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Geometry
Volume 1
TIMOTHY D. KANOLD
EDWARD B. BURGER
JULI K. DIXON
MATTHEW R. LARSON
STEVEN J. LEINWAND
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Authors
Timothy D. Kanold, Ph.D., is an
award-winning international educator,
author, and consultant. He is a former
superintendent and director of
mathematics and science at Adlai E.
Stevenson High School District 125
in Lincolnshire, Illinois. He is a past
president of the National Council of
Supervisors of Mathematics (NCSM)
and the Council for the Presidential
Awardees of Mathematics (CPAM). He
has served on several writing and leadership commissions
for NCTM during the past decade. He presents motivational
professional development seminars with a focus on developing
professional learning communities (PLC’s) to improve the
teaching, assessing, and learning of students. He has recently
authored nationally recognized articles, books, and textbooks
for mathematics education and school leadership, including
What Every Principal Needs to Know about the Teaching and
Learning of Mathematics.
Edward B. Burger, Ph.D., is the
President of Southwestern University,
a former Francis Christopher Oakley
Third Century Professor of Mathematics
at Williams College, and a former vice
provost at Baylor University. He has
authored or coauthored more than sixtyfive articles, books, and video series;
delivered over five hundred addresses
and workshops throughout the world;
and made more than fifty radio and
television appearances. He is a Fellow of the American
Mathematical Society as well as having earned many national
honors, including the Robert Foster Cherry Award for Great
Teaching in 2010. In 2012, Microsoft Education named him a
“Global Hero in Education.”
Juli K. Dixon, Ph.D., is a Professor
of Mathematics Education at the
University of Central Florida. She has
taught mathematics in urban schools
at the elementary, middle, secondary,
and post-secondary levels. She is an
active researcher and speaker with
numerous publications and conference
presentations. Key areas of focus are
deepening teachers’ content knowledge
and communicating and justifying
mathematical ideas. She is a past chair of the NCTM Student
Explorations in Mathematics Editorial Panel and member of the
Board of Directors for the Association of Mathematics Teacher
Educators.
Matthew R. Larson, Ph.D., is the K-12
mathematics curriculum specialist for
the Lincoln Public Schools and served on
the Board of Directors for the National
Council of Teachers of Mathematics from
2010 to 2013. He is a past chair of
NCTM’s Research Committee and
was a member of NCTM’s Task Force
on Linking Research and Practice.
He is the author of several books on
implementing the Common Core
Standards for Mathematics. He has taught mathematics at
the secondary and college levels and held an appointment as
an honorary visiting associate professor at Teachers College,
Columbia University.
Steven J. Leinwand is a Principal
Research Analyst at the American
Institutes for Research (AIR) in
Washington, D.C., and has over 30 years
in leadership positions in mathematics
education. He is past president of the
National Council of Supervisors of
Mathematics and served on the NCTM
Board of Directors. He is the author of
numerous articles, books, and textbooks
and has made countless presentations
with topics including student achievement, reasoning,
effective assessment, and successful
implementation of standards.
Performance Task Consultant
Robert Kaplinsky
Teacher Specialist, Mathematics
Downey Unified School District
Downey, California
STEM Consultants
Science, Technology, Engineering, and Mathematics
Michael A. DiSpezio
Michael R. Heithaus
Global Educator
North Falmouth, Massachusetts
Executive Director, School of Environment, Arts,
and Society
Professor, Department of Biological Sciences
Florida International University
North Miami, Florida
Reviewers
Mindy Eden
Ted Ryan
Richwoods High School
Peoria School District
Peoria, IL
Madison LaFollette High School
Madison Metropolitan School District
Madison, WI
Dustin Johnson
Tony Scoles
Badger High School Math Teacher
Department Chair
Lake Geneva-Genoa City Union High
School District
Lake Geneva, WI
Fort Zumwalt School District
O’Fallon, MO
Ashley D. McSwain
Murray High School
Murray City School District
Salt Lake City, UT
Rebecca Quinn
Doherty Memorial High School
Worcester Public Schools District
Worcester, MA
Cynthia L. Smith
Higley Unified School District
Gilbert, AZ
Phillip E. Spellane
Doherty Memorial High School
Worcester Public Schools District
Worcester, MA
Mona Toncheff
Math Content Specialist
Phoenix Union High School District
Phoenix, AZ
Transformations and
Congruence
Math in Careers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Reading Start-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
MODULE
1
Volume 1
1
Tools of Geometry
Real-World Video . . . . . . . . . . . . . 3
Are You Ready? . . . . . . . . . . . . . . . 4
1.1
1.2
1.3
1.4
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UNIT
Segment Length and Midpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Angle Measures and Angle Bisectors. . . . . . . . . . . . . . . . . . . . . . . . 19
Representing and Describing Transformations . . . . . . . . . . . . . . 31
Reasoning and Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Module Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Ready to Go On?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
MODULE
2
Transformations and Symmetry
Real-World Video . . . . . . . . . . . . 61
Are You Ready? . . . . . . . . . . . . . . 62
2.1
2.2
2.3
2.4
Translations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Reflections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Investigating Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Module Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Ready to Go On?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
MODULE
3
Congruent Figures
Real-World Video . . . . . . . . . . .113
Are You Ready? . . . . . . . . . . . . .114
3.1 Sequences of Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.2 Proving Figures Are Congruent Using Rigid Motions. . . . . . . . 127
3.3 Corresponding Parts of Congruent Figures Are Congruent. . 139
Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Module Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Ready to Go On?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Unit 1 Assessment Readiness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
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Lines, Angles, and
Triangles
Math in Careers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Reading Start-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
MODULE
4
UNIT
Volume 1
2
Lines and Angles
Real-World Video . . . . . . . . . . .161
Are You Ready? . . . . . . . . . . . . .162
4.1
4.2
4.3
4.4
4.5
Angles Formed by Intersecting Lines. . . . . . . . . . . . . . . . . . . . . . . 163
Transversals and Parallel Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Proving Lines Are Parallel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Perpendicular Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Equations of Parallel and Perpendicular Lines . . . . . . . . . . . . . . 205
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Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
Module Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
Ready to Go On?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
MODULE
5
Triangle Congruence Criteria
Real-World Video . . . . . . . . . . .217
Are You Ready? . . . . . . . . . . . . .218
5.1
5.2
5.3
5.4
Exploring What Makes Triangles Congruent . . . . . . . . . . . . . . . . 219
ASA Triangle Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
SAS Triangle Congruence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
SSS Triangle Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
Module Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
Ready to Go On?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
MODULE
6
Applications of Triangle
Congruence
Real-World Video . . . . . . . . . . .271
Are You Ready? . . . . . . . . . . . . .272
6.1 Justifying Constructions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
6.2 AAS Triangle Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
6.3 HL Triangle Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
MODULE
7
Properties of Triangles
Real-World Video . . . . . . . . . . .311
Are You Ready? . . . . . . . . . . . . .312
7.1 Interior and Exterior Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
7.2 Isosceles and Equilateral Triangles . . . . . . . . . . . . . . . . . . . . . . . . . 327
7.3 Triangle Inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
Module Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
Ready to Go On?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
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Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
Module Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
Ready to Go On?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
MODULE
8
Special Segments in Triangles
Real-World Video . . . . . . . . . . .357
Are You Ready? . . . . . . . . . . . . .358
8.1
8.2
8.3
8.4
Perpendicular Bisectors of Triangles . . . . . . . . . . . . . . . . . . . . . . . 359
Angle Bisectors of Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
Medians and Altitudes of Triangles . . . . . . . . . . . . . . . . . . . . . . . . 381
Midsegments of Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
Module Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
Ready to Go On?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
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Unit 2 Assessment Readiness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
UNIT
Volume 1
3
MODULE
9
Quadrilaterals and
Coordinate Proof
Math in Careers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
Reading Start-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
Properties of Quadrilaterals
Real-World Video . . . . . . . . . . .417
Are You Ready? . . . . . . . . . . . . .418
9.1
9.2
9.3
9.4
9.5
Properties of Parallelograms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
Conditions for Parallelograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
Properties of Rectangles, Rhombuses, and Squares. . . . . . . . . 447
Conditions for Rectangles, Rhombuses, and Squares . . . . . . . 459
Properties and Conditions for Kites and Trapezoids. . . . . . . . . 471
Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
Module Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488
Ready to Go On?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
MODULE
Real-World Video . . . . . . . . . . .493
Are You Ready? . . . . . . . . . . . . .494
10.1 Slope and Parallel Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
10.2 Slope and Perpendicular Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
10.3 Coordinate Proof Using Distance with Segments
and Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521
10.4 Coordinate Proof Using Distance with Quadrilaterals . . . . . . 537
10.5 Perimeter and Area on the Coordinate Plane . . . . . . . . . . . . . . 549
Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565
Module Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566
Ready to Go On?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567
Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568
Unit 3 Assessment Readiness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569
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10
Coordinate Proof Using Slope
and Distance
Similarity
UNIT
Math in Careers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573
Reading Start-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574
MODULE
11
Volume 2
4
Similarity and Transformations
Real-World Video . . . . . . . . . . .575
Are You Ready? . . . . . . . . . . . . .576
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11.1
11.2
11.3
11.4
MODULE
12
Dilations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577
Proving Figures Are Similar Using Transformations . . . . . . . . 587
Corresponding Parts of Similar Figures. . . . . . . . . . . . . . . . . . . . 601
AA Similarity of Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623
Module Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624
Ready to Go On?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627
Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628
Using Similar Triangles
Real-World Video . . . . . . . . . . .629
Are You Ready? . . . . . . . . . . . . .630
12.1
12.2
12.3
12.4
Triangle Proportionality Theorem. . . . . . . . . . . . . . . . . . . . . . . . . 631
Subdividing a Segment in a Given Ratio . . . . . . . . . . . . . . . . . . 641
Using Proportional Relationships . . . . . . . . . . . . . . . . . . . . . . . . . 653
Similarity in Right Triangles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663
Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
Module Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676
Ready to Go On?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677
Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678
Unit 4 Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679
UNIT
Volume 2
5
MODULE
13
Trigonometry
Math in Careers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683
Reading Start-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684
Trigonometry with Right Triangles
Real-World Video . . . . . . . . . . .685
Are You Ready? . . . . . . . . . . . . .686
13.1
13.2
13.3
13.4
Tangent Ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687
Sine and Cosine Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697
Special Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709
Problem Solving with Trigonometry . . . . . . . . . . . . . . . . . . . . . . 723
Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737
Module Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738
Ready to Go On?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739
Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740
14
Trigonometry with All Triangles
Real-World Video . . . . . . . . . . .741
Are You Ready? . . . . . . . . . . . . .742
14.1 Law of Sines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743
14.2 Law of Cosines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755
Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767
Module Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768
Ready to Go On?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769
Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 770
Unit 5 Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771
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Shutterstock; (b)©Carol Kohen/Cultura RM/Alamy
MODULE
Properties of Circles
Math in Careers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775
Reading Start-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776
MODULE
15
UNIT
Volume 2
6
Angles and Segments in Circles
Real-World Video . . . . . . . . . . .777
Are You Ready? . . . . . . . . . . . . .778
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iStockPhoto.com; (b) ©primopiano/Shutterstock
15.1
15.2
15.3
15.4
15.5
MODULE
16
Central Angles and Inscribed Angles. . . . . . . . . . . . . . . . . . . . . . 779
Angles in Inscribed Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . 793
Tangents and Circumscribed Angles . . . . . . . . . . . . . . . . . . . . . . 805
Segment Relationships in Circles . . . . . . . . . . . . . . . . . . . . . . . . . 815
Angle Relationships in Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829
Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843
Module Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844
Ready to Go On?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847
Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 848
Arc Length and Sector Area
Real-World Video . . . . . . . . . . .849
Are You Ready? . . . . . . . . . . . . .850
16.1 Justifying Circumference and Area of a Circle . . . . . . . . . . . . . 851
16.2 Arc Length and Radian Measure . . . . . . . . . . . . . . . . . . . . . . . . . . 863
16.3 Sector Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873
Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883
Module Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884
Ready to Go On?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885
Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886
MODULE
17
Equations of Circles and Parabolas
Real-World Video . . . . . . . . . . .887
Are You Ready? . . . . . . . . . . . . .888
17.1 Equation of a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 889
17.2 Equation of a Parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 901
Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913
Module Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914
Ready to Go On?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915
Module 17 Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916
Unit 6 Assessment Readiness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917
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Eggers/U.S. Coast Guard
Measurement and Modeling
in Two and Three Dimensions
Math in Careers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 921
Reading Start-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 922
MODULE
18
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Stringer/
Reuters/Corbis; (b) ©Stocktrek Images, Inc./Getty Images
19
Volume 2
7
Volume Formulas
Real-World Video . . . . . . . . . . .923
Are You Ready? . . . . . . . . . . . . .924
18.1
18.2
18.3
18.4
MODULE
UNIT
Volume of Prisms and Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . 925
Volume of Pyramids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937
Volume of Cones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949
Volume of Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963
Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975
Module Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976
Ready to Go On?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977
Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 978
Visualizing Solids
Real-World Video . . . . . . . . . . .979
Are You Ready? . . . . . . . . . . . . .980
19.1
19.2
19.3
19.4
Cross-Sections and Solids of Rotation. . . . . . . . . . . . . . . . . . . . . 981
Surface Area of Prisms and Cylinders . . . . . . . . . . . . . . . . . . . . . 991
Surface Area of Pyramids and Cones. . . . . . . . . . . . . . . . . . . . .1005
Surface Area of Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1019
Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1031
Module Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1032
Ready to Go On?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1033
Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1034
MODULE
20
Modeling and Problem Solving
Real-World Video . . . . . . . . . 1035
Are You Ready? . . . . . . . . . . . 1036
20.1 Scale Factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1037
20.2 Modeling and Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1047
20.3 Problem Solving with Constraints . . . . . . . . . . . . . . . . . . . . . . .1057
Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1071
Module Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1072
Ready to Go On?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1073
Module 17 Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1074
Unit 7 Assessment Readiness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1075
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Berry/iStockPhoto.com
Probability
UNIT
Math in Careers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1079
Reading Start-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1080
MODULE
21
Volume 2
8
Introduction to Probability
Real-World Video . . . . . . . . . 1081
Are You Ready? . . . . . . . . . . . 1082
21.1
21.2
21.3
21.4
Probability and Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1083
Permutations and Probability . . . . . . . . . . . . . . . . . . . . . . . . . . .1095
Combinations and Probability . . . . . . . . . . . . . . . . . . . . . . . . . . .1107
Mutually Exclusive and Overlapping Events . . . . . . . . . . . . . .1119
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Images/Alamy; (b) ©Sergey Nivens/Shutterstock
Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1131
Module Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1132
Ready to Go On?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1133
Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1134
MODULE
22
Conditional Probability and
Independence of Events
Real-World Video . . . . . . . . . 1135
Are You Ready? . . . . . . . . . . . 1136
22.1 Conditional Probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1137
22.2 Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1149
22.3 Dependent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1165
Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1177
Module Performance Task. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1178
Ready to Go On? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1179
Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1180
MODULE
23
Probability and Decision Making
Real-World Video . . . . . . . . . 1181
Are You Ready? . . . . . . . . . . . 1182
23.1 Using Probability to Make Fair Decisions . . . . . . . . . . . . . . . . .1183
23.2 Analyzing Decisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1193
Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1205
Module Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1206
Ready to Go On?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1207
Module 17 Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1208
Unit 8 Assessment Readiness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1209
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Graphics/Alamy
© Houghton Mifflin Harcourt Publishing Company Image Credits: (California
Poppies) Photodisc/Getty Images; (Nashville, TN at night) Photodisc/Getty
Images; (Smoky Mountains, TN) Digital Vision/Getty Images; (Salt Lake City, UT)
Douglas Pulsipher/Alamy.
HMH Geometry
Online State Resources
Scan the QR code or visit:
my.hrw.com/nsmedia/osp/2015/ma/hs/tempaga
for correlations and other state-specific resources.
xix
Succeeding with HMH
Geometry
HMH Geometry is built on the 5E
instructional model--Engage, Explore,
Explain, Elaborate, Evaluate--to develop
strong conceptual understanding and
mastery of key mathematics standards.
Lesson 19.2 Precision and Accuracy
Engage
Essential Question
How do you use significant digits
when reporting the results of
calculations involving measurement?
ENGAGE
Preview the Lesson
Performance Task in the
Interactive Student Edition.
Preview
Lesson Performance Task
The sun is an excellent source of electrical energy. Suppose a company owns
a field of solar panels. How much electricity is produced by the field? The an
answer depends on the amount of power the field yields per square foot, as well
as the size of the field.
Lesson 19.2 Precision and Accuracy
Explore
Concept 1
Comparing Precision of
Measurements
EXPLORE
Eric is a technician in a pharmaceutical lab. Every week, he needs to test the
scales in the lab to make sure that they are. He uses a that is exactly 12.000 g
Scale
Mass
Scale 1
12.03 g
Scale 2
12.029 g
Scale 3
11.98 g
Definition of Precision: The level of detail of a, determined by the smallest
unit or fraction of a unit that can be reasonably measured.
Scan the QR code to access
engaging videos, activities, and
more in the Resource Locker for
each lesson.
Definition of Accuracy: The closeness of a given of value to the actual
measurement or value.
Which measuring tool is the most precise?
Scale 2
DO NOT EDIT--Changes must be made through “File info”
CorrectionKey=NL-A;CA-A
Name
Which scale is the most accurate?
Class
1.3
Date
Reporting with Precision
and Accuracy
Scale 2
Essential Question: How do you use significant digits when reporting the results of
calculations involving measurement?
Resource
Locker
My answer
Explore
Precision is the level of detail of a measurement, determined by the smallest unit or fraction of a unit that can be
reasonably measured.
Given two measurements, is it possible that the more precise one is not the
Accuracy is the closeness of a given measurement or value to the actual measurement or value. Suppose you know the
actual measure of a quantity, and someone else measures it. You can find the accuracy of the measurement by finding
the absolute value of the difference of the two.
most accurate? Why?

Send to Notebook
XX
Engage
Explain
n Mifflin Harcourt Publishing Company
Formula
Comparing Precision of Measurements.
Numbers are values without units. They can be used to compute or to describe measurements. Quantities are realword values that represent specific amounts. For instance, 15 is a number, but 15 grams is a quantity.
Reflect

Complete the table to choose the more precise measurement.
Measurement 1
Measurement 2
4g
4.3 g
5.71 oz
5.7 oz
4.2 m
422 cm
7 ft 2 in.
7.2 in.
Smaller Unit
More Precise Measurement
Eric is a lab technician. Every week, he needs to test the scales in the lab to make sure that they
are accurate. He uses a standard mass that is exactly 8.000 grams and gets the following results.
Scale
Mass
Scale 1
8.02 g
Scale 2
7.9 g
Scale 3
8.029 g
Scale 1
Scale 2
Scale 3
© Houghton Mifflin Harcourt Publishing Company
Explore and interact with new concepts
to develop
a deeper understanding
We’ll look at this challenge during the lesson using significant digits!
of mathematics in your book and the
Interactive Student Edition.
and gets the following results:
Explore
1
Lesson 19.2 Precision and Accuracy
Explain
Concept 2
Determining Precision
As you have seen, measurements are given to a certain precision. Therefore,
the value reported does not necessarily represent the actual value of the
measurement. For example, a measurement of 5 centimeters, which is
given to the nearest whole unit, can actually range from 0.5 units below the
reported value, 4.5 centimeters, up to, but not including, 0.5 units above
it, 5.5 centimeters. The actual length, l, is within a range of possible values:
centimeters. Similarly, a length given to the nearest tenth can actually range
EXPLAIN
from 0.05 units below the reported value up to, but not including, 0.05 units
above it. So a length reported as 4.5 cm could actually be as low as 4.45 cm or
as high as nearly 4.55 cm.
Learn concepts with step-bystep interactive examples. Every
example is also supported by a
Math On the Spot video tutorial.
DO NOT EDIT--Changes must be made through “File info”
CorrectionKey=NL-B;CA-B

Lesson 19.2 Precision and Accuracy
Your Turn
Scale 1: Accuracy = ⎜8.000 -
Concept 2
Calculate the minimum and maximum possible areas.
your
ScaleRound
2: Accuracy
= ⎜answer
8.000 - to
the nearest square centimeters.
1
2
3
4
Question 3 of 17
Find the accuracy of each of the measurements in Step B.
5
6
7
8
9
10
11 - 17
Video Tutor
View Step by Step
⎟=
⎟=
Complete each statement: the measurement for Scale
The width and length of a rectangle are 8 cm and 19.5 cm, respectively.
Personal Math Trainer
Textbook
Scale 3: Accuracy = ⎜8.000 -
⎟=
is the most accurate because
X2 Animated Math
, which is
grams,
.
Reflect
1.
Discussion Given two measurements of the same quantity, is it possible that the more precise
Find the range of values for the actual length and width of the rectangle.measurement is not the more accurate? Why do you think that is so?
Solve the quadratic equation by factoring.
Minimum width = 7.5 cm and maximum width < 8.5 cm
7x + 44x = 7x − 10
x=
Explain 1
,
Check
Find the range of values for the actual length and width of the rectangle.
Example 1
Save & Close
?
Determining Precision of Calculated Measurements
As you have seen, measurements are reported to a certain precision. The reported value does not necessarily represent
My answer
the actual value of the measurement. When you measure
to the nearest unit, the actual length can be 0.5 unit less than
the measured length or less than 0.5 unit greater than the measured length. So, a length reported as 4.5 centimeters
could actually be anywhere between 4.45 centimeters and 4.55 centimeters, but not including 4.55 centimeters.
It cannot include 4.55 centimeters because 4.55 centimeters reported to the nearest tenth would round up to 4.6
centimeters.
Turn It In
!
Minimum length = 19.45
Calculate the minimum and maximum possible areas. Round your
answers to the nearest square centimeter.
cm and maximum
length
< and19.55
length
width of cm
a book cover are 28.3 centimeters and 21 centimeters, respectively.
 The
Minimum length = (28.3 - 0.05) cm and maximum length = (28.3 + 0.05) cm,
so 28.25 cm ≤ length < 28.35 cm.
© Houghton Mifflin Harcourt Publishing Company
Explain
Minimum width = (21 - 0.5) cm and maximum width = (21 + 0.5) cm, so 20.5 cm ≤ width < 21.5 cm.
Elaborate
Find the minimum and maximum areas.
Minimum area = minimum length ⋅ minimum width
Find the range of values for the actual length and width of the rectangle.
Check your understanding of new
concepts and skills with Your Turn
exercises in your book or online
with Personal Math Trainer.
= 28.25 cm ⋅ 20.5 cm ≈ 579 cm 2
Maximum area = maximum length ⋅ maximum width
Minimum Area = Minimum width × Minimum length= 28.35 cm ⋅ 21.5 cm ≈ 610 cm 2
=
7.5 cm × 19.45 cm
So 579 cm 2 ≤ area < 610 cm 2.
Module 1
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CorrectionKey=NL-B;CA-B
Lesson 3
28
DO NOT EDIT--Changes must be made through “File info”
CorrectionKey=NL-B;CA-B
15
A1_MNLESE368170_U1M01L3.indd 28
Reflect
13. In the example, why wouldn’t it be wise to
use the length of a shorter car?
Reflect
14. Critical Thinking How else might the number
of cars be estimated? Would you expect the
estimate to be the same? Explain.
Lesson 19.2 Precision and Accuracy
In the above exercise, the location of the uncertainty in the linear
measurements results in different amounts of uncertainty in the calculated
measurement. Explain how
to fix this problem.
Elaborate
ELABORATE
Your Turn
Given two measurements, is it possible that the more precise measurement
may not be the more accurate?
Estimate the quantity needed in the following situations. Use the correct number of significant digits.
15. Claire and Juan are decorating a
rectangular wall of 433 square feet with
two types of rectangular pieces of fabric.
One type has an area of 9.4 square feet
and the other has an area of 17.2 square
feet. About how many decorative pieces
can Claire and Juan fit in the given area?
16. An artist is making a mosaic
and has pieces of smooth glass
ranging in area from 0.25
square inch to 3.75 square
inches. Suppose the mosaic
is 34.1 inches wide and 50.0
inches long. About how many pieces of glass
will the artist need?
Show your understanding and reasoning
with Reflect and Elaborate questions.
18. What is the relationship between the range of possible error in the measurements used in a calculation and
the range of possible error in the calculated measurement?
19. Essential Question Check-In How do you use significant digits to determine how to report a sum or
product of two measurements?
Module 1
34
Lesson 3
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Michael
Freeman/Corbis
Elaborate
17. Given two measurements, is it possible that the more accurate measurement is not the more precise?
Justify your answer.
© Houghton Mifflin Harcourt Publishing Company
Find the range of values for the actual length and width of the book cover.
Formula
Send to Notebook
Formula
Explore
Send to
Notebook
Your
Turn
What is the relationship between the precision used in the length and width of
the rectangle and the precision of the resulting area measurement?
XXI
Formula
Send to Notebook
How are the significant digits related to the calculations using measurements?
Lesson 19.2 Precision and Accuracy
Evaluate
1
2
3
4
Question 3 of 17
5
6
7
8
9
View Step by Step
10
11 - 17
Video Tutor
Personal Math Trainer
X2 Animated Math
Textbook
Solve the quadratic equation by factoring.
7x + 44x = 7x − 10
EVALUATE
Practice and apply skills and
concepts with Evaluate exercises
and a Lesson Performance Task
in your book with plenty of
workspace, or complete these
exercises online with
Personal Math Trainer.
x=
,
Check
Save & Close
?
Turn It In
!
Evaluate: Homework and Practice
Elaborate
1.
Look Back
• Online Homework
• Hints and Help
• Extra Practice
The diagram represents the expression
x 2 + 4x + c with the constant term missing.
Complete the square by filling in the bottom
right corner with 1-tiles, and write the
expression as a trinomial and in factored form.
Complete the square to form a perfect square trinomial. Then factor the trinomial.
2.
m 2 + 10m +
4.
y 2 + 2y +
3.
g 2 - 20g +
Lesson Performance
Task
5. w - 11w +
2
The quarterback of a football team is practicing throwing a 50-yard pass to a wide
receiver. The quarterback can throw a pass with an initial vertical velocity of 40 feet per
second and an initial height of 6 feet. He wants to throw the ball so it lands in the wide
receiver’s hands at a height of 6 feet at exactly the right time.
The wide receiver can run 40 yards in 4.4 seconds and begins running at top speed when
8.
y 2 + 19y + 78 = 0
x 2 - 19x + 84 = 0
LOOK BACK
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Lesson 8
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468
Lesson 8
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Review what you have learned and
prepare for high-stakes tests with a
variety of resources, including Study
Guide Reviews, Performance Tasks,
and Assessment Readiness test
preparation.
Module 8
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Module 8
9.
© Houghton Mifflin Harcourt Publishing Company
thesquare.
quarterback hikes the ball. How long should the quarterback wait between hiking the
Solve the equation by completing the
ball and throwing it?
6. s 2 + 15s = -56
7. r 2 - 4r = 165