfile
1 2 3 4 5 048 09 08 07 06
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5/25/06 4:07:27 PM
AUTHORS
Edward B. Burger, Ph.D.
is Professor of Mathematics
and Chair at Williams College
and is the author of numerous
articles, books, and videos.
He has won several of the
most prestigious writing and
teaching awards offered by
the Mathematical Association
of America. Dr. Burger has
appeared on NBC TV, National
Public Radio, and has given
innumerable mathematical
performances around the world.
Steven J. Leinwand spent
22 years as the Mathematics
Supervisor with the Connecticut
Department of Education.
He is currently a Principal
Research Analyst at the
American Institutes for
Research.
David J. Chard, Ph.D.,
is an Associate Dean of
Curriculum and Academic
Programs at the University of
Oregon. He is the President
of the Division for Research
at the Council for Exceptional
Children, is a member of
the International Academy
for Research on Learning
Disabilities, and is the Principal
Investigator on two major
research projects for the U.S.
Department of Education.
Freddie L. Renfro,
BA, MA, has 35 years of
experience in Texas education
as a classroom teacher
and director/coordinator of
Mathematics PreK-12 for school
districts in the Houston area.
She has served as TEA TAAS/
TAKS reviewer, team trainer
for Texas Math Institutes, TEKS
Algebra Institute writer, and
presenter at math workshops.
Earlene J. Hall, Ed.D.,
is the middle school
mathematics supervisor for
Detroit Public Schools, and an
adjunct professor at Wayne
State University in Detroit
Michigan where she teaches
graduate courses in the
College of Education.
Dale G. Seymour is a
retired mathematics teacher,
author, speaker and publisher.
Dale founded Creative
Publications in 1968, and
went on to found two other
mathematics publishing
companies. Creating
mathematical sculptures is
one of his many hobbies.
Paul A. Kennedy,
Ph.D. is a professor in the
Department of Mathematics
at Colorado State University.
Dr. Kennedy is a leader in
mathematics education. His
research focuses on developing
algebraic thinking by using
multiple representations and
technology. He is the author of
numerous publications.
Bert K. Waits, Ph.D.,
is a Professor Emeritus of
Mathematics at The Ohio State
University and co-founder
of T3 (Teachers Teaching
with Technology), a national
professional development
program.
CONTRIBUTING AUTHORS
Linda Antinone
Fort Worth, TX
Ms. Antinone teaches mathematics at R. L.
Paschal High School in Fort Worth, Texas.
She has received the Presidential Award for
Excellence in Teaching Mathematics and
the National Radio Shack Teacher award.
She has coauthored several books for Texas
Instruments on the use of technology in
mathematics.
Carmen Whitman
Pflugerville, TX
Ms. Whitman travels nationally helping
districts improve mathematics education.
She has been a program coordinator on the
mathematics team at the Charles A. Dana
Center, and has served as a secondary math
specialist for the Austin Independent School
District.
REVIEWERS
Robert Brouhle
Mathematics Department Chair, retired
Marina High School
Huntington Beach, CA
Carey Carter
Mathematics Teacher
Everman Joe C. Bean High School
Everman, TX
Greg Davis
Department Chair, retired
Lodi High School
Lodi, WI
Roger Fuller
Mathematics Department Chair
Grand Prairie High School
Grand Prairie, TX
Anthony Gugliotta
Supervisor of Math & Science
Rumson-Fair Haven Regional HS
Rumson, NJ
Marieta W. Harris
Mathematics Specialist
Memphis, TN
Debbie Hecky
Geometry Teacher
Scott High School
Covington, KY
Kathleen Kelly
Mathematics Department Chair,
retired
Lawrence High School
Fairfield, ME
Mike Kingery
Mathematics Teacher
Mayfield High School
Las Cruces, NM
Joy Lindsay
Mathematics Instructor
Bonita High School
LaVerne, CA
Kim Loggins
Geometry Teacher
Los Alamitos High School
Los Alamitos, CA
Elaine Pappas
Mathematics Department Chair
Cedar Shoals High School
Athens, GA
Terri Salas
Mathematics Consultant
Corpus Christi, TX
Jane Schneider
Mathematics Department Chair
Parkway West High School
Ballwin, MO
Cynthia Hodges
Department Chair
Shoemaker High School
Killeen, TX
ge07se_FM_i_c3.indd iv
5/25/06 4:08:09 PM
Jamae Sellari
Mathematics Instructor
Forest Hill High School
Jackson, MS
Anna Valdez
Geometry Teacher
Nikki Rowe High School
McAllen, TX
Caren Sorrells
Mathematics Coordinator
Birdville ISD
Haltom City, TX
Lauralea Wright
Mathematics Teacher
Mauldin High School
Mauldin, SC
E. Robin Staudenmeier
Middle/High School Math Coordinator
Olympia Community USD 16
Stanford, IL
Denise Young
Mathematics Teacher
Blue Valley West High School
Overland Park, KS
Maureen “Marnie” Stockman
Geometry Specialist and Consultant
Cordova, MD
CONTRIBUTING WRITER
Karen Droga Campe
Instructor
Yale University
New Haven, CT
FIELD TEST PARTICIPANTS
ge07se_FM_i_c3.indd v
Jill Morris
Navasota High School
Navasota, TX
Carey Carter
Alvarado High School
Alvarado, TX
Ruth Stutzman
Jefferson Forest High School
Forest, VA
Walter Babst
Bonita High School
La Verne, CA
5/25/06 4:16:08 PM
Preparing for Standardized Tests
Holt Geometry provides many opportunities for you to
prepare for standardized tests.
Test Prep Exercises
Use the Test Prep Exercises for daily
practice of standardized test questions
in various formats.
41. What is the
value
19
52
42. Find the
value of s.
23
28
of x?
57
71
��
���
34
56
����
���
���
���������
43. ∠A and ∠B
�
are the remo
te interior angl
equations must
es of ∠BCD in
be true?
�ABC. Which
of these
m∠A - 180°
= m∠B
m∠A = 90° m∠B
CD = m∠BCA
m∠B
- m∠A
m∠B = m∠B
44. Extende
CD - m∠A
d Response
The measure
2 : 3 : 4. Describe
s of the angles
how to use alge
in a triangle
the measure
bra to find the
are in the ratio
of each angle
measures of
and classify the
these angles.
Then find
triangle.
Multiple Choice—choose your answer.
CHALLENGE
Gridded Response—write your answer
in a grid and fill in the corresponding
bubbles.
AND
EXTEND
45. An exter
ior angle of a
triangle measure
(2y 2 + 7)° and (61
s 117°. Its remo
- y 2)°. Find the
te interior angl
value of y.
es measure
46. Two para
llel lines are inter
sected by a trans
by the intersecti
versal. What
on
type of triangle
(Hint: Use geom of the angle bisectors of two
is formed
etry software
same-side inter
or construct
same-side inter
ior angles? Expl
a diagram of
ior angles.)
ain.
the angle bisec
tors of two
47. Critical
Thinking Expl
ain why an exter
to a remote inter
ior
angle of a trian
ior angle.
gle cannot be
48. Probabil
congruent
ity The mea
sure of each angl
What is the prob
e in a triangle
ability that the
is a multiple
triangle has at
of 30°.
49. In �ABC,
least two cong
m∠B is 5° less
ruent angles?
than 1 __1 time
What is m∠A
s m∠A. m∠C
2
in degrees?
is 5° less than __1
2 2 times m∠A
.
Short Response—write open-ended
responses that are scored with a
2-point rubric.
SPIRAL REVIE
W
Make a table
to show the valu
e of each func
(Previous cour
tion when x is
se)
-2, 0, 1, and
50. f(x) = 3x
4.
-4
51. f(x) = x 2
+1
53. Find the
−−−
length of NQ.
52. f(x) = (x
Name the theo
- 3) 2 + 5
or postulate that
rem
justifies your
�����
answer.
(Lesson 2-7)
�
Extended Response—write openended responses that are scored with a
4-point rubric.
�
Classify each
triangle by its
side
�
�����
�
���
230
e area
nd th
se Fi
C
spon
Item
ers.
ed Re
imet
Gridd re cent
ua
in sq
����
��� �
�
e to
to us
ethod d come
rent m
n an
a diffe e questio
th
ink of
n’t th er, circle
ca
u
If yo
answ
your
check it later.
to
back
(
)
(
)
Te
Method 2: To
check this answ
er, use a diffe
By the Centroid
rent method.
Theorem, the
centroid of a
of the distance
triangle is __2
from each verte
x to the midp
opposite side. −−
oint of the3
CD is vertical
with a length
and the coord
of 6 units. __2 ( )
inates of the
point that is
3 6 = 4,
4 units up from
C is (1, 3).
This meth
od confirms the
C2
�
� � �
�� �
les to
iang
ght tr
ial ri
spec
by
u use on?
swer
n yo
ti
ur an
k yo .
ow ca this ques
H
ec
ch
5.
er
rem
can
answ
you ean Theo
e
how
or
� is th 3n + 1,
plain e Pythag
Ex
at
th −− AC =
6.
th
n?
,
iven
using
A
ce G or of AB value of
Item
4),
Choi
ct
e
Any Qu
is th
(-8, ion
tiple
r bise
irs Aest
Mul
cula 11, what
?
Type: Check
d pa
endi
deisre t triangle
perp C = 6n with a Differ
orIt
impo
�
B
o the
d
ent Method
a righ rtant to check all of your
an
D
se D
�
diffe
answers on a
form
rent method
Item Respon
)
4
test. An effec
to
8,
answer the ques
(
t
C
tive way to do
tion a second
���
Shor 2), and
er. two different methods, then
this is to use
sw
time. If you get
�����
if
a
an
your answer
the same answ
ine
B(0, n your
is probably corre
��
er with
term
ai
����
ct.
pl
de
Ex
to
ope
�
use sl
w to triangle.
the swer?
ho
of
�
an
erse
ght
plain
�
Conv eck your
7. Ex BC is a ri
4
theSho
_
rtch
Response Wha
�A
u use rem to
t are the coor
3
n yo
eo �ABC with A(-2,
dinates of the
ow ca ean Th
4), B(4, 6), and
4
8. H
C(1, -1)? Show centroid of
agor
your work.
-4
Pyth
�
Method 1: The
.
centroid of
�aRST
triangle is the
�
of
3
�
of
r
_
the medians.
te
point of conc
ces
urrency
ocen Write the equations
4
choi
th
�
find their
or
er
of
two medians
e
point of inter
sw
and
section.
nd th
n an
Fi
ve
E
se
e gi
.
Item Respon Let D be the midpoint
−−
rectly
use thm?
of
k.
AB
t
di
u
and
let E be the midp
� �
le
Shor your wor
lem
n yo
−−
oint of BC.
-2 + 4 4 +
prob
prob
ow ca
D= _
6
this
Show
,_
1. H solve this
= (1,5)
4+1 _
6 + (-1)
solve
� 2
E= _
2
�� �
to
,
w to
�
�
= (2.5, 2.5)
2
e ho
2
The median
�
escrib
�
� from C to D
�
2. D
contains C(1,
It is vertical, �
-1
)
and
D(1, 5).
so its equa
� tion
a
is x = 1.
�
rms
fo
�
r
be
��
The medi
num d 17?
an
from
ch
��
A to E contains
hi
��
B
A(-2, 4) and
ce W ith 15 an
−− _
E(2.5, 2.5).
8
Item
w
Choi
slope of AE
= 4 - 2.5 = _
1.5
tiple an triple
1 .
Mul
-2 - 2.5
-_
gore
-4.5 = ob
lem
10
y - y1 = m x
Pytha
is pr 3
( - x 1)
5
lve th
-slope form
soPoint
ld
of
1
y - 4 = - _(ux wou
oices
tude swer
7
+ 2)
Subs
altititute
3
w yo
er ch
1
4 for y , - _
an
ird
e ho
1
for m,
answ
your
the th atand
3
ven
escrib
-2 for x .
u use nfirm th
9. D
1
the gi
n yo
use to
Solveca
cox = 1
u use ?
ow the syste
le tom 
n yo
u can
er
10. H e triang
d yo
to
1
y
ow ca the answ
__
find

4
ho
=
the
H
point of inter
(
et
th
3.
t?
3 x + 2)
section.
nd
ent m
rr4ec
1
er
to fi
co
y
_
ff
=
is
- (1 + 2)
e a di
Substitute 1
er.
3
for x.
escrib ur answ
y=3
4. D
k yo
Problem
Simplify.
chec
Solving
The coordina
tes of the cent
373
Strategies
roid are (1, 3
). st Tackler
• Draw
372
�
re
Test Tackler
s
The second page
guides you through
applications of the
test-taking strategy.
�
e squa
of th
estion
e qu
th
swer
Chapter 4 Trian
gle Congruen
ce
d an
em an
test it
each
Read llow.
fo
that
�
�����
lengths. (Less
on 4-1)
55. �BCD
56. �ABD
57. What if…?
If CA = 8, Wha
t is the effect
classification
on the
of �ACD?
54. �ACD
first answer.
Chapter 5 Prop
erties and Attri
butes of Trian
gles
Preparing For Standardized Tests
Use the Test Tackler to
become familiar with
and practice test-taking
strategies.
�
�
a Diagram
• Make a Mode
l
• Guess and
Test
• Work Backw
ard
• Find a Patte
rn
• Make a Table
• Solve a Simpl
er Problem
• Use Logical
Reasoning
• Use a Venn
Diagr
• Make an Organ am
ized List
The first page of this
feature explains and
shows an example of
a test-taking strategy.
Standardized
Test Prep
Short Response
12 and 13.
am for Items
Use this diagr
�
�
∠2
explain why
transversal n,
m with
20. Given � �
lementary.
and ∠3 are comp
�
����
�
�
�
Use the Standardized Test Prep to
apply test-taking strategies.
?
ure of ∠ACD
100°
meas
12. What is the
40°
�
supplementary
140°
80°
of
13. What type
�
�
�
�
angles.
are
21. ∠G and ∠H 12)°, and m∠H = x°.
m∠G = (2x +
be used to
ion that can
ion
a. Write an equatvalue of x. Solve the equat
determine the
?
triangle is �ABC
and justify each
Isosceles acute
Equilateral acute
∠H
b. Explain why
step.
t but ∠G
has a complemen
KEYWORD: MG7
TestPr
ep
does not.
parts
CUMcture
for every 1000
ULA
s that
TIV
Etive.
ASSESSMENT,
conje
22. A manager
CHAPTERS 1–4
ces, 60 are defec
Multiple
a factory produ
Choice parts in one day,
ry produces 1500
ons for filling
ted to be
If the facto
directi
a.
expec
the
be
Use
can
learn
to
thethem am
are
6. Which condi
cture?
Take some time
for
how many of diagrthe
make sure you
ger’s1conje
Items
tiona
mana
and 2.
and recheck to
on
value as its invers l statement has the same
only get credit
in a grid. Check
defective based
truth
your answer.
e?
properly. You will
in correctly.
in how you found
filling in the grid
�
filled
Expla
that
are
If n < 0, then 2
to show
the boxes
n > 0.
if the ovals below r, solve the problem using
in the table below
b. Use the data conjecture is false.
If a triangle has
To check your answe
you originally
three congruent
the manager’s
d from the one
�
metho
it
nt
you
is
sides, then
an
time,
isosceles triang
a differe
� 5
4
le.
a mistake the first
3�
2
used. If you made the same mistake when
1
If an angle meas
Day
make
ures less than
1500 2500
are unlikely to
an acute angle
90°, then it is
500
nt way.
.
1000 2000
�
you solve a differe
Parts
If n is a negat
1. Which of these
150
ive integer, then
90
30
n < 0.
150congruence
Defective
statements can
proved60
from the inform
be
7. On a map,
onse
ation given in
Parts
an island has
the figure?
Gridded Resp
�AEB
coord
and
°,
)
−−
4
�
a
inate
+
reef
�CED
s (3, 5), and
has coordinate
AC.�ABD �
. m∠E = (3x
s (6, 8). If each
�BCA
lar bisector of
represents 1
14. �CDE � �JKL5)°. What is the value of x?
ndicu
�BAC
−−
map unit
perpe
� �DAC
mile, what is
make from
23. BD is the
the distance
m∠L = (6x the island and
usions you can �DEC � �DEA
between
the reef to the
are the concl
on
mile?
a. What
nearest tenth
2. What
−−
and Frank live
other
of a
ent? information
statem
do, Carmen,
this�CEB
why BD
−−
into
is halfway
Explad
15. Lucy, Eduar
� �AED by AC at isD.neede
−−
prove that
4.2 miles
. Eduardo’s house house.
−−
the HL Cong
BD intersects
’s
the same street
Suppose−−−
AC.
ruenc
Frank
to
b.
9.0
B
e
and
−−
miles
Theor
house
em?
AD est
path from
� AB
6.0 miles
−− −−−
Carmen’s house
between Lucy’s
is the short
ay between
CB
−−
�
halfw
15.8
is
AD
−−
miles
BE � AE
ce between
Lucy’s house
−− −−
. If the distan
ft,
8.
DE
A
150
line
is
�
CE
has
and Frank’s house
an x-intercept
and Lucy’s house
en’s
of -8 and a
of 3. What is
3. Which bicononse
Eduardo’s house ce in feet between Carm
y-intercept
−− −−
the equation
ditional statem
Extended Resp
BC � EF,
of the line?
ent is les.
what is the distan
true?
isosceles triang
y = -8x + 3
do’s house?
Tomo
�DEF
rroware
and
8
is Mond
m∠E = 95°.
house and Eduar
24. �ABC
y=_
if and
−− not−−
x-8
= 42.5°,ayand
3
only if today
m∠C
y=_
Satur
and
day.
3
x+3
and AC � DF.
mined is
- 2n. XY = 2,
8
y = 3x - 8
in how you deter
, and JK = 10
Next montExpla
h is January if
a. What is m∠D?
16. �JKL �2 �XYZKL.
9. �JK
and only if this
�� passes throu
her.
answ
is December.
yourmont
YZ = n . Find
gh points J(1,
are congruent.
3) and K(-3,
Which of these
11).
is
Today �ABC and �DEF
lines is perpe
ement. What
b. Show thatis a weekend day ifAB
+ 2, find
= 3x
ndicular to �JK
is its own suppl
yesterday
only if
+ 7 and and
1
��?
= 2x
1
y = -_
EFwas
17. An angle
Friday
x+_
.
you determined
c. Given that
2
1
y = -2x - _
Explain how
3
its measure?
This
forh x.
mont
value
the
had 31 days
1
y=_
5
if
and
x
s.
+
mont
6
only
er.
inche
e
h
if
had 30 days.
last
your answ
2
154 squar
y = 2x - 4
of a circle is
nearest inch?
the
to
18. The area
ce
feren
10. If PQ = 2(
4. What must
RS) + 4
What is its circum
be true if PQ
equation is true and RS = TU + 1, which
��� intersects
than one point
�ST
measure of ∠Q.
�� at more
by the Subst
?
3_1_ times the
itution Prope
of Equality?
ure of ∠P is 2
rty
m∠P
19. The meas
P, Q, S, and T
tary, what is
are collinear.
are complemen
PQ = TU + 5
If ∠P and ∠Q
P, Q, S, and T
are noncoplana
in degrees?
PQ
=
TU
r.
+6
��� and ST
PQ
��� are oppo
site rays.
PQ
=
293
(
2
TU) + 5
��� and �ST
PQ
Chapters 1–4
�� are lative
Assessment,
PQ = 2(TU) +
Cumu perpendicu
lar.
6
5. �ABC � �DEF
, EF = x 2 - 7,
11. Which of
and BC = 4x
Find the value
the following
- 2.
s of x.
is NOT valid
that triangles
for proving
are congruent?
-1 and 5
AAA
1 and 5
-1 and 6
SAS
ASA
2 and 3
HL
e
Isosceles obtus
Scalene acute
The Hot Tip provides testtaking tips to help you
suceed on your tests.
These pages include practice with
multiple choice, gridded response,
short response, and extended
response test items.
292
Countdown
9
WEEK
to Testing
DA Y 2
4, 1)
��� If A(-4,
��� and BC.
ed by AB
∠ABC is form
inates for C will
, what coord
and B(-4, 6)
se angle?
result in an obtu
)
2
(-1,
DA Y 1
,
��� If A(3, 5)
ular to XY.
��� is perpendic
AB
are the
X(-2, -5), what
B(9, 3), and
of Y?
coordinates
(-2, 9)
(0, 5)
(2, 6)
(6, 8)
(1, 4)
(3, 1)
(2, 5)
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t to ∠5.
∠1 is congruen
ntary to ∠14.
∠2 is suppleme
t to ∠11.
∠8 is congruen
ntary to ∠13.
∠6 is suppleme
DA Y 5
DA Y 4
segment to
h of the given
What is the lengt
the nearest unit?
Which equation
the graph?
the line in
best represents
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Countdown to Testing
Use the Countdown to Testing to practice
for your state test every day.
DA Y 3
is true?
the following
���. Which of
��� � BD
��� and AC
��� � CD
AB
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Chapter 4 Triang
le Congruenc
e
There are 24 pages of practice for
your state test. Each page is designed
to be used in a week so that all
practice will be completed before
your state test is given.
�
��
�
�
�
��
��
1
8
10
Countdown
�
�
y = -2x - 3
1
_
y = - 2x + 3
5
C12
�
1
_
y= x+3
2
y = 3x + 1
to Testing
Each week’s page has five
practice test items, one for
each day of the week.
Test-Taking Tips
✔ Get plenty of sleep the night before the
❑
test. A rested mind thinks more clearly
and you won’t feel like falling asleep
while taking the test.
✔
❑
Draw a figure when one is not provided
with the problem. If a figure is given,
write any details from the problem on
the figure.
✔ Read each problem carefully. As you
❑
finish each problem, read it again to
make sure your answer is reasonable.
✔ Review the formula sheet that will be
❑
supplied with the test. Make sure you
know when to use each formula.
✔ First answer problems that you know
❑
how to solve. If you do not know how to
solve a problem, skip it and come back
to it when you have finished the others.
✔ Use other test-taking strategies that can
❑
be found throughout this book, such
as working backward and eliminating
answer choices.
Preparing For Standardized Tests
C3
COUNTDOWN
TO
TESTING
WEEK
1
DAY 1
DAY 2
Which statement about a number line is
true?
If a = b and b = c, which statement must
be true?
Values increase toward the right.
a>c
Values increase toward the left.
-a - c = 0
Whole numbers are toward the right
and decimal numbers are toward the
left.
a+c=0
a=c
Negative numbers are toward the
right and positive numbers are
toward the left.
DAY 3
If the width of each square in the grid is 1 centimeter,
what is the diameter of the circle?
1 centimeter
3 centimeters
6 centimeters
12 centimeters
DAY 4
DAY 5
Which shape is NOT included in the figure?
Which statement best describes these two
figures?
Circle
Square
Triangle
They cover the same area.
Trapezoid
They are the same size.
They have the same number of sides.
The distance around each figure is
the same.
C4
Countdown to Testing
Geometry 2
Countdown to Testing
DAY 1
WEEK
2
DAY 2
−−
What is the length of FD?
∠ABC is an obtuse angle. Which of these
could be the measure of ∠ABC?
0°
{ Î Ó £
ä
£
Ó
Î
{
53°
0
90°
3
108°
6
9
DAY 3
Which point is described by the coordinates (-2, 3)?
{
Þ
Ý
ä
{
{
{
A
B
C
D
DAY 4
DAY 5
An architect is sketching a blueprint of a
patio for a new home. On the blueprint,
−−−
C is the midpoint of AD, which represents
one side of the patio. Point B is the
−−
midpoint of AC. If BC = 8 feet, what is the
−−−
length of AD?
bisects ∠AOC, and m∠AOC = 60°.
OB
What is m∠BOE?
ÎäÂ
8 feet
30°
16 feet
60°
24 feet
120°
32 feet
150°
"
Countdown to Testing
ge07se_FM_C04_C27.indd C5
C5
5/25/06 4:23:47 PM
Countdown to Testing
3
WEEK
DAY 1
The figure below shows the first three elements in a pattern. The area of the white
region in the first element is 8 cm 2, and the area of the white region in the second
element is 16 cm 2. What will the area of the white region be when an element
contains six circles?
���� ���������
���� ����������
���� ����������
36 square centimeters
48 square centimeters
144 square centimeters
168 square centimeters
DAY 2
DAY 3
Which of these is a unit that can describe
the perimeter of a figure?
̶̶
Point X is the midpoint of HI. What is the
coordinate of the point X?
Meters
�
Square centimeters
�� �� �� ��
Cubic inches
�
�
�
�
�
�
-4
Seconds
0
1
3
DAY 4
DAY 5
Which expression best represents the
perimeter of the figure below?
A line segment is drawn between the
points (5, 8) and (-1, 6). What are the
coordinates of the midpoint of the
segment?
��
������
27x
5x + 11
9x + 9
11x + 5
C6
Countdown to Testing
�
(3, 1)
(4, 14)
(2, 7)
(-_12 , 3)
Geometry 2
2007 SE
Countdown to Testing
4
WEEK
DAY 1
DAY 2
Which equation below represents the
second step of the solution process?
Which conjecture best describes a rule for
the pattern below?
Step 1: 6x - 12 = 3(5 - x)
Step 2: ?
�
Step 3: 9x - 12 = 15
�
�
Step 4: 9x = 27
Rotate counterclockwise 90°
Step 5: x = 3
Rotate clockwise 90°
6x - 12 = 15 - x
Rotate counterclockwise 180°
6x - 12 = 15 - 3x
Rotate clockwise 180°
���
6x - 12 = 5 - 3x
6x = 3(5 - x) - 12
DAY 3
Given: A triangle is a right triangle.
Conclusion: Two of the sides are congruent.
This conclusion—
is true because right triangles have exactly one angle that measures 90°.
is true because all right triangles have two congruent angles.
is false because, for example, the sides of a 30°-60°-90° right triangle have
different lengths.
is false because a right triangle cannot have two congruent angles.
DAY 4
DAY 5
Which of the following best describes the
value of 4n + 1 when n is an integer?
 bisects ∠LMO. Which statement must
MN
be true?
The value is always negative.
m∠LMN = m∠OMN
The value is always positive.
m∠LMO = m∠OMN
The value is always even.
m∠LMN = m∠OML
The value is always odd.
m∠LMO = m∠ONM
Countdown to Testing
C7
Countdown to Testing
WEEK
5
DAY 1
Which statement is the converse of the conditional statement “If m∠A = 48°,
then ∠A is acute?”
If ∠A is not acute, then m∠A ≠ 48º.
If ∠A is acute, then m∠A = 48º.
If m∠A ≠ 48º, then ∠A is not acute.
If ∠A is not acute, then it must be obtuse.
DAY 2
DAY 3
Which of the following
statements is true,
based on the figure?
Let a represent “Three points are not
collinear,” and let b represent “The three
points lie in exactly one plane.” Which
symbolic sentence represents the statement
“If three points lie in exactly one plane,
then the three points are not collinear”?
∠2 and ∠4 are not
adjacent but
form a linear pair.
�
��
�
∠2 and ∠4 are
adjacent angles that
form a linear pair.
a→b
∠1 and ∠3 are adjacent angles and
form a linear pair.
∼b → ∼a
b→a
∼a → ∼b
∠1 and ∠3 are not adjacent angles
but form a linear pair.
DAY 4
DAY 5
The figure below shows a pattern of right
triangles and their areas, A. Based on the
pattern, what will be the area of a right
triangle with a height of 64 units?
How many pairs of
vertical angles are
in the diagram?
2
3
6
��
�
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�����
�
��
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�����
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4 square units
100 square units
364 square units
1536 square units
C8
12
�
�
Countdown to Testing
Geometry 2
2007 SE
Countdown to Testing
WEEK
6
DAY 1
DAY 2
A transversal crosses two parallel lines.
If two angles are on opposite sides of
the transversal and inside the two parallel
lines, then they are alternate interior
angles. If two angles are alternate interior
angles, then they are congruent. ∠1 and
∠2 are alternate interior angles.
Two angles are labeled in the figure below.
Which of the following statements best
describes this angle pair?
�
Which conclusion can be drawn from the
given information?
�
∠1 and ∠2 are parallel.
∠1 and ∠2 are alternate interior
angles.
∠1 and ∠2 are complementary.
They are complementary angles.
∠1 and ∠2 are congruent.
They are congruent angles.
They are supplementary angles.
They are parallel angles.
DAY 3
If line a is parallel to line b, and m∠8 = 62º, what is m∠1?
�
�
�
�
�
�
�
�
�
�
28°
62°
118°
180°
DAY 4
DAY 5
The area of a circle is about 7 cm 2. By how
many times will the area increase if the
radius of the circle is tripled?
B is in the interior of ∠AOC. Which of the
following statements must be true?
m∠AOB + m∠BOC = m∠AOC
1.5
m∠AOB = m∠BOC
3
m∠AOB + m∠AOC = m∠BOC
6
m∠BOC + m∠AOC = m∠AOB
9
Countdown to Testing
C9
Countdown to Testing
WEEK
7
DAY 1
Four rays are drawn from the origin to each of the following points: S(-2, 5),
T(0, 4), U(-1, -3), and V(2, 6). Which point is on the ray that forms an acute angle
with the ray in the figure?
�
�
�
��
�
�
��
S
T
U
V
DAY 2
DAY 3
What must be true if two nonvertical lines
are perpendicular?
Which line is parallel to y = 2x + 3?
y = 2x - 8
Their slopes add to 0.
y = 3x + 2
The product of their slopes is -1.
2y = -4x + 6
Their slopes are equal.
y = -2x + 3
Their y-intercepts are equal.
DAY 4
DAY 5
Which expression best represents the
perimeter of the rectangle?
Sheena is drawing a line graph to relate
the side length of a square to the area of
the square. Which of the following best
describes the graph?
�����
��
4x + 1
6x + 4
8x + 2
3x 2 + 3x
C10
Countdown to Testing
steep downward straight line
steep upward curve
horizontal line
upward straight line
Geometry 2
2007 SE
Countdown to Testing
WEEK
8
DAY 1
What is the slope of the given line segment?
�
-2
1
-_
2
�
1
�
2
��
�
�
��
DAY 2
DAY 3
Which two lines are perpendicular?
Which of the following is the best
classification for the given triangle?
y = -5x + 2 and 2y - 10x = 4
1 x + 1 and y = 4x + 2
y=_
4
y = 3x + 1 and y - 4x = 6
1 x + 2 and y + 2x = -4
y=_
2
Equilateral
Isosceles
Scalene
Right
DAY 4
DAY 5
̶̶
△SQT is an equilateral triangle. QR bisects
∠SQT. What are the measures of the
angles of △SQR?
What is the area of a circle with a radius
of 2y?
�
2πy
4πy
4πy 2
8πy 2
�
�
�
30°-30°-30°
30°-60°-90°
30°-60°-60°
60°-60°-60°
Countdown to Testing
C11
Countdown to Testing
WEEK
9
DAY 1
DAY 2
is perpendicular to XY
. If A(3, 5),
AB
B(9, 3), and X(-2, -5), what are the
coordinates of Y?
. If A(-4, 1)
and BC
∠ABC is formed by BA
and B(-4, 6), what coordinates for C will
result in an obtuse angle?
(6, 8)
(1, 4)
(3, 1)
(2, 5)
(-1, 2)
(-2, 9)
(0, 5)
(2, 6)
DAY 3
and AC
BD
CD
. Which of the following is true?
AB
£
Ó
Î
{
x
È
Ç
£Ó
£{
£x
£Î
££
n
£ä
™
∠1 is congruent to ∠5.
∠2 is supplementary to ∠14.
∠8 is congruent to ∠11.
∠6 is supplementary to ∠13.
DAY 4
DAY 5
What is the length of the given segment to
the nearest unit?
Which equation best represents the line in
the graph?
{
Þ
Þ
È
Ý
{
ä
{
1
5
8
10
C12
{
{
ä
Î
Ý
1x + 3
y=_
2
y = 3x + 1
y = -2x - 3
1x + 3
y = -_
2
Countdown to Testing
ge07se_FM_C04_C27.indd C12
5/25/06 4:31:05 PM
Countdown to Testing
WEEK
10
DAY 1
Which set of angle measures can be used to conclude that lines x and y are parallel?
Ý
£
Ó
Þ
Î
{
m∠1 = 87° and m∠3 = 93°
m∠1 = 82° and m∠4 = 98°
m∠1 = 80° and m∠2 = 100°
m∠3 = 88° and m∠4 = 92°
DAY 2
DAY 3
Which postulate or theorem can be used to
prove that these triangles are congruent?
Which of the following conjectures is false?
The product of an even number and
an odd number is even.
The difference of two negative
numbers is a positive number.
If x is negative, then -x is positive.
If x is even, then x + 1 is odd.
SAS
ASA
AAS
SSS
DAY 4
DAY 5
How many line segments can be
determined by four points, no three of
which are collinear?
Timothy sketches a sphere with a circle
around the middle. He labels the radius of
the circle, which is the same as the radius
of the sphere. Which problem might he be
trying to solve?
4
6
8
Determining the angle at which Earth
tilts
10
Calculating the mass of Earth
Measuring the surface area of Earth
Finding the distance around the
equator
Countdown to Testing
ge07se_FM_C04_C27.indd C13
C13
5/25/06 4:32:03 PM
Countdown to Testing
WEEK
11
DAY 1
DAY 2
What conclusion can you draw from the
figure?
Jan drew the figure below and claims that
line is parallel to line m. Which of the
following proves her statement true?
Ű
£xÊV“
££äÂ
££äÂ
ÇäÂ
“
ÇäÂ
££äÂ
ABC is isosceles.
The perimeter of ABC is
45 centimeters.
Angles on opposite sides of the
transversal are equal.
DE = 10 centimeters
Corresponding angles on the same
side of the transversal are congruent.
1 AB
DE = _
2
More than two angles in the diagram
have the same value.
Two straight lines pass through the
same transversal.
DAY 3
Which of the following can you use to prove that two angles are complementary?
The sum of their measures is 90°.
The sum of their measures is 180°.
The angles have the same measure.
The measure of one angle is twice the other measure.
DAY 4
DAY 5
If X(5, 5) and Y(0, 0), what are the
coordinates of Z so that m∠XYZ = 90°?
is a bisector of ∠XOY. Which of the
OZ
following statements is NOT true?
(5, -5)
(-5, -5)
(5, 0)
(0, 5)
C14
2m∠ZOY = m∠XOY
2m∠XOZ = m∠XOY
m∠ZOY = m∠XOY
1 m∠XOY
m∠XOZ = _
2
Countdown to Testing
ge07se_FM_C04_C27.indd C14
5/25/06 4:32:51 PM
Countdown to Testing
WEEK
12
DAY 1
DAY 2
Which of the following correctly completes
the congruence statement?
̶̶
AB ≅
?
̶̶̶̶̶
Based on the figure, which inequality is
correct?
�
�
�
�
������
�
̶̶
FD
̶̶
AF
̶̶
EF
̶̶
ED
��
2x > x + 10
2x < 10
x < 10
x>8
DAY 3
Roberta is attaching wooden trim around a stained glass window. The window is
made up of eight congruent isosceles triangles.
�����
�����
What length of trim does Roberta need in order to surround the entire window?
22 centimeters
78 centimeters
176 centimeters
624 centimeters
DAY 4
DAY 5
How many different segments can be
created from eight points on a given
segment (including the segment’s
endpoints)?
Which of these conditional statements is
true?
8
13
28
36
If two angles are vertical angles, then
they are congruent.
If two angles are congruent, then
they are right angles.
If four points are given, then they lie
in exactly one plane.
If one angle of a triangle measures
60º, then the triangle is a right
triangle.
Countdown to Testing
C15
Countdown to Testing
WEEK
13
DAY 1
DAY 2
What are the coordinates of point P?
Which postulate or theorem can be used
to verify the congruence of these two
triangles?
�
�
�
�
�
��
�
��
(3, -2)
SSS
(-2, 3)
ASA
(3, 2)
AAS
(2, -3)
SAS
DAY 3
Which conjecture is true?
If a figure is a rectangle, its perimeter is equal to its area.
If a figure is a triangle, all three sides are congruent.
If a figure is a quadrilateral, then it has four sides.
If a figure is a circle, its area is always greater than its circumference.
DAY 4
DAY 5
The layout of a swimming pool is plotted
on the coordinate grid below. If each unit
on the grid represents 2 meters, what is
the length of the pool?
△LMN is shown on the grid. What is the
̶̶̶
slope of MN?
�
�
�
�
�
�
��
�
�
1
-_
2
8 meters
10 meters
2
25 meters
C16
�
-4
1
_
2
5 meters
Countdown to Testing
�
�
�
�
Countdown to Testing
14
WEEK
DAY 1
A ceramic tile is in the shape of a 30°-60°-90° triangle. The side across from the 30°
angle is 6.25 centimeters long. How long is the hypotenuse of the tile?
3.125 centimeters
 centimeters
6.25 √3
12.5 centimeters
15 centimeters
DAY 2
DAY 3
What is the slope of this line?
Which equation should Aretha use to find
the distance between two points across
a river?
�
�
�
�
��
�
�
��
�
�
1
c = a2 + b2
1
_
3
3
c=a+b
c 2 = √
a+b
1
-_
3
c = √
a2 + b2
DAY 4
DAY 5
The sums of the angle measures of three
polygons are given. Based on the pattern,
what will be the sum of the measures of
a hexagon?
Which line in the graph is described by the
equation y = x + 2?
�
�
�
�
�
�
�
����
����
�
����
�
��
240°
420°
ℓ
600°
m
720°
n
o
Countdown to Testing
C17
Countdown to Testing
WEEK
15
DAY 1
DAY 2
Three coordinates of ABCD are A(4, 5),
C(7, 3), and D(1, 3). Which coordinates
could represent point B?
Which two lines are parallel?
1x = 3
y = 6x + 8 and y + _
6
1 x - 1 and y = 3x + 1
y=_
3
y - 2x = 2 and y = 2 - 2x
(1, 5)
(3, 7)
(5, 1)
(10, 5)
1 x and y - _
1x = 1
y=_
4
4
DAY 3
̶̶
What is the midpoint of QR?
(1, -2)
(-2, 1)
(1, 2)
(-1, -2)
�
�
�
�
��
�
�
�
��
DAY 4
DAY 5
Which of these statements is true?
Which expression describes the total
number of diagonals in a polygon with
n sides?
All quadrilaterals are parallelograms.
Every rectangle is a parallelogram.
Every parallelogram is also a
rectangle.
The diagonals of a rhombus are
congruent.
No. of sides
3
4
5
6
7
No. of
diagonals
0
2
5
9
14
n(n - 3)
_
2
2n
3n
_
2
2n + 6
_
3
C18
Countdown to Testing
Countdown to Testing
16
WEEK
DAY 1
DAY 2
The coordinates of the vertices of △ABC
are (1, 1), (6, 1) and (1, 8). Which of the
following could be the coordinates of the
vertices of a triangle congruent to △ABC?
Natalia is using indirect measurement to
find the distance across a pond. Which
Pythagorean triple is represented by the
triangle?
(-8, -2), (-3, -2), (-3, -9)
(4, 1), (6, 2), (8, 10)
(-2, 5), (-2, -9), (-8, 3)
(0, 0), (-1, 8), (5, 2)
����
����
3-4-5
����
5-12-13
8-15-17
7-24-25
DAY 3
Which of the following sets of measurements could represent the side lengths
of a right triangle?
3, 5, 9
4.5, 12, 8.5
6, 7, 10
2.5, 6, 6.5
DAY 4
DAY 5
What is the area of the composite figure?
What is the measure of ∠3 in the regular
hexagon?
���
���
�
���
�
���
���
�
���
8 square meters
30°
21 square meters
60°
25 square meters
90°
45 square meters
120°
Countdown to Testing
C19
Countdown to Testing
WEEK
17
DAY 1
Which two lines are perpendicular?
y = x + 6 and y = x - 6
3x - 4
2 x = 1 and y = _
y+_
3
2
1 x - 2 and y = -_
1x + 3
y=_
2
2
y - 2x = 5 and y = 2x + 2
DAY 2
DAY 3
What is the perimeter of the composite
figure to the nearest centimeter?
What is the measure of ∠1 in the triangle
below?
£ÓÊvÌ
£ÓÊV“
£
£ÓÊvÌ
30°
£äÊV“
xÊV“
xÊV“
Ó°xÊV“
45°
Ó°xÊV“
{ÊV“
60°
90°
xÊV“
44 centimeters
52 centimeters
60 centimeters
83 centimeters
DAY 4
DAY 5
What is the sixth item in the pattern
below?
The vertices of polygon ABCD are A(1, 5),
B(8, 5), C(8, 3), and D(1, 3). Which of the
following statements about this polygon
is true?
64, 32, 16, 8, …
0
1
_
2
2
4
C20
It is a square.
Its width is 2 units.
Its perimeter is 6 units.
Its area is 9 square units.
Countdown to Testing
ge07se_FM_C04_C27.indd C20
5/25/06 4:33:30 PM
Countdown to Testing
18
WEEK
DAY 1
Based on the pattern of similar triangles below, what is the value of x?
2
4

4 √3
�
8
�
�
���
�
�
���
���
���
������
������
DAY 2
DAY 3
Which ratio is equivalent to sin B?
What is the value of x to the nearest tenth
of a millimeter?
�
���
���
�
�
�
�
��
��
�
�
�
���
52.0 millimeters

2 √3
_
3
61.6 millimeters
√3

140.4 millimeters
78.8 millimeters
√3

_
2
1
_
2
DAY 4
DAY 5
What is the value of x in the regular
pentagon below?
Which conjecture about
polygons is NOT true?
����
The area of a parallelogram is the
product of its base and height.
A rhombus has four right angles.
A square has four congruent sides.
��
A trapezoid has exactly one pair of
parallel sides.
54°
90°
108°
180°
Countdown to Testing
C21
Countdown to Testing
WEEK
19
DAY 1
Which two line segments are congruent?
̶̶
̶̶
AB and DF
̶̶
̶̶̶
CE and GH
̶̶̶
̶̶
GH and AB
̶̶
̶̶
CD and DE
�
�
�
�
�
�
�
�
��
�
�
�
�
�
��
DAY 2
DAY 3
Based on the table, which algebraic
expression best represents the number
of triangles formed by drawing all of the
diagonals from one vertex in a polygon
with n sides?
At a certain time of the day, a 24-foot tree
casts an 18-foot shadow. How long is the
shadow cast by a 4-foot mailbox at the
same time of day?
No. of sides
3
4
5
8
No. of
triangles
formed
1
2
3
6
�����
����
�����
n
2n - 1
1.3 feet
n-2
3 feet
n+2
_
2
4.5 feet
5 feet
DAY 4
DAY 5
A school increases the width of its
rectangular playground from 25 meters to
40 meters and the length from 45 meters
to 60 meters. By how much does the
perimeter of the playground increase?
What is x?
�
60 meters
2
200 meters
5
225 meters
10
30
Countdown to Testing
� ���
���
30 meters
C22
���
Countdown to Testing
WEEK
20
DAY 1
The figure shows the measure of each interior angle for several regular polygons.
���
���
����
����
Which algebraic expression best represents the measure of an interior angle of a
regular polygon with n sides?
(n - 2)180
__
n
360n
_
n+2
(n - 2)180
180n
_
2
DAY 2
DAY 3
Which coordinates represent a vertex of
the hexagon?
The two triangles in the figure are similar.
̶̶̶
What is the length of MN?
�
�
�
��
�
�
�
��
��
(0, 2)
(4, -2)
(3, 2)
(-2, 2)
�
�
��
��
�
�
�
�
3.5
7
6
17.5
DAY 4
DAY 5
Two regular pentagons have perimeters of
30 and 75 respectively. What scale factor
relates the smaller figure to the larger
one?
Alissa is painting a diagonal line across a
square tile. What is the length of the line?
����
1 : 2.5
1:6
1 : 15
1 : 21
2 √
8 centimeters
6 centimeters
8 centimeters
8 √
2 centimeters
Countdown to Testing
C23
Countdown to Testing
21
WEEK
DAY 1
The table lists the measure of an exterior angle for the given regular polygon.
Which expression best represents the measure of an exterior angle of a regular
polygon with n sides?
Figure
Quadrilateral
Pentagon
Decagon
90°
72°
36°
Exterior angle
360
_
n-2
360 + n
_
2+n
360n
360
_
n
DAY 2
DAY 3
Carrie is building a skateboard ramp
with the dimensions below. What is the
approximate measure of x?
What is the value of z?
��
��
�����
����
�
�
4°
12
8°

12 √2
12°

12 √3
15°
17
DAY 4
DAY 5
Which equation best describes the line
containing the hypotenuse of this triangle?
The center of circle C is the midpoint
̶̶
of AB. What are the coordinates of the
midpoint?
�
�
�
�
�
��
�
�
�
��
�
1x + 3
y=_
2
y=5
y=x+3
1x - 3
y = -_
2
C24
Countdown to Testing
�
�
(0, 4)
(1, 4)
(2, 4)
(3, 3)
�
Countdown to Testing
WEEK
22
DAY 1
If this pattern is continued, how many shaded triangles will there be in the fourth
element of the pattern?
�
9
27
13
40
�
�
DAY 2
DAY 3
What is the slope of the line?
A delivery truck travels 13.5 mi east and
then 18 mi north. How far is the truck
from its starting point?
�
�
�
�
��
�
�
�����
��
1
-_
2
1
_
3
1
_
2
3
�������
4.5 miles
20.25 miles
22.5 miles
31.5 miles
DAY 4
DAY 5
What are the side lengths of the triangle?
An 18-foot ladder reaches the top of
a building when placed at an angle of
45° with the horizontal. What is the
approximate height of the building?
�
�
�
9.0 feet
12.7 feet
3, 4, and 5
14.4 feet
2, 3, and 5
30.9 feet
3, 3, and 3

3, 3, and 3 √2
Countdown to Testing
C25
Countdown to Testing
23
WEEK
DAY 1
DAY 2
RST is a 30°-60°-90° triangle. What is the
y-coordinate of R if a = -5 and c = -2?
What is x if y is 12.8 and z is 16 in the right
triangle below?
,­>]ÊÞ®
â
Þ
Ý
/­V]Êä®
-­>]Êä®
3.2
4.0
9.6
3
12.8
3 √2
3 √3
6
DAY 3
How does the slope of the hypotenuse of ABC compare with that of DBC?
They have the same value.
Þ
They have opposite signs.
x
They have the same sign.
They are reciprocals.
ä
{
Ý
{
Ó
DAY 4
DAY 5
How many sides does a regular polygon
have if each interior angle measures 120°?
An electrician is standing at the top of
a tower. He sees a truck at an angle of
depression of 3°. If the tower is 300 feet
tall, about how far away is the truck?
3
4
6
8
16 feet
300 feet
1052 feet
5724 feet
C26
Countdown to Testing
ge07se_FM_C04_C27.indd C26
5/25/06 4:34:52 PM
Countdown to Testing
WEEK
24
DAY 1
Quadrilaterals ABCD and WXYZ are similar. What is XY?
�
�
�
�
��
�
�
�
�
�
�
�
��
�
3.5
21
24.5
35
DAY 2
DAY 3
What is the second term in a proportion in
which the first, third, and fourth terms are
3, 9, and 12, respectively?
The endpoints of a segment are
Q(-2, 6) and R(5, -4). What is
the length of the segment to the
nearest tenth?
3
4
3.6 units
6
4.1 units
8
8.5 units
12.2 units
DAY 5
Which Pythagorean triple would be most
helpful in finding the value of a?
What is the perimeter of the square?
���
��
DAY 4
��
�
��
6
3-4-5
12
5-12-14
24
8-15-17
36
7-24-25
Countdown to Testing
C27
Foundations
for Geometry
KEYWORD: MG7 TOC
ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Euclidean and Construction Tools
1-1 Understanding Points, Lines, and Planes . . . . . . . . . . . . . . . . . . . 6
Explore Properties Associated with Points
. . . . . . . 12
1-2 Measuring and Constructing Segments . . . . . . . . . . . . . . . . . . . 13
1-3 Measuring and Constructing Angles . . . . . . . . . . . . . . . . . . . . . . 20
1-4 Pairs of Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Coordinate and Transformation Tools
1-5 Using Formulas in Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Connecting Geometry to Algebra:
Graphing in the Coordinate Plane . . . . . . . . . . . . . . . . . . . . . .
1-6 Midpoint and Distance in the Coordinate Plane . . . . . . . . . . .
1-7 Transformations in the Coordinate Plane. . . . . . . . . . . . . . . . . .
Explore Transformations
.........................
MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
42
43
50
56
58
59
Study Guide: Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Tools for Success
Test Prep Exercises 11, 19, 26, 33,
Reading Math 5
Writing Math 10, 18, 26, 33, 40, 48, 54
Vocabulary 3, 4, 9, 17, 24, 31, 38, 47,
53, 60
Know-It Notes 6, 7, 8, 13, 14, 16, 20, 21,
22, 24, 28, 29, 31, 36, 37, 43, 44, 45, 46,
50, 52
Graphic Organizers 8, 16, 24, 31, 37,
46, 52
Homework Help Online 9, 17, 24, 31,
38, 47, 53
ge07se_FM_vi.indd vi
40–41, 49, 55
Multi-Step Test Prep 10, 18, 26, 32,
34, 39, 48, 54, 58
College Entrance Exam Practice 65
Test Tackler 66
Standardized Test Prep 68
5/25/06 4:21:55 PM
Geometric Reasoning
ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Inductive and Deductive Reasoning
2-1 Using Inductive Reasoning to Make Conjectures . . . . . . . . . . . 74
Connecting Geometry to Number Theory:
Venn Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2-2 Conditional Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2-3 Using Deductive Reasoning to Verify Conjectures . . . . . . . . . . 88
Solve Logic Puzzles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
2-4 Biconditional Statements and Definitions . . . . . . . . . . . . . . . . . 96
MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
KEYWORD: MG7 TOC
Table of Contents
Mathematical Proof
2-5 Algebraic Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-6 Geometric Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Design Plans for Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-7 Flowchart and Paragraph Proofs . . . . . . . . . . . . . . . . . . . . . . . .
MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
EXT Introduction to Symbolic Logic . . . . . . . . . . . . . . . . . . . . . . . . . .
104
110
117
118
126
127
128
Study Guide: Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Problem Solving on Location: South Carolina . . . . . . . . . . . . . . . 140
Tools for Success
Test Prep Exercises 79, 86, 93, 101,
Reading Math 73
Writing Math 78, 81, 86, 92, 96, 100,
109, 111, 115, 125
Vocabulary 71, 72, 77, 84, 91, 99, 107,
113, 122, 130
Know-It Notes 75, 76, 81, 83, 84, 89,
90, 98, 104, 106, 107, 110, 111, 112, 113,
118, 120, 122, 128
Graphic Organizers 76, 84, 90, 98, 107,
113, 122
Homework Help Online 77, 84, 91, 99,
107, 113, 122
109, 116, 125
Multi-Step Test Prep 78, 85, 92, 100,
102, 109, 115, 124, 126
College Entrance Exam Practice 135
Test Tackler 136
Standardized Test Prep 138
Parallel and
Perpendicular Lines
ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Lines with Transversals
KEYWORD: MG7 TOC
3-1 Lines and Angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Connecting Geometry to Algebra:
Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Explore Parallel Lines and Transversals
.........
3-2 Angles Formed by Parallel Lines and Transversals . . . . . . . . .
3-3 Proving Lines Parallel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Construct Parallel Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-4 Perpendicular Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Construct Perpendicular Lines. . . . . . . . . . . . . . . . . . . . . . . .
MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
146
152
154
155
162
170
172
179
180
181
Coordinate Geometry
3-5 Slopes of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Explore Parallel and Perpendicular Lines
.........
3-6 Lines in the Coordinate Plane . . . . . . . . . . . . . . . . . . . . . . . . . . .
Connecting Geometry to Data Analysis:
Scatter Plots and Lines of Best Fit . . . . . . . . . . . . . . . . . . . . .
MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Study Guide: Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
182
188
190
198
200
201
144
145
202
206
Tools for Success
Writing Math 150, 160, 168, 177,
186, 196
Vocabulary 143, 144, 148, 175, 185,
194, 202
Study Strategy 145
Test Prep Exercises 150–151,
Know-It Notes 146, 147, 148, 155, 156,
160–161, 168–169, 177–178, 187,
196–197
157, 162, 163, 173, 174, 182, 184, 185,
190, 192, 193
Multi-Step Test Prep 150, 160, 168,
Graphic Organizers 148, 157, 165, 174,
185, 193
Homework Help Online 148, 158, 166,
175, 185, 194
176, 180, 186, 196, 200
College Entrance Exam Practice 207
Test Tackler 208
Standardized Test Prep 210
©2007 Artists Rights Society (ARS),
New York/ADAGP, Paris
ge07se_FM_vii_xii.indd viii
5/25/06 4:37:16 PM
Triangle Congruence
ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
Triangles and Congruence
4-1 Classifying Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Develop the Triangle Sum Theorem . . . . . . . . . . . . . . . . . .
4-2 Angle Relationships in Triangles . . . . . . . . . . . . . . . . . . . . . . . .
4-3 Congruent Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
216
KEYWORD: MG7 TOC
222
223
231
238
239
Proving Triangle Congruence
4-4
4-5
4-6
4-7
4-8
EXT
Explore SSS and SAS Triangle Congruence . . . . . . . . . . .
Triangle Congruence: SSS and SAS . . . . . . . . . . . . . . . . . . . . . .
Predict Other Triangle Congruence Relationships . . . .
Triangle Congruence: ASA, AAS, and HL . . . . . . . . . . . . . . . . .
Triangle Congruence: CPCTC . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Connecting Geometry to Algebra:
Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction to Coordinate Proof . . . . . . . . . . . . . . . . . . . . . . . .
Isosceles and Equilateral Triangles . . . . . . . . . . . . . . . . . . . . . . .
MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Proving Constructions Valid . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
240
Study Guide: Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problem Solving on Location: Michigan . . . . . . . . . . . . . . . . . . . .
214
215
284
288
294
242
250
252
260
266
267
273
280
281
282
Tools for Success
Reading Math 215, 273
Writing Math 220, 229, 236, 248, 258,
264, 271, 278
Vocabulary 213, 214, 219, 227, 234,
245, 256, 262, 270, 276, 284
Know-It Notes 216, 217, 218, 223, 224,
225, 226, 231, 233, 242, 243, 245, 252,
254, 255, 262, 267, 269, 273, 274, 275,
276
Graphic Organizers 218, 226, 233, 245,
255, 262, 269, 276
Homework Help Online 219, 227, 234,
245, 256, 262, 270, 276
Test Prep Exercises 221, 230, 236,
248, 258–259, 264–265, 272, 279
Multi-Step Test Prep 220, 229, 236,
238, 247, 258, 264, 271, 278, 280
College Entrance Exam Practice 289
Test Tackler 290
Standardized Test Prep 292
Properties and
Attributes of Triangles
ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
Segments in Triangles
KEYWORD: MG7 TOC
5-1 Perpendicular and Angle Bisectors . . . . . . . . . . . . . . . . . . . . . .
5-2 Bisectors of Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-3 Medians and Altitudes of Triangles . . . . . . . . . . . . . . . . . . . . . .
Special Points in Triangles
......................
5-4 The Triangle Midsegment Theorem . . . . . . . . . . . . . . . . . . . . . .
MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
300
307
314
321
322
328
329
Relationships in Triangles
5-5
5-6
5-7
5-8
Connecting Geometry to Algebra:
Solving Compound Inequalities . . . . . . . . . . . . . . . . . . . . . . .
Explore Triangle Inequalities . . . . . . . . . . . . . . . . . . . . . . . . .
Indirect Proof and Inequalities in One Triangle . . . . . . . . . . .
Inequalities in Two Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Connecting Geometry to Algebra:
Simplest Radical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hands-on Proof of the Pythagorean Theorem . . . . . . . .
The Pythagorean Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Applying Special Right Triangles . . . . . . . . . . . . . . . . . . . . . . . .
Graph Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Study Guide: Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
330
331
332
340
346
347
348
356
363
364
365
298
299
366
370
Tools for Success
Test Prep Exercises 306, 313, 319,
Reading Math 299, 300
Writing Math 306, 313, 318, 325, 338,
344, 354, 361
Vocabulary 297, 298, 304, 311, 317, 324,
336, 352, 366
Know-It Notes 300, 301, 303, 307, 309,
310, 314, 317, 323, 324, 333, 334, 335,
340, 342, 350, 351, 352, 356, 358, 359
Graphic Organizers 303, 310, 317, 324,
335, 342, 352, 359
Homework Help Online 304, 311, 317,
324, 336, 343, 352, 360
326, 339, 345, 355, 362
Multi-Step Test Prep 305, 312, 319,
326, 328, 338, 344, 354, 361, 364
College Entrance Exam Practice 371
Test Tackler 372
Standardized Test Prep 374
Polygons and
Quadrilaterals
ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
Polygons and Parallelograms
Construct Regular Polygons . . . . . . . . . . . . . . . . . . . . . . . . . .
6-1 Properties and Attributes of Polygons . . . . . . . . . . . . . . . . . . .
Connecting Geometry to Algebra:
Relations and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Explore Properties of Parallelograms . . . . . . . . . . . . . . . .
6-2 Properties of Parallelograms . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-3 Conditions for Parallelograms. . . . . . . . . . . . . . . . . . . . . . . . . . .
MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
380
KEYWORD: MG7 TOC
382
389
390
391
398
406
407
Other Special Quadrilaterals
6-4 Properties of Special Parallelograms . . . . . . . . . . . . . . . . . . . . .
Predict Conditions for Special Parallelograms
...
6-5 Conditions for Special Parallelograms . . . . . . . . . . . . . . . . . . .
Explore Isosceles Trapezoids
....................
6-6 Properties of Kites and Trapezoids . . . . . . . . . . . . . . . . . . . . . .
MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
426
Study Guide: Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problem Solving on Location: Ohio . . . . . . . . . . . . . . . . . . . . . . . .
378
379
438
442
448
408
416
418
427
436
437
Tools for Success
Test Prep Exercises 388, 397, 405,
Writing Math 379, 388, 397, 404, 405,
414, 424, 434
Vocabulary 377, 378, 386, 395, 412, 432,
438
Know-It Notes 383, 384, 385, 391, 392,
394, 398, 399, 401, 408, 409, 411, 418,
419, 421, 427, 429, 431
Graphic Organizers 385, 394, 401, 411,
421, 431
Homework Help Online 386, 395, 402,
412, 422, 432
ge07se_FM_vii_xii.indd xi
414–415, 425, 434–435
Multi-Step Test Prep 387, 396, 404,
406, 414, 424, 434, 436
College Entrance Exam Practice 443
Test Tackler 444
Standardized Test Prep 446
5/25/06 4:37:59 PM
Similarity
KEYWORD: MG7 TOC
ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
Similarity Relationships
7-1 Ratio and Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Explore the Golden Ratio
.......................
7-2 Ratios in Similar Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Predict Triangle Similarity Relationships
........
7-3 Triangle Similarity: AA, SSS, and SAS . . . . . . . . . . . . . . . . . . . .
MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
454
460
462
468
470
478
479
Applying Similarity
Investigate Angle Bisectors of a Triangle
........
7-4 Applying Properties of Similar Triangles . . . . . . . . . . . . . . . . .
7-5 Using Proportional Relationships . . . . . . . . . . . . . . . . . . . . . . . .
7-6 Dilations and Similarity in the Coordinate Plane . . . . . . . . . .
Connecting Geometry to Algebra:
Direct Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
480
Study Guide: Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
452
453
504
508
481
488
495
501
502
503
Tools for Success
Test Prep Exercises 459, 467, 477,
Reading Math 453, 455, 456
Writing Math 459, 463, 466, 476, 486,
493, 499
Vocabulary 451, 452, 457, 465, 491, 498,
504
Know-It Notes 455, 457, 462, 464, 470,
471, 473, 481, 482, 483, 484, 490, 497
Graphic Organizers 457, 464, 473, 484,
490, 497
Homework Help Online 457, 465, 474,
484, 491, 498
487, 493, 500
Multi-Step Test Prep 458, 466, 476,
478, 486, 492, 499, 502
College Entrance Exam Practice 509
Test Tackler 510
Standardized Test Prep 512
Right Triangles and
Trigonometry
ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
Trigonometric Ratios
8-1 Similarity in Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Explore Trigonometric Ratios
...................
8-2 Trigonometric Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Connecting Geometry to Algebra:
Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8-3 Solving Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
518
KEYWORD: MG7 TOC
524
525
533
534
542
543
Applying Trigonometric Ratios
8-4 Angles of Elevation and Depression . . . . . . . . . . . . . . . . . . . . .
Indirect Measurement Using Trigonometry . . . . . . . . . .
8-5 Law of Sines and Law of Cosines . . . . . . . . . . . . . . . . . . . . . . .
8-6 Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
EXT Trigonometry and the Unit Circle . . . . . . . . . . . . . . . . . . . . . . . .
544
Study Guide: Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problem Solving on Location: Illinois . . . . . . . . . . . . . . . . . . . . . .
516
517
572
576
582
550
551
559
568
569
570
Tools for Success
Test Prep Exercises 523, 532, 540,
Reading Math 517, 534, 570
Writing Math 523, 525, 531, 540, 548,
557, 566, 571
Vocabulary 515, 516, 521, 529, 547, 563,
572
Know-It Notes 518, 519, 520, 525, 528,
537, 546, 552, 553, 554, 561, 563
Graphic Organizers 520, 528, 537, 546,
554, 563
Homework Help Online 521, 529, 537,
547, 555, 563
549, 558, 567
Multi-Step Test Prep 522, 530, 539,
542, 548, 557, 565, 568
College Entrance Exam Practice 577
Test Tackler 578
Standardized Test Prep 580
Extending Perimeter,
Circumference, and Area
ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585
Developing Geometric Formulas
KEYWORD: MG7 TOC
Connecting Geometry to Algebra:
Literal Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9-1 Developing Formulas for Triangles and Quadrilaterals . . . .
Develop π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9-2 Developing Formulas for Circles and Regular Polygons . . . .
9-3 Composite Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Develop Pick’s Theorem for Area of
Lattice Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
588
589
598
600
606
613
614
615
Applying Geometric Formulas
9-4 Perimeter and Area in the Coordinate Plane . . . . . . . . . . . . . .
9-5 Effects of Changing Dimensions Proportionally . . . . . . . . . .
Connecting Geometry to Probability:
Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9-6 Geometric Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Use Geometric Probability to Estimate π . . . . . . . . . . . . .
MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
616
Study Guide: Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
586
587
640
644
622
628
630
637
638
639
Tools for Success
Test Prep Exercises 596–597, 605,
Writing Math 596, 605, 611, 620,
626, 635
Vocabulary 585, 586, 603, 609, 633, 640
Study Strategy 587
Know-It Notes 589, 590, 591, 593, 600,
601, 602, 608, 619, 623, 624, 630, 633
Graphic Organizers 593, 602, 608, 619,
624, 633
Homework Help Online 593, 603, 609,
619, 625, 633
611–612, 621, 627, 636
Multi-Step Test Prep 595, 604, 610,
614, 620, 626, 635, 638
College Entrance Exam Practice 645
Test Tackler 646
Standardized Test Prep 648
Spatial Reasoning
ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651
Three-Dimensional Figures
10-1 Solid Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10-2 Representations of Three-Dimensional Figures . . . . . . . . . . .
Use Nets to Create Polyhedrons . . . . . . . . . . . . . . . . . . . . . .
10-3 Formulas in Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . .
MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
654
KEYWORD: MG7 TOC
661
669
670
678
679
Surface Area and Volume
10-4 Surface Area of Prisms and Cylinders . . . . . . . . . . . . . . . . . . . .
Model Right and Oblique Cylinders . . . . . . . . . . . . . . . . . .
10-5 Surface Area of Pyramids and Cones . . . . . . . . . . . . . . . . . . . .
10-6 Volume of Prisms and Cylinders . . . . . . . . . . . . . . . . . . . . . . . . .
10-7 Volume of Pyramids and Cones . . . . . . . . . . . . . . . . . . . . . . . . .
Connecting Geometry to Algebra:
Functional Relationships in Formulas . . . . . . . . . . . . . . . . . .
10-8 Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Compare Surface Areas and Volumes
...........
MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
EXT Spherical Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
680
Study Guide: Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problem Solving on Location: Pennsylvania . . . . . . . . . . . . . . . . .
652
653
730
734
740
688
689
697
705
713
714
722
724
725
726
Tools for Success
Writing Math 653, 659, 667, 676, 686,
695, 703, 711, 720
Vocabulary 651, 652, 657, 665, 674, 684,
693, 701, 709, 718, 730
Know-It Notes 654, 656, 664, 670, 671,
672, 673, 680, 681, 683, 689, 690, 692,
697, 699, 700, 705, 707, 708, 714, 716,
717, 726, 727
Graphic Organizers 656, 664, 673, 683,
692, 700, 708, 717
Homework Help Online 657, 665, 674,
684, 693, 701, 709, 718
ge07se_FM_xiii_xvii.indd xv
Test Prep Exercises 659, 667, 677,
687, 695, 703–704, 712, 721
Multi-Step Test Prep 658, 666, 675,
678, 686, 695, 703, 711, 720, 724
College Entrance Exam Practice 735
Test Tackler 736
Standardized Test Prep 738
5/25/06 4:41:26 PM
Circles
ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743
Lines and Arcs in Circles
KEYWORD: MG7 TOC
11-1 Lines That Intersect Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Connecting Geometry to Data Analysis:
Circle Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11-2 Arcs and Chords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11-3 Sector Area and Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
746
755
756
764
770
771
Angles and Segments in Circles
11-4 Inscribed Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Explore Angle Relationships in Circles
..........
11-5 Angle Relationships in Circles . . . . . . . . . . . . . . . . . . . . . . . . . . .
Explore Segment Relationships in Circles
.......
11-6 Segment Relationships in Circles . . . . . . . . . . . . . . . . . . . . . . . .
11-7 Circles in the Coordinate Plane . . . . . . . . . . . . . . . . . . . . . . . . . .
MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
EXT Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
772
Study Guide: Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
744
745
810
814
780
782
790
792
799
806
807
808
Tools for Success
Reading Math 745, 748
Writing Math 754, 756, 762, 769, 778,
788, 797, 804
Vocabulary 743, 744, 751, 760, 767,
776, 810
Know-It Notes 746, 747, 748, 749, 750,
756, 757, 759, 764, 765, 766, 772, 773,
774, 775, 782, 783, 784, 785, 786, 792,
793, 794, 795, 799, 801
Graphic Organizers 750, 759, 766, 775,
786, 795, 801
Homework Help Online 751, 760, 767,
776, 786, 795, 802
Test Prep Exercises 754, 763, 769,
778, 789, 798, 804
Multi-Step Test Prep 753, 762, 768,
770, 777, 788, 797, 803, 806
College Entrance Exam Practice 815
Test Tackler 816
Standardized Test Prep 818
Extending
Transformational
Geometry
ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821
Congruence Transformations
12-1 Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12-2 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Connecting Geometry to Algebra:
Transformations of Functions . . . . . . . . . . . . . . . . . . . . . . . . .
12-3 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Explore Transformations with Matrices . . . . . . . . . . . .
12-4 Compositions of Transformations . . . . . . . . . . . . . . . . . . . . . . .
MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
KEYWORD: MG7 TOC
824
831
838
839
846
848
854
855
Patterns
12-5 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12-6 Tessellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Use Transformations to Extend Tessellations . . . . . . . . .
12-7 Dilations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
EXT Using Patterns to Generate Fractals . . . . . . . . . . . . . . . . . . . . .
856
Study Guide: Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problem Solving on Location: New Jersey . . . . . . . . . . . . . . . . . .
822
823
884
888
894
863
870
872
880
881
882
Tools for Success
Writing Math 829, 836, 844, 852, 861,
868, 878, 883
Vocabulary 821, 822, 827, 851, 859,
866, 875, 884
Study Strategy 823
Know-It Notes 825, 826, 832, 833, 840,
841, 848, 849, 850, 856, 857, 858, 866,
873, 874
Graphic Organizers 826, 833, 841, 850,
858, 866, 874
Homework Help Online 827, 834, 842,
851, 859, 866, 875
Test Prep Exercises 829–830,
836–837, 845, 853, 862, 869, 878
Multi-Step Test Prep 829, 835, 843,
853, 854, 861, 868, 876, 880
College Entrance Exam Practice 889
Test Tackler 890
Standardized Test Prep 892
WHO USES MATHEMATICS?
The Career Path features are a set of interviews with young adults who are
either preparing for or just beginning in different career fields. These people
share what math courses they studied in high school, how math is used in
their field, and what options the future holds. Also, many exercises throughout
the book highlight skills used in various career fields.
KEYWORD: MG7 Career
Career Applications
Advertising 499
Agriculture 765
Animation 53, 835, 842
Anthropology 802
Archaeology 262, 787, 793
Architecture 47, 467, 667
Art 483, 657, 873
Aviation 277, 546, 564
Business 108, 194, 625
Carpentry 168, 418, 836
City Planning 305, 827
Communication 634, 802
Computer Graphics 495
Design 311, 317, 318
Electronics 692
Engineering 260, 554
Finance 108, 522
Forestry 548
Graphic Design 498, 752
Health 343
Industry 344
Interior Decorating 609, 867
Landscaping 607, 686, 702
Manufacturing 38, 754
Marine Biology 698, 720
Mechanics 434
Meteorology 801
Music 24
Navigation 228, 567, 729
Nutrition 107
Oceanography 174
Optometry 877
Photography 385, 459, 475
Political Science 79, 93
Real Estate 486
Surveying 25, 263, 556
xviii
Who Uses Mathematics?
ELECTRICIAN p. 320
Electricians install and
maintain the systems that
provide many of the modernday comforts we rely on, such
as climate control, lighting,
and technology. Look on
page 320 to find out how
Alex Peralta got started on
this career path.
TECHNICAL WRITER p. 612
Have you ever wondered who
writes manuals for operating
televisions or stereos? A technical
writer not only writes manuals
for operating electronics, but
also documents maintenance
procedures for airplanes. Look
at the Career Path on page 612
to find out how to become a
technical writer.
FURNITURE MAKER p. 805
A furniture maker must take
precise measurements and be
aware of spatial relationships
in order to build a quality
finished product. The Career
Path on page 805 describes
the kind of experience
needed to be successful as
a furniture maker.
WHY LEARN MATHEMATICS?
Find a counterexample to show that the converse of each conditional is false.
38. If x = -5, then x 2 = 25.
Links to interesting topics may
accompany real-world applications
in the examples or exercises.
These links help you see how math
is used in the real world. For a
complete list of all applications
in Holt Geometry, see page S162
in the Index.
39. If two angles are vertical angles, then they are congruent.
40. If two angles are adjacent, then they share a vertex.
41. If you use sunscreen, then you will not get sunburned.
Geology
Use the table and the statements below for Exercises 42–47.
Write each conditional and find its truth value.
p: calcite
Animation
q: not apatite
r: a hardness of 3
Diamond is four times
as hard as the next
mineral on Mohs’ scale,
corundum (ruby and
sapphire).
Real-World
Animation 835
Geology Mohs’ scale is used to identify minerals.
A mineral with a higher number is harder than a
mineral with a lower number.
s: a hardness less than 5
42. p → r
43. s → q
44. q → s
45. q → p
46. r → q
47. p → s
48. Critical Thinking Consider the conditional
“If two angles are congruent, then they have the same
measure.” Write the converse, inverse, and contrapositive
and find the truth value of each. Use the related conditionals
to draw a Venn diagram that represents the relationship
between congruent angles and their measures.
Mohs’ Scale
Hardness
Mineral
1
2
3
4
5
6
7
8
9
10
Talc
Gypsum
Calcite
Fluorite
Apatite
Orthoclase
Quartz
Topaz
Corundum
Diamond
49. Write About It When is a conditional statement false? Explain why a true
conditional statement can have a hypothesis that is false.
Food 195
Navigation
278
50. What is the inverse of “If
it is Saturday, then it
is the weekend”?
If it is the weekend, then it is Saturday.
Geography 626
Recreation
92
If it is not Saturday, then it is the weekend.
If it is not Saturday,
then it is not the weekend.
Geology 86, 804
Shuffleboard
305
If it is not the weekend, then it is not Saturday.
History 48, 413, 531, 566, 595
Space Shuttle 548
51. Let a represent “Two lines are parallel to the same line,” and let b represent
Marine Biology 720 “The two lines are parallel.” Which symbolic statement represents the conditional
“If two lines are NOT parallel, then they are parallel to the same line”?
b → ∼a
∼b → a
b→a
Math History 41, 78, 257, a → b
52. Which statement is a counterexample for the conditional statement
318, 493, 611, 703, 768
���
“If f(x) = √25
- x , then f(x) is positive”?
Shuttle
Measurement 404
x=5
x=4
xSpace
=3
x=0
53. Which statement has the same truth value as its converse?
Mechanics 434
If a triangle has a right angle, its side lengths are 3 centimeters, 4 centimeters,
Meteorology 476, 675, 797 and 5 centimeters.
If an angle measures 104°, then the angle is obtuse.
Monument 466
If a number is an integer, then it is a natural number.
2
If an angle measures 90°, then it is an acute angle.
Each frame of a
computer-animated
1
feature represents __
24
of a second of film.
86
Chapter 2 Geometric Reasoning
Monument
Source: www.pixar.com
Architecture 159, 220, 695
Art 876
Astronomy 752
Bicycles 337
Biology 100, 604
Chemistry 828
Conservation 271
Design 313
Ecology 248
Electronics 692
Engineering 115
Entertainment 149
Fitness 539
During its launch, a
space shuttle accelerates
to more than 27,359
km/h in just over 8
minutes. So the shuttle
travels 3219 km/h faster
each minute.
The height of the
Statue of Liberty from
the foundation of the
pedestal to the torch
is 305 ft. Her index
finger measures 8 ft,
and the fingernail is
13 in. by 10 in.
Sports 19, 635
Surveying 353, 556
Travel 458
Source:
libertystatepark.org
Why Learn Mathematics
xix
HOW
TO
STUDY GEOMETRY
This book has many features designed to help you learn and study
effectively. Becoming familiar with these features will prepare you
for greater success on your exams.
Learn
Triangle Congruence:
SSS and SAS
4-4
The vocabulary is
listed at the beginning
of every lesson.
Who uses this?
Engineers used the property of
triangle rigidity to design the
internal support for the Statue
of Liberty and to build bridges,
towers, and other structures.
(See Example 2.)
Objectives
Apply SSS and SAS to
construct triangles and
to solve problems.
Prove triangles congruent
by using SSS and SAS.
Vocabulary
triangle rigidity
included angle
Study the examples to
apply new concepts and
skills. Examples include
stepped out solutions.
In Lesson 4-3, you proved triangles congruent
by showing that all six pairs of corresponding
parts were congruent.
The property of triangle rigidity gives you a shortcut
for proving two triangles congruent. It states that if
the side lengths of a triangle are given, the triangle
can have only one shape.
For example, you only need to know that two triangles
have three pairs of congruent corresponding sides.
This can be expressed as the following postulate.
Postulate 4-4-1
Look for the Know-It-Note
icons to identify important
information.
Side-Side-Side (SSS) Congruence
POSTULATE
HYPOTHESIS
�
If three sides of one
triangle are congruent
to three sides of
another triangle, then
the triangles
are congruent.
1
EXAMPLE
�
CONCLUSION
�
����
����
�ABC � �FDE
�
����
�
����
����
����
�
Using SSS to Prove Triangle Congruence
Use SSS to explain why �PQR � �PSR.
−− −−
−− −−
It is given that PQ � PS and that QR � SR. By
−− −−
the Reflexive Property of Congruence, PR � PR.
Therefore �PQR � �PSR by SSS.
Adjacent triangles
share a side, so
you can apply the
Reflexive Property
to get a pair of
congruent parts.
�
�
�
�
1. Use SSS to explain why
�ABC � �CDA.
�
�
�
�
An included angle is an angle formed by two
adjacent sides of a polygon. ∠B is the included
−−
−−
angle between sides AB and BC.
Practice
242
�
�
�
1. Describe three ways you could
prove that �ABC � �DEF.
DEF
DEF.
2. Explain why the SSS and SAS
Postulates are shortcuts for
proving triangles congruent.
�
�
3. GET ORGANIZED Copy and
complete the graphic organizer.
Use it to compare the SSS and
SAS postulates.
4-4
�
Chapter 4 Triangle Congruence
THINK AND DISCUSS
Use a graphic organizer
to summarize each lesson.
Test your understanding
of examples by trying the
Check It Out problems.
Check your work in the
Selected Answers.
�
�
�
���
���
�������������������
�����������������������
Exercises
KEYWORD: MG7 4-4
KEYWORD: MG7 Parent
GUIDED PRACTICE
Refer to the examples
from the lesson to
solve the Guided
Practice exercises.
−−
−−
1. Vocabulary In �RST which angle is the included angle of sides ST and TR?
SEE EXAMPLE
1
p. 242
If you get stuck, use the
internet for Homework
Help Online.
Use SSS to explain why the triangles in each pair are congruent.
2. �ABD � �CDB
3. �MNP � �MQP
�
�
�
�
�
�
�
SEE EXAMPLE
2
p. 243
�
H
J
4. Sailing Signal flags are used to communicate
messages when radio silence is required.
The Zulu signal flag means, “I require a tug.”
GJ = GH = GL = GK = 20 in. Use SAS to
explain why �JGK � �LGH.
G
L
K
SEE EXAMPLE
p. 244
3
Show that the triangles are congruent for the given value of the variable.
5. �GHJ � �IHJ, x = 4
�
�
6. �RST � �TUR, x = 18
�
��
�
������
�
���
�
�
���
�������
�
������
�
�
Review
4-4 Triangle Congruence: SSS and SAS
245
For a complete
list of the
postulates and
theorems in
this chapter,
see p. S82.
Vocabulary
Study and review
vocabulary from
the entire chapter.
acute triangle . . . . . . . . . . . . . . 216
CPCTC . . . . . . . . . . . . . . . . . . . . . 260
isosceles triangle . . . . . . . . . . . 217
auxiliary line . . . . . . . . . . . . . . . 223
equiangular triangle . . . . . . . . 216
legs of an isosceles triangle . . 273
base . . . . . . . . . . . . . . . . . . . . . . . 273
equilateral triangle . . . . . . . . . 217
obtuse triangle . . . . . . . . . . . . . 216
base angle . . . . . . . . . . . . . . . . . . 273
exterior . . . . . . . . . . . . . . . . . . . . 225
remote interior angle . . . . . . . 225
congruent polygons . . . . . . . . . 231
exterior angle . . . . . . . . . . . . . . 225
right triangle . . . . . . . . . . . . . . . 216
coordinate proof . . . . . . . . . . . . 267
included angle. . . . . . . . . . . . . . 242
scalene triangle . . . . . . . . . . . . . 217
corollary . . . . . . . . . . . . . . . . . . . 224
included side . . . . . . . . . . . . . . . 252
triangle rigidity . . . . . . . . . . . . . 242
corresponding angles . . . . . . . 231
interior . . . . . . . . . . . . . . . . . . . . 225
vertex angle . . . . . . . . . . . . . . . . 273
corresponding sides. . . . . . . . . 231
interior angle . . . . . . . . . . . . . . . 225
Complete the sentences below with vocabulary words from the list above.
Use the list on
p. S82 to review the
postulates and
theorems found in
the chapter.
1. A(n) ? is a triangle with at least two congruent sides.
−−−−
2. A name given to matching angles of congruent triangles is
3. A(n)
? .
−−−−
? is the common side of two consecutive angles in a polygon.
−−−−
4-1 Classifying Triangles (pp. 216–221)
EXERCISES
EXAMPLE
�
Classify the triangle by its angle measures
and side lengths.
isosceles right triangle
Classify each triangle by its angle measures and
side lengths.
4.
5.
���
���
���
����
4-2 Angle Relationships in Triangles (pp. 223–230)
EXERCISES
EXAMPLE
�
Find m∠S.
12x = 3x + 42 + 6x
12x = 9x + 42
���������� �
����
�
���
�
Find m∠N.
�
6.
��
3x = 42
x = 14
m∠S = 6 (14) = 84°
�
��
����
�
7. In�LMN, m∠L = 8x °, m∠M = (2x + 1)°, and
m∠N = (6x - 1)°.
284
xx
How To Study Geometry
Chapter 4 Triangle Congruence
Test yourself with
practice problems
from every lesson
in the chapter.
TOOLS
GEOMETRY
OF
In geometry, it is important to use tools correctly in order to measure
accurately and produce accurate figures. One important tool is your pencil.
Always use a sharp pencil with a good eraser.
Ruler
Protractor
The ruler shown has a mark every __18 inch, so the
accuracy is to the nearest __18 inch.
�
�
�
�
�
Line up one
end with 0,
not the edge.
�
�
�
�
To use a protractor to
measure an angle, you
may need to extend
the sides of the angle.
�
�
�
�
�
��
��
��
For acute angles, use the
smaller measurement.
For obtuse angles, use
the larger measurement.
��
�
��
��
Straightedge
A compass is used to draw arcs and circles. If
you have trouble keeping the point in place, try
keeping the compass still and turning the paper.
A straightedge is used to draw a line through
two points. If you use a ruler as a straightedge,
do not use the marks on the ruler.
Tilt the
compass
slightly.
�
��
�
Place the center of your
protractor on the vertex.
Compass
Keep your wrist
flexible. Turn the
compass with
your index finger
and thumb.
��
��
Line up one
� ray with 0.�
�
Choose the
measurement
that is the closest.
��
��
First place
your pencil
on one
points.
Place the straightedge
against your pencil
and the other point.
Draw the line.
�
�
�
�
�
�
�
�
�
�
�
�
��
��
��
��
�
�
��
��
��
��
��
�
�
��
��
�
�
�
Geometry Software
Geometry software can be used to create figures and explore their properties.
Use the toolbar
to select, draw;
and label figures.
Drag points
to explore
properties
of a figure.
Use the menus to
construct, transform,
and measure figures.
The parts of each figure are
linked. To avoid deleting
a whole figure, hide parts
instead of deleting them.
Tools of Geometry
xxi
Scavenger H
Use this scavenger hunt to discover a few of the
many tools in Holt Geometry that you can use to
become an independent learner. On a separate sheet of paper, write
the answers to each question below. Within each answer, one letter
will be in a yellow box. After you have answered every question,
identify the letters that would be in yellow boxes and rearrange them
to reveal the answer to the question at the bottom of the page.
1. ■ ■ ■ ■ ■
What is the first Vocabulary term in the Study Guide: Preview
for Chapter 1?
2. ■ ■ ■ ■ ■ ■
3. ■ ■ ■ ■ ■ ■ ■
4. ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
5. ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
6. ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
7. ■ ■ ■ ■ ■ ■
8. ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
What keyword should you enter for Homework Help for Lesson 3-3?
In Lesson 8-2, what is Example 4 teaching you to find?
What theorem are you asked about in the Know-It Note on page 352?
What mathematician is featured in the Math History link on page 318?
Whose job is described in the Career Path on page 612?
In the Study Guide: Review for Lesson 11-1, what do
the lines intersect?
What advice does Chapter 1’s Test Tackler give about how to
answer a multiple choice test item you don’t know how to solve?
What did the little acorn say when it grew up?
■■■■■■■■
xxii
Scavenger Hunt
The Problem Solving Plan
Mathematical problems are a part of daily life. You need to use a good
problem-solving plan to be a good problem solver. The plan used in
this textbook is outlined below.
UNDERSTAND the Problem
First make sure you understand the problem you are asked to solve.
What are you asked to find?
What information is given?
What information do you need?
Do you have all the information
needed?
Do you have too much
information?
Restate the question in your own words.
Identify the key facts given in the problem.
Determine what information you need
to solve the problem.
Determine if you need more information.
Determine if there is unnecessary
information and eliminate it from
your list of important facts.
Make a PLAN
Plan how to use the information you are given.
Have you solved similar problems?
What problem solving strategy
or strategies could you use to
solve this problem?
Think about similar problems you have
solved successfully.
Choose an appropriate problem solving
strategy and decide how you will use it.
SOLVE
Use your plan to solve the problem. Show the steps in the solution,
and write a final statement that gives the solution to the problem.
LOOK BACK
Check your answer against the original problem.
Have you answered the
question?
Is the answer reasonable?
Are your calculations correct?
Can you use another strategy or
solve the problem in another way?
Did you learn anyting that could
help you solve similar problems
in the future?
Make sure you have answered the
original question.
The answer must make sense in relation
to the question.
Check to make sure your calculations
are accurate.
Using another strategy is a good way to
check your answer.
Try to remember the types of problems
you have solved and the strategies
you applied.
Focus on Problem Solving
xxiii