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2A Chapter 1
RESOU RCE OPTIONS • RESOURCE OPTIONS • RESOU RCE OPTIONS • RESOU RCEE
Foundations for Geometry
Section 1A Section 1B
Euclidean and Construction Tools Coordinate and Transformation Tools
1-1 Understanding Points, Lines, and Planes
1-2 Technology Lab Explore Properties Associated with Points
1-2 Measuring and Constructing Segments
1-3 Measuring and Constructing Angles
1-4 Pairs of Angles
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
1-7 Technology Lab Explore Transformations
Pacing Guide for 45-Minute Classes
Chapter 1 Countdown to Testing Weeks^1 ,^2
DAY 1 DAY 2 DAY 3 DAY 4 DAY 5
1-1 Lesson 1-2 Technology Lab
1-2 Lesson
1-3 Lesson 1-4 Lesson Multi-Step Test Prep
Ready to Go On?
DAY 6 DAY 7 DAY 8 DAY 9 DAY 10
1-5 Lesson Connecting Geometry
to Algebra
1-6 Lesson
1-7 Lesson 1-7 Technology Lab Multi-Step Test Prep
Ready to Go On?
DAY 11 DAY 12
Chapter 1 Review Chapter 1 Test
Pacing Guide for 90-Minute Classes
Chapter 1
DAY 1 DAY 2 DAY 3 DAY 4 DAY 5
1-1 Lesson
1-2 Technology Lab
1-2 Lesson
1-3 Lesson
1-4 Lesson
Multi-Step Test Prep
Ready to Go On?
1-5 Lesson
Connecting Geometry
to Algebra
1-6 Lesson
1-7 Lesson
1-7 Technology Lab
Multi-Step Test Prep
Ready to Go On?
DAY 6
Chapter 1 Review
Chapter 1 Test
2B
OPTIONS • RESOU RCE OPTIONS • RESOURCE OPTIONS • RESOURCE OPTIONS
Before Chapter 1 Testing
Diagnose mastery of concepts in the chapter.
Ready to Go On? SE pp. 35, 59
Multi-Step Test Prep SE pp. 34, 58
Section Quizzes AR pp. 5–
Test and Practice Generator
Prescribe intervention.
Ready to Go On? Intervention pp. 2–
Scaffolding Questions TE pp. 34, 58
Before High Stakes Testing
Diagnose mastery of benchmark concepts.
College Entrance Exam Practice SE p. 65
New York Test Prep SE pp. 68–
Prescribe intervention.
College Entrance Exam Practice
New York State Test Prep Workbook
Before Every Lesson
Diagnose readiness for the lesson.
Warm Up TE, every lesson
Prescribe intervention.
Skills Bank SE pp. S50–S
Reteach CRB, Ch. 1
During Every Lesson
Diagnose understanding of lesson concepts.
Check It Out! SE, every example
Think and Discuss SE, every lesson
Write About It SE, every lesson
Journal TE, every lesson
Prescribe intervention.
Questioning Strategies TE, every example
Reading Strategies CRB, every lesson
Success for ELL pp. 1–
After Every Lesson
Diagnose mastery of lesson concepts.
Lesson Quiz TE, every lesson
Alternative Assessment TE, every lesson
Test Prep SE, every lesson
Test and Practice Generator
Prescribe intervention.
Reteach CRB, every lesson
Problem Solving CRB, every lesson
Test Prep Doctor TE, every lesson
Homework Help Online
DIAGNOSE PRESCRIBE
KEY: SE = Student Edition TE = Teacher’s Edition CRB = Chapter Resource Book AR = Assessment Resources Available online Available on CD-ROM
Assess
Prior
Knowledge
After Chapter 1
Check mastery of chapter concepts.
Multiple-Choice Tests (Forms A, B, C)
Free-Response Tests (Forms A, B, C)
Performance Assessment AR pp. 7–
Test and Practice Generator
Check mastery of benchmark concepts.
New York State Regents Exam
College Entrance Exams
Prescribe intervention.
Reteach CRB, every lesson
Lesson Tutorial Videos Chapter 1
Prescribe intervention.
New York State Test Prep Workbook
College Entrance Exam Practice
Summative
Assessment
Before Chapter 1
Diagnose readiness for the chapter.
Are You Ready? SE p. 3
Prescribe intervention.
Are You Ready? Intervention Skills 20, 57, 60, 79
Formative
Assessment
2D
OPTIONS • RESOU RCE OPTIONS • RESOURCE OPTIONS • RESOURCE OPTIONS
Lesson Tutorial Videos Multilingual Glossary Online Interactivities
Starring Holt authors Ed Burger and Freddie
Renfro! Live tutorials to support every
lesson in Chapter 1.
Searchable glossary includes definitions
in English, Spanish, Vietnamese, Chinese,
Hmong, Korean, and 4 other languages.
Interactive tutorials provide visually engaging
alternative opportunities to learn concepts and
master skills.
Technology Highlights for Reaching All Learners
KEY: SE = Student Edition TE = Teacher’s Edition CRB = Chapter Resource Book JG8E@J? Spanish version available Available online Available on CD-ROM
Reaching All Learners
Resources for All Learners
Geometry Lab Activities ....................................... Chapter 1
Technology Lab Activities ..................................... Chapter 1
New York Homework and Practice Workbook .........pp. 1–
Know-It Notebook .............................................. Chapter 1
New York Problem Solving Workbook ....................pp. 1–
DEVELOPING LEARNERS
Practice A .............................................. CRB, every lesson
Reteach ................................................. CRB, every lesson
Inclusion ..........................TE pp. 18, 25, 30, 36, 42, 45, 51
Questioning Strategies ............................ TE, every example
Modifi ed Chapter 1 Resources ................. IDEA Works!
Homework Help Online
ON-LEVEL LEARNERS
Practice B ............................................... CRB, every lesson
Multiple Representations ........................................TE p. 18
Cognitive Strategies ........................................TE pp. 29, 44
ADVANCED LEARNERS
Practice C ............................................... CRB, every lesson
Challenge ............................................... CRB, every lesson
Reading and Writing Math EXTENSION ...........................TE p. 5
Multi-Step Test Prep EXTENSION ..........................TE pp. 34, 58
Critical Thinking .....................................................TE p. 14
English Language Learners
Are You Ready? Vocabulary...................................... SE p. 3
Vocabulary Connections ........................................... SE p. 4
Lesson Vocabulary .....................................SE, every lesson
Vocabulary Exercises .........................SE, every exercise set
Vocabulary Review ................................................ SE p. 60
English Language Learners ..........TE pp. 5, 7, 14, 22, 23, 39
Reading Strategies .................................. CRB, every lesson
Success for English Language Learners ...................pp. 1–
Multilingual Glossary
Reaching All Learners Through...
Inclusion ..........................TE pp. 18, 25, 30, 36, 42, 45, 51
Visual Cues ........................................TE pp. 21, 22, 27, 30
Kinesthetic Experience ....................................TE pp. 18, 26
Concrete Manipulatives...................................TE pp. 14, 51
Multiple Representations ........................................TE p. 18
Cognitive Strategies ........................................TE pp. 29, 44
Cooperative Learning ..............................................TE p. 29
Modeling .................................................................TE p. 7
Critical Thinking .....................................................TE p. 14
Test Prep Doctor .................................TE pp. 11, 26, 33, 40
49, 55, 65, 66, 68
Common Error Alerts .........................TE pp. 15, 19, 21, 23,
25, 29, 39, 45, 49
Scaffolding Questions .....................................TE pp. 34, 58
E NGLISH
L ANGUAGE
L EARNERS
2E Chapter 1
RESOU RCE OPTIONS • RESOURCE OPTIONS • RESOU RCE OPTIONS • RESOU RCEE
Ongoing Assessment
KEY: SE = Student Edition TE = Teacher’s Edition AR = Assessment Resources JG8E@J? Spanish version available Available online Available on CD-ROM
Are You Ready? JG8E@J? Ready to Go On? Test and Practice Generator
Automatically assess readiness and
prescribe intervention for Chapter 1
prerequisite skills.
Automatically assess understanding
of and prescribe intervention for
Sections 1A and 1B.
Use Chapter 1 problem banks to create
assessments and worksheets to print out or
deliver online. Includes dynamic problems.
Technology Highlights for Ongoing Assessment
Assessing Prior Knowledge
Determine whether students have the prerequisite concepts
and skills for success in Chapter 1.
Are You Ready? JG8E@J? ........................... SE p. 3
Warm Up ................................... TE, every lesson
Test Preparation
Provide review and practice for Chapter 1 and standardized
tests.
Multi-Step Test Prep ...................................... SE pp. 34, 58
Study Guide: Review ...................................... SE pp. 60–
Test Tackler ................................................... SE pp. 66–
New York Test Prep ....................................... SE pp. 68–
College Entrance Exam Practice .............................. SE p. 65
New York State Test Prep Workbook
IDEA Works!
Alternative Assessment
Assess students’ understanding of Chapter 1 concepts
and combined problem-solving skills.
Chapter 1 Project ..................................................... SE p. 2
Alternative Assessment .............................. TE, every lesson
Performance Assessment ............................... AR pp. 19–
Portfolio Assessment ......................................... AR p. xxxiv
Daily Assessment
Provide formative assessment for each day of Chapter 1.
Questioning Strategies ............................ TE, every example
Think and Discuss ......................................SE, every lesson
Check It Out! Exercises ............................SE, every example
Write About It ...........................................SE, every lesson
Journal...................................................... TE, every lesson
Lesson Quiz ............................... TE, every lesson
Alternative Assessment .............................. TE, every lesson
Modifi ed Lesson Quizzes ......................... IDEA Works!
Weekly Assessment
Provide formative assessment for each week of Chapter 1.
Multi-Step Test Prep ...................................... SE pp. 34, 58
Ready to Go On? ............................ SE pp. 35, 59
Cumulative Assessment ................................. SE pp. 68–
Test and Practice Generator ............. One-Stop Planner
Formal Assessment
Provide summative assessment of Chapter 1 mastery.
Section Quizzes ................................................. AR pp. 5–
Chapter 1 Test ....................................................... SE p. 64
Chapter Test (Levels A, B, C) ............................. AR pp. 7–
- Multiple Choice • Free Response
Cumulative Test ............................................. AR pp. 21–
Test and Practice Generator ............. One-Stop Planner
Modifi ed Chapter 1 Test .......................... IDEA Works!
2 Chapter 1
1A Euclidean and
Construction Tools
1-1 Understanding Points, Lines,
and Planes
Lab Explore Properties Associated
with Points
1-2 Measuring and Constructing
Segments
1-3 Measuring and Constructing
Angles
1-4 Pairs of Angles
1B Coordinate and
Transformation Tools
1-5 Using Formulas in Geometry
1-6 Midpoint and Distance in the
Coordinate Plane
1-7 Transformations in the
Coordinate Plane
Lab Explore Transformations
Picture This!
Many geometric concepts and shapes
may be used in creating works of art.
Unique designs can be made using
only points, lines, planes, or circles.
KEYWORD: MG7 ChProj
Foundations
for Geometry
ge07se_c01_0002_0005.indd 1 11/23/06 1:22:27 PM
2 Chapter 1
S E C T I O N 1A
Euclidean and
Construction Tools
On page 34, stu-
dents analyze a
diagram of an
archaeological dig by
applying definitions, using the dis-
tance formula, and classifying angles.
Exercises designed to prepare stu-
dents for success on the
Multi-Step Test Prep can
be found on pages 10,
18, 26, and 32.
S E C T I O N 1B
Coordinate and
Transformation Tools
On page 58, stu-
dents find the area
and perimeter of a
patio to determine
the total cost of the paving stones
used. They use distance, midpoint,
and transformations to create the
construction plans for the patio.
Exercises designed to prepare stu-
dents for success on the
Multi-Step Test Prep can
be found on pages 39,
48, and 54.
About the Project
Students begin by using geoboards to
explore designs based on line segments.
Then they use a compass and paper folding
to make star designs. Finally, students use
everything they’ve learned to produce an
original piece of string art.
Picture This
Project Resources
All project resources for teachers and stu-
dents are provided online.
Materials:
- Activity 1: geoboard, colored rubber
bands or colored string
- Activity 2: compass, straightedge - Activity 3: straightedge - Activity 4: wooden board, small nails,
hammer, colored string or thread
KEYWORD: MG7 ProjectTS
Foundations for Geometry 3
Vocabulary
Match each term on the left with a definition on the right.
1. coordinate
2. metric system
of measurement
3. expression
4. order of operations
Measure with Customary and Metric Units
For each object tell which is the better measurement.
5. length of an unsharpened pencil 6. the diameter of a quarter
__^1
in. or 9
3 __
in. 1 m or 2
__^1
cm
7. length of a soccer field 8. height of a classroom
100 yd or 40 yd 5 ft or 10 ft
9. height of a student’s desk 10. length of a dollar bill
30 in. or 4 ft 15.6 cm or 35.5 cm
Combine Like Terms
Simplify each expression.
11. - y + 3 y - 6 y + 12 y 12. 63 + 2 x - 7 - 4 x
13. - 5 - 9 - 7 x + 6 x 14. 24 - 3 y + y + 7
Evaluate Expressions
Evaluate each expression for the given value of the variable.
15. x + 3 x + 7 x for x = - 5 16. 5 p + 10 for p = 78
17. 2 a - 8 a for a = 12 18. 3 n - 3 for n = 16
Ordered Pairs
Write the ordered pair for each point.
{ n
n
n
Þ
Ý
{ ä
19. A 20. B
21. C 22. D
23. E 24. F
A. a mathematical phrase that contains operations, numbers,
and/or variables
B. the measurement system often used in the United States
C. one of the numbers of an ordered pair that locates a point
on a coordinate graph
D. a list of rules for evaluating expressions
E. a decimal system of weights and measures that is used
universally in science and commonly throughout the world
C
E
A
D
_ 1
in. 2
_ 1
cm
100 yd 10 ft
30 in. 15.6 cm
8 y - 2 x + 56
- x - 14 - 2 y + 31 **- 55 400
(0, 7) (-5, 4)
(6, 3) (-8, - 2 )
(3, - 5 ) (6, - 4 )
ge07se_c01_0002_0005.indd 3 4/27/06 11:56:23 AM
Are You Ready? 3
NO
INTERVENE (^) Diagnose and Prescribe
ARE Y OU R EADY? Intervention, Chapter 1
Prerequisite Skill Worksheets CD-ROM Online
Measure with Customary and Metric Units Skill 20 Activity 20
Diagnose and
Prescribe Online
Combine Like Terms Skill 57 Activity 57
Evaluate Expressions Skill 60 Activity 60
Ordered Pairs Skill 79 Activity 79
YES
ENRICH
ARE YOU READY?
Enrichment, Chapter 1
Worksheets
CD-ROM
Online
Organizer
Objective: Assess students’
understanding of prerequisite skills.
Prerequisite Skills
Measure with Customary and
Metric Units
Combine Like Terms
Evaluate Expressions
Ordered Pairs
Assessing Prior
Knowledge
INTERVENTION
Diagnose and Prescribe
Use this page to determine
whether intervention is necessary
or whether enrichment is
appropriate.
Resources
Are You Ready?
Intervention and
Enrichment Worksheets
Are You Ready? CD-ROM
Are You Ready? Online
Foundations for Geometry 5
Reading Strategy: Use Your Book for Success
Understanding how your textbook is organized will help you locate and use
helpful information.
Try This
Use your textbook for the following problems.
1. Use the index to find the page where right angle is defined.
2. What formula does the Know-It Note on the first page of
Lesson 1-6 refer to?
3. Use the glossary to find the definition of congruent segments.
4. In what part of the textbook can you find help for solving equations?
As you read through an example problem, pay attention to the notes in
the margin. These notes highlight key information about the concept
and will help you to avoid common mistakes.
The Glossary is found
in the back of your
textbook. Use it when
you need a definition
of an unfamiliar word
or phrase.
The Index is located at the
end of your textbook. If
you need to locate the page
where a particular concept
is explained, use the Index
to find the corresponding
page number.
The Skills Bank is located in
the back of your textbook.
Look in the Skills Bank for
help with math topics that
were taught in previous
courses, such as the order
of operations.
ge07se_c01_0002_0005.indd 5 12/13/06 12:09:55 PM
New York Performance Indicators
Performance
Indicators 1-
LAB
LAB
G.G.17 (^) ★
G.G.48 ★
G.G.54 ★
G.G.55 ★
G.G.56 ★
G.G.66 ★
G.G.67 ★
G.PS.5 ★ ★
G.CM.11 ★ ★
G.CN.6 ★ ★
New York Mathematics Performance Indicators are written out completely on pp NY28-NY35.
Reading and Writing Math 5
Organizer
Objective: Help students apply
strategies to understand and retain
key concepts.
G
I
D
@<I
Online Edition
Resources
Chapter 1 Resource Book
Reading Strategies
Reading Strategy:
E NGLISH
L ANGUAGE
L EARNERS
Use Your Book for
Success
Discuss The index, glossary, margin
notes, and Skills Bank can provide
students with a great deal of use-
ful information as they use this
textbook. Have students divide
their journal into sections such as
research, reflections, constructions,
postulates, and so on. Encourage
them to use all the resources pro-
vided in this book.
Extend Ask students to find a term
they do not know and look it up
in the glossary. Then ask them to
enter the definition in their jour-
nal. Challenge the students to use
information in the book that was
not mentioned. For example, look
at the table of contents, highlighted
words, words in italics, or words in
boldface.
Answers to Try This
1. p. 21
2. Mdpt. Formula
3. segs. that have the same length
4. Skills Bank
C H A P T E R
1
6A Chapter 1
Euclidean and Construction Tools
One-Minute Section Planner
Lesson Lab Resources Materials
Lesson 1-1 Understanding Points, Lines, and Planes
-
Identify, name, and draw points, lines, segments, rays, and planes.
-
Apply basic facts about points, lines, and planes.
□
NY Regents Exam
□✔^
SAT-
□✔^
NAEP
□✔^
ACT
□✔^
SAT
Required
straightedge (MK)
Optional
raw spaghetti, index cards,
straw, clay, gumdrops,
toothpicks
1-2 Technology Lab Explore Properties Associated with Points
-
Use geometry software to measure distances and explore properties
of points on segments.
□
NY Regents Exam
□✔^
SAT-
□✔^
NAEP
□✔^
ACT
□
SAT
Technology Lab Activities
1-2 Lab Recording Sheet
Required
geometry software
Optional
ruler (MK)
Lesson 1-2 Measuring and Constructing Segments
-
Use length and midpoint of a segment.
-
Construct midpoints and congruent segments.
□
NY Regents Exam
□✔^
SAT-
□✔^
NAEP
□✔^
ACT
□✔^
SAT
Geometry Lab Activities
1-2 Geometry Lab
Required
compass (MK), straightedge
Optional
road map, masking tape,
butcher paper, small plastic
disks, meter stick
Lesson 1-3 Measuring and Constructing Angles
-
Name and classify angles.
-
Measure and construct angles and angle bisectors.
□✔^ NY Regents Exam^ □ SAT-10^ □✔^ NAEP^ □✔^ ACT^ □✔^ SAT
Geometry Lab Activities
1-3 Geometry Lab
Required
compass (MK), straightedge,
protractor (MK), geometry
software
Optional
yarn, index cards, pictures of
angles, origami paper, sticky
notes, acetate or tracing paper,
clock (MK), Mira
Lesson 1-4 Pairs of Angles
-
Identify adjacent, vertical, complementary, and supplementary
angles.
-
Find measures of pairs of angles.
□ NY Regents Exam^ □ SAT-10^ □✔^ NAEP^ □✔^ ACT^ □✔^ SAT
Required
protractor (MK)
MK = Manipulatives Kit
6 Chapter 1 Foundations for Geometry
The most basic figures in geometry are undefined
terms , which cannot be defined by using other
figures. The undefined terms point, line, and plane
are the building blocks of geometry.
TERM NAME DIAGRAM
A point names a location
and has no size. It is
represented by a dot.
A capital letter
point P
A line is a straight path
that has no thickness and
extends forever.
A lowercase letter or
two points on the line
line , XY or YX
A plane is a flat surface
that has no thickness and
extends forever.
A script capital letter
or three points not
on a line
plane R or plane ABC
Undefined Terms
Points that lie on the same line are collinear. K , L ,
and M are collinear. K, L, and N are noncollinear.
Points that lie in the same plane are coplanar.
Otherwise they are noncoplanar.
E X A M P L E 1 Naming Points, Lines, and Planes
A Name four coplanar points.
K, L, M, and N all lie in plane R.
B Name three lines.
AB , BC , and CA .
1. Use the diagram to
name two planes.
1-
Understanding Points,
Lines, and Planes
A plane may be
named by any three
noncollinear points
on that plane. Plane
ABC may also be
named BCA , CAB ,
CBA , ACB , or BAC.
Objectives
Identify, name, and draw
points, lines, segments,
rays, and planes.
Apply basic facts about
points, lines, and planes.
Vocabulary
undefined term
point
line
plane
collinear
coplanar
segment
endpoint
ray
opposite rays
postulate
Who uses this?
Architects use representations of points, lines, and
planes to create models of buildings. Interwoven
segments were used to model the beams of
Beijing’s National Stadium for the 2008 Olympics.
Refer to the design in the roof of
Beijing’s National Stadium.
NY Performance
Indicators
G.CM.11 Understand and
use appropriate language,
representations, and terminology
when describing objects,
relationships, mathematical
solutions, and geometric
diagrams. Also, G.CN.6.
Possible answer:
plane R and plane ABC
ge07se_c01_0006_0011.indd 1 11/23/06 1:23:25 PM
6 Chapter 1
New York Performance
Indicators
Process
G.CM.11 Understand and use
appropriate language, representations,
and terminology when describing
objects, relationships, mathematical
solutions, and geometric diagrams
G.CN.6 Recognize and apply
mathematics to situations in the
outside world
Motivate
Point out different objects in the classroom that
are representations of points, segments, and
planes, such as the tips of pushpins on the bul-
letin board, rulers, and desktops. Discuss with stu-
dents what these items have in common.
Ask students to give examples of other objects
that have these same characteristics and can be
found in the world around them, such as the loca-
tions of cities on a map, the lines on a football
field, and the bases on a baseball field.
Introduce
1
Explorations and answers are provided in the
Explorations binder.
1-1 Organizer
Pacing: Traditional 1 day
Block
__^1
day
Objectives: Identify, name, and
draw points, lines, segments, rays,
and planes.
Apply basic facts about points,
lines, and planes.
G
I
D
@<I
Online Edition
Tutorial Videos, Interactivity
Countdown to
Testing Week 1
Warm Up
Graph each inequality.
1. x ≥ 3
Ó ä Ó {
2. 2 ≤ x ≤ 6
ä Ó { È
3. x < 1 or x > 0
ä £
Also available on transparency
Q: What did the little acorn say
when it grew up?
A: “Geometry”
E X P L O R AT I O N
4HE FOLLOWING ARE SOME OF THE TERMS USED AS THE BASIC
BUILDING BLOCKS OF GEOMETRY $ESCRIBE A REAL WORLD OBJECT THAT
IS SUGGESTED BY EACH TERM
POINT
LINE
PLANE
4ELL WHETHER EACH OF THE FOLLOWING IS MOST LIKE A POINT A LINE
OR A PLANE
A A DESKTOP
B A SPECK OF DUST
C A JET CONTRAIL
/ Ê Ê - 1--
%XPLAIN WHAT IS REPRESENTED BY THE ARROWHEAD AT EACH END
OF THE LINE IN 0ROBLEM
$ISCUSS WHETHER THE SURFACE OF %ARTH CAN BE AN EXAMPLE OF
A PLANE
1iÀÃÌ>}Ê*ÌÃ]Ê
iÃ]Ê>`Ê*>iÃ
1- 1 Understanding Points, Lines, and Planes 7
DEFINITION NAME DIAGRAM
A segment , or line segment,
is the part of a line consisting
of two points and all points
between them.
The two endpoints
AB or
BA
An endpoint is a point at
one end of a segment or the
starting point of a ray.
A capital letter
C and D
A ray is a part of a line
that starts at an endpoint
and extends forever in one
direction.
Its endpoint and any other
point on the ray
RS
- ,
Opposite rays are two
rays that have a common
endpoint and form a line.
The common endpoint and
any other point on each ray
EF and EG
Segments and Rays
E X A M P L E 2 Drawing Segments and Rays
Draw and label each of the following.
A a segment with endpoints U and V
B opposite rays with a common endpoint Q
2. Draw and label a ray with endpoint M that contains N.
A postulate , or axiom, is a statement that is accepted as true without proof.
Postulates about points, lines, and planes help describe geometric properties.
1-1-1 Through any two points there is exactly
one line.
1-1-2 Through any three noncollinear points there
is exactly one plane containing them.
1-1-3 If two points lie in a plane, then the line
containing those points lies in the plane.
Postulates Points, Lines, and Planes
E X A M P L E 3 Identifying Points and Lines in a Plane
Name a line that passes through two points.
There is exactly one line n passing through
G and H.
3. Name a plane that contains three noncollinear points.
Possible answer: plane GHF
M N
ge07se_c01_0006_0011.indd 7 12/2/05 7:08:00 PM
Lesson 1-1 7
Guided Instruction
Show students how to draw a plane.
Discuss all the different ways of labeling a
line, segment, ray, and plane. Explain that
CD and DC are not the same ray.
Visual Point out to students that
the diagrams of planes, lines, and
rays extend forever, but we can-
not draw them that way.
Teach
2
Through Modeling
Have students work in small groups to use
physical models, such as raw spaghetti and
index cards, to illustrate Postulates 1-1-3,
1-1-4, and 1-1-5. Use contrasting colors of
paper to emphasize the different planes in
Postulate 1-1-5.
Example 1
A. Name four coplanar points.
A, B, C, D
B. Name three lines.
Possible answer: AE , BE , CE
Example 2
Draw and label each of the
following.
A. segment with endpoints M
and N
B. opposite rays with common
endpoint T
Example 3
Name a line that passes through
two points. XY
Also available on transparency
Additional Examples
Reading Math
The phrase
“exactly one”
means that one exists
and it is unique.
E NGLISH
L ANGUAGE
L EARNERS
INTERVENTION
Questioning Strategies
EX AM P LE 1
- What are the different ways you
can name the planes represented
in the stadium roof?
- How else could you name the
lines?
EX AM P LE 2
- What figure is formed by two
opposite rays? Is there a way to
name the opposite rays? Give an
example.
EX AM P LE 3
- Are there other ways to name the
line determined by the points in
the diagram? Explain.
1- 1 Understanding Points, Lines, and Planes 9
Exercises Exercises
1-
GUIDED PRACTICE
Vocabulary Apply the vocabulary from this lesson to answer each question.
1. Give an example from your classroom of three collinear points.
2. Make use of the fact that endpoint is a compound of end and point and name
the endpoint of ST .
S E E E X A M P L E 1
p. 6
Use the figure to name each of the following.
3. five points
4. two lines
5. two planes
6. point on BD
S E E E X A M P L E 2
p. 7
Draw and label each of the following.
7. a segment with endpoints M and N
8. a ray with endpoint F that passes through G
S E E E X A M P L E 3
p. 7
Use the figure to name each of the following.
9. a line that contains A and C
10. a plane that contains A, D, and C
S E E E X A M P L E 4
p. 8
Sketch a figure that shows each of the following.
11. three coplanar lines that intersect in a common point
12. two lines that do not intersect
PRACTICE AND PROBLEM SOLVING
For See
Exercises Example
Independent Practice Use the figure to name each of the following.
13. three collinear points
14. four coplanar points
15. a plane containing E
Draw and label each of the following.
16. a line containing X and Y
17. a pair of opposite rays that both contain R
Use the figure to name each of the following.
18. two points and a line that lie in plane T
19. two planes that contain
Sketch a figure that shows each of the following.
20. a line that intersects two nonintersecting planes
21. three coplanar lines that intersect in three
different points
KEYWORD: MG7 1-
KEYWORD: MG7 Parent
Skills Practice p. S
Application Practice p. S
Extra Practice
Possible answer: the intersection of 2 floor tiles
S
A, B, C, D, E
Possible answer: AC , BD
Possible answer: ABC and N
Possible answer: B, C, or D
Possible answer:
AB
Possible answer: plane ABD
B, E, A
Possible answer:
B, C, D, E
Possible answer: plane ABC
Possible answer: G, J, and
Possible answer: planes T and S
M N
F G
A B
C D
X
Y
R
R
S
ge07se_c01_0006_0011.indd 9 12/2/05 7:08:05 PM
Lesson 1-1 9
Exercises Exercises
1-
Assignment Guide
Assign Guided Practice exercises
as necessary.
If you finished Examples 1–
Basic 13–17, 33
Average 13–17, 33, 36, 43
Advanced 13–17, 33, 34, 36,
If you finished Examples 1–
Basic 13–28, 31–33, 39–42,
Average 13–28, 30–33, 35–43,
Advanced 14–28 even, 29–
Homework Quick Check
Quickly check key concepts.
Exercises: 14–20 even, 24, 26,
Communicating Math In
Exercises 3–6, students
sometimes forget to place
a symbol above the letters that are
used to name lines, segments, and
rays. Remind students that two let-
ters without a symbol represent
a distance.
KEYWORD: MG7 Resources
10 Chapter 1 Foundations for Geometry
Draw each of the following.
23. plane H containing two lines that intersect at M
24. ST intersecting plane M at R
Use the figure to name each of the following.
25. the intersection of TV and US
26. the intersection of US and plane R
27. the intersection of
TU and
UV
Write the postulate that justifies each statement.
28. The line connecting two dots on a sheet of paper lies on the same sheet of paper as
the dots.
29. If two ants are walking in straight lines but in different directions, their paths cannot
cross more than once.
30. Critical Thinking Is it possible to draw three points that are noncoplanar? Explain.
Tell whether each statement is sometimes, always, or never true. Support your
answer with a sketch.
31. If two planes intersect, they intersect in a straight line.
32. If two lines intersect, they intersect at two different points.
33. AB is another name for BA .
34. If two rays share a common endpoint, then they form a line.
35. Art Pointillism is a technique in which tiny dots of
complementary colors are combined to form a picture.
Which postulate ensures that a line connecting two of
these points also lies in the plane containing the points?
36. Probability Three of the labeled
points are chosen at random.
What is the probability that
they are collinear?
37. Campers often use a cooking stove with three legs.
Which postulate explains why they might prefer this
design to a stove that has four legs?
38. Write About It Explain why three coplanar lines may
have zero, one, two, or three points of intersection.
Support your answer with a sketch.
© 2004 Artists’ Rights Society (ARS), New York/ADAGP,Paris/Reunion des Musees Nationaux/Art Resource, NY/Detail
22. This problem will prepare you for the
Multi-Step Test Prep on page 34. Name an
object at the archaeological site shown that
is represented by each of the following.
a. a point
b. a segment
c. a plane
U
22. Possible
answers:
a. tip of a stake
b. string
c. grid formed
by string
28. If 2 pts. lie in
a plane, then the
line containing
those pts. lies
in the plane.
If 2 lines intersect, then they intersect in exactly 1 pt.
U
A
N
A
S
35. Post. 1-1-
_^1
U
30. It is not
possible. By
Post. 1-1-2, any
3 noncollinear pts.
are contained in
a unique plane.
If the 3 pts. are
collinear, they
are contained in
infinitely many
planes. In either
case, the 3 pts.
will be coplanar.
M
S A N
Post. 1-1-
ge07se_c01_0006_0011.indd 10 4/27/06 11:59:35 AM
10 Chapter 1
££
,%33/. ,iÌi>V
5NDERSTANDING 0OINTS ,INES AND 0LANES
! POINT HAS NO SIZE )T IS NAMED USING A CAPITAL LETTER
!LL THE FIGURES BELOW CONTAIN POINTS
&IGURE #HARACTERISTICS $IAGRAM 7ORDS AND 3YMBOLS
LINE ENDPOINTS
EXTENDS FOREVER
IN TWO DIRECTIONS
! "
LINE !" OR
j
??
k
LINE SEGMENT
OR SEGMENT
ENDPOINTS
HAS A FINITE LENGTH
8 9
SEGMENT 89 OR
?
RAY ENDPOINT
EXTENDS FOREVER
IN ONE DIRECTION
1 2
RAY 21 OR
???
k
! RAY IS NAMED STARTING WITH ITS ENDPOINT
PLANE EXTENDS FOREVER
IN ALL DIRECTIONS
& ('
PLANE &'( OR PLANE
$RAW AND LABEL A DIAGRAM FOR EACH FIGURE
POINT 7 LINE -.
s 7
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?
??
k
*** + % &**
.AME EACH FIGURE USING WORDS AND SYMBOLS
#
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4 3
LINE ^ #$ OR
j
k #$ (^) RAY 34 OR
k 34
.AME THE PLANE IN TWO DIFFERENT WAYS
7
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,
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.
PLANE ^ ,-.^ PLANE ^ SEGMENT ^78 ^
78
(^) s 0
POINT 0
££^ 1-1 RETEACH
,%33/. ,i>`}Ê-ÌÀ>Ìi}ià )DENTIFY 2ELATIONSHIPS
)N GEOMETRY MANY FIGURES CAN BE BROKEN DOWN INTO THREE BASIC BUILDING
BLOCKS POINTS LINES AND PLANES
#ONSIDER THE FOLLOWING
s ! POINT NAMES A LOCATION AND HAS NO SIZE s 0
s ! LINE IS A STRAIGHT PATH THAT EXTENDS FOREVER IN TWO DIRECTIONS
! "
s ! PLANE IS A FLAT SURFACE THAT EXTENDS FOREVER .
/
-
5SE THAT INFORMATION TO THINK ABOUT THE FOLLOWING DEFINITIONS
s ! SEGMENT IS PART OF A LINE AND CONSISTS OF TWO POINTS AND
&
'
ENDPOINT
ENDPOINT
ALL THE POINTS IN BETWEEN
s !N ENDPOINT IS A POINT AT THE END OF A SEGMENT OR A RAY
s ! RAY IS A PART OF A LINE THAT STARTS AT AN ENDPOINT AND
+
*
EXTENDS FOREVER IN ONE DIRECTION
!NSWER EACH QUESTION
(OW ARE A LINE AND A LINE SEGMENT THE SAME
! LINE SEGMENT IS A SPECIFIC PORTION OF A LINE THAT BEGINS AND ENDS
(OW ARE A LINE AND A LINE SEGMENT DIFFERENT
! LINE GOES ON FOREVER IN BOTH DIRECTIONS WHILE A SEGMENT HAS ENDPOINTS
(OW ARE A LINE SEGMENT AND A RAY THE SAME
! RAY IS A PART OF A LINE THAT STARTS AT THE ENDPOINT AND EXTENDS IN ONE
(OW ARE A LINE SEGMENT AND A RAY DIFFERENT
! LINE SEGMENT HAS ENDPOINTS A RAY EXTENDS FOREVER IN ONE DIRECTION
)DENTIFY THE FOLLOWING
s 9 !
"
^1
0
POINT^ SEGMENT^ RAY
3
4
-.
+
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ENDPOINT LINE PLANE
DIRECTION FOREVER
1-1 READING STRATEGIES
££
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.AME A PLANE 0OSSIBLE ANSWER PLANE ^ "#$
"
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.AME A SEGMENT ^
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? "#
? "% OR
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.AME A LINE 0OSSIBLE ANSWER
j
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.AME THREE COLLINEAR POINTS
POINTS ^ "^ #^ AND ^ %^
.AME THREE NONCOLLINEAR POINTS
0OSSIBLE ANSWER POINTS ^ "^ #^ AND ^ $
.AME THE INTERSECTION OF A LINE AND A SEGMENT NOT ON THE LINE POINT ^ "
.AME A PAIR OF OPPOSITE RAYS
k "# AND
k "%
5SE THE FIGURE FOR %XERCISES n
.AME THE POINTS THAT DETERMINE PLANE
M
7
8 9
:
POINTS 8 9 AND :
.AME THE POINT AT WHICH LINE M INTERSECTS
PLANE POINT :
.AME TWO LINES IN PLANE THAT INTERSECT LINE M
j
k 8: AND
j
k 9:
.AME A LINE IN PLANE THAT DOES NOT INTERSECT
LINE M ^
j
k 89
$RAW YOUR ANSWERS IN THE SPACE PROVIDED
-ICHELLE +WAN WON A BRONZE MEDAL IN FIGURE SKATING AT THE 3ALT ,AKE
#ITY 7INTER /LYMPIC 'AMES
-ICHELLE SKATES STRAIGHT AHEAD FROM POINT , AND STOPS
, -
AT POINT - $RAW HER PATH
-ICHELLE SKATES STRAIGHT AHEAD FROM POINT , AND CONTINUES
,
RAY
-
THROUGH POINT - .AME A FIGURE THAT REPRESENTS HER PATH
$RAW HER PATH
-ICHELLE AND HER FRIEND !LEXEI START BACK TO BACK AT POINT ,
+ , -
AND SKATE IN OPPOSITE DIRECTIONS -ICHELLE SKATES THROUGH
POINT - AND !LEXEI SKATES THROUGH POINT + $RAW THEIR PATHS
*À>VÌViÊ 5NDERSTANDING 0OINTS ,INES AND 0LANES
1-1 PRACTICE C
1-1 PRACTICE B
1-1 PRACTICE A
Exercise 22 involves
identifying geometric
shapes at an archae-
ological site. This exercise prepares
students for the Multi-Step Test Prep
on page 34.
Visual For Exercise 36,
have students make an
organized list of all the
combinations of three points that
could be chosen and then select
those that are collinear.
Answers
R
S
T
A B
38. Lines may not
intersect: 0 pts.
of intersection.
All 3 lines may
intersect in 1 pt.
Two of the lines
may not intersect,
but they might each
intersect a third line.
Each line may
intersect each of
the other lines.
12 Chapter 1 Foundations for Geometry
1-
Use with Lesson 1-
Activity
1 Construct a segment and label its endpoints A and C.
2 Create point B on
AC.
3 Measure the distances from A to B and from B to C.
Use the Calculate tool to calculate the sum of AB
and BC.
4 Measure the length of
AC. What do you notice
about this length compared with the measurements
found in Step 3?
5 Drag point B along
AC. Drag one of the endpoints
of
AC. What relationships do you think are true
about the three measurements?
6 Construct the midpoint of
AC and label it M.
7 Measure
AM and
MC. What relationships do you
think are true about the lengths of
AC ,
AM , and
MC?
Use the Calculate tool to confirm your findings.
8 How many midpoints of
AC exist?
Try This
1. Repeat the activity with a new segment. Drag each of the points in your figure
(the endpoints, the point on the segment, and the midpoint). Write down any
relationships you observe about the measurements.
2. Create a point D not on
AC. Measure
AD ,
DC , and
AC. Does AD + DC = AC?
What do you think has to be true about D for the relationship to always be true?
Explore Properties
Associated with Points
The two endpoints of a segment determine its length. Other points on the
segment are between the endpoints. Only one of these points is the midpoint
of the segment. In this lab, you will use geometry software to measure
lengths of segments and explore properties of points on segments.
KEYWORD: MG7 Lab
NY Performance
Indicators
G.RP.3, G.R.
Check students’ work.
No; D must be between A and C.
ge07se_c01_0012.indd 1 11/23/06 1:24:18 PM
12 Chapter 1
New York Performance
Indicators
Process
G.RP.3 Investigate and evaluate
conjectures in mathematical terms,
using mathematical strategies to reach
a conclusion
G.R.1 Use physical objects, diagrams,
charts, tables, graphs, symbols,
equations, or objects created using
technology as representations of
mathematical concepts
Pacing:
Traditional
__ 1
day
Block
__ 1
day
Objective: Use geometry
software to measure distances and
explore properties of points on
segments.
Materials: geometry software
G
I
D
@<I
Online Edition
TechKeys
Countdown to
Testing Week 1
Resources
Technology Lab Activities
1-2 Lab Recording Sheet
Teach
Discuss
Emphasize that there are infinitely
many points on a line segment
between its endpoints, but only one
of these points is the midpoint.
Close
Key Concept
When three points A , B , and C are
collinear, then AB + BC = AC. The
midpoint of a segment is equidistant
from each endpoint.
Assessment
Journal Have students summarize
how they determine, given three col-
linear points, if the point between
the endpoints is the midpoint. Then
have them support their answer with
a sketch.
To introduce the concept of
midpoint, I like to tell the fol-
lowing riddle to the class.
Q: A hunter walks into the
woods and keeps walking
in the same direction. At
what point does the hunter
begin to leave the woods?
A: The midpoint.
Then I can explain to students
that once the hunter is half-
way through the woods, she is
as close to the end as to the
beginning. After passing the
midpoint, the hunter is going
out instead of in. This helps
students remember that the
midpoint is not just any point
between the endpoints, but
exactly one point in the middle.
Teresa Salas
Corpus Christi, TX
Teacher to Teacher
Organizer
Use with Lesson 1-
1- 2 Measuring and Constructing Segments 13
Why learn this?
You can measure a segment to calculate the
distance between two locations. Maps of a
race are used to show the distance between
stations on the course. (See Example 4.)
A ruler can be used to measure the distance between
two points. A point corresponds to one and
only one number on the ruler. This number
is called a coordinate. The following
postulate summarizes this concept.
The points on a line can be put
CM
into a one-to-one correspondence
with the real numbers.
Postulate 1-2-1 Ruler Postulate
The distance between any two points is the absolute value of the difference
of the coordinates. If the coordinates of points A and B are a and b , then the
distance between A and B is a - b ^ or b - a . The distance between A and B
is also called the length of
AB , or AB.
Ê Ê ] >Ê ÊL ] Ê Ê ] LÊ Ê> ]
> L
E X A M P L E 1 Finding the Length of a Segment
{°x
{ Î Ó £ ä £ Ó Î { x
Find each length.
A DC B EF
DC = 4.5 - 2 EF = - 4 - (- 1 )
= 2.5^ = - 4 + 1
Find each length.
Ó
£ÊÊ
£
pÊ Ê
Ó
ÊÊ
1a. XY Î £ ä £ Ó Î { x È
1b. XZ
Congruent segments are segments that have
/VÊ>ÀÃ
the same length. In the diagram, PQ = RS , so you
can write
_
PQ
_
RS. This is read as “segment PQ
is congruent to segment RS .” Tick marks are
used in a figure to show congruent segments.
Objectives
Use length and midpoint
of a segment.
Construct midpoints and
congruent segments.
Vocabulary
coordinate
distance
length
congruent segments
construction
between
midpoint
bisect
segment bisector
PQ represents a
number, while
_
PQ
represents a geometric
figure. Be sure to
use equality for
numbers ( PQ = RS )
and congruence for
figures (
_
PQ
_
RS ).
1-
Measuring and
Constructing Segments
NY Performance
Indicators
G.PS.5 Choose an effective
approach to solve a problem
from a variety of strategies
(numeric, graphic, algebraic).
Also, G.RP.2.
_ 1
_^1
ge07se_c01_0013_0019.indd 1 11/23/06 1:25:08 PM
Lesson 1-2 13
1-2 Organizer
Pacing: Traditional
(^1) __
day
Block
__^1
day
Objectives: Use length and
midpoint of a segment.
Construct midpoints and congruent
segments.
Geometry Lab
In Geometry Lab Activities
G
I
D
@<I
Online Edition
Tutorial Videos
Countdown to
Testing Week 1
Warm Up
Simplify.
1. 7 - (- 3 )^ 10 2. - 1 - (- 13 )^ 12 3. - 7 - 1 ^8
Solve each equation.
4. 2 x + 3 = 9 x - 11 2
5. 3 x = 4 x - 5 5
6. How many real numbers are
there between
__^1
and
__^3
infinitely many
Also available on transparency
Carpenter 1: Why did you leave your
tape measure in the truck?
Carpenter 2: Because we’re entering
a construction zone.
E X P L O R AT I O N
5SE GEOMETRY SOFTWARE OR A RULER FOR THE FOLLOWING
%XPLORATION
$RAW
!# 4HEN PLACE " ON THE SEGMENT BETWEEN ! AND #
-EASURE
?
"# AND
!# IN INCHES 4HEN MEASURE EACH
SEGMENT IN CENTIMETERS AND MILLIMETERS 2ECORD THE LENGTHS IN
A TABLE LIKE THE ONE SHOWN
5NIT ,ENGTH OF !" ,ENGTH OF "# ,ENGTH OF !#
)NCH
#ENTIMETER
-ILLIMETER
/ Ê Ê - 1--
$ESCRIBE THE RELATIONSHIP AMONG THE LENGTHS OF !" "# AND
!# )S THIS RELATIONSHIP THE SAME NO MATTER WHAT MEASURE
MENT UNIT YOU USE
$ISCUSS WHAT WOULD HAPPEN IF YOU LOCATED $ BETWEEN ! AND
" 7HAT WOULD BE THE RELATIONSHIP AMONG THE LENGTHS OF !$
$" "# AND !#
i>ÃÕÀ}Ê>`Ê
ÃÌÀÕVÌ}Ê-i}iÌÃ
£Ó
Motivate
Show students a road map. Ask them to locate a
place that is midway between two other places.
Ask questions about distance between locations
on a straight road. “Suppose you are driving to a
relative’s house. The only place to stop for food is
75 miles from home, and it is
__ 1
of the total trip.
How far from the food stop does your relative
live?” 150 mi
Introduce
1
Explorations and answers are provided in the
Explorations binder.
New York Performance
Indicators
Process
G.PS.5 Choose an effective approach
to solve a problem from a variety of
strategies (numeric, graphic, algebraic)
G.RP.2 Recognize and verify, where
appropriate, geometric relationships
of perpendicularity, parallelism,
congruence, and similarity, using
algebraic strategies
