Segments, Rays, 1-4 Parallel Lines and Planes
1-4
Segments, Rays,
Parallel Lines and Planes
What You'll Learn
? To identify segments and
rays
? To recognize parallel lines
. . . And Why
To identify compass directions that can be represented by opposite rays, as in Exercise 36
Check Skills You'll Need
GO for Help Lesson 1-3
Judging by appearances, will the lines intersect?
1.
no
2.
yes
3. no
Name the plane represented by each
P
surface of the box.
K
4. the bottom NMR 5. the top PQL
6. the front NKL 7. the back PQR
S
8. the left side PKN 9. the right side LQR N
Q L
R M
New Vocabulary ? segment ? ray ? opposite rays ? parallel lines ? skew lines ? parallel planes
1 Identifying Segments and Rays
Many geometric figures, such as squares and angles, are formed by parts of lines called segments or rays. A segment is the part of a line consisting of two endpoints and all points between them.
A ray is the part of a line consisting of one endpoint and all the points of the line on one side of the endpoint.
Segment AB
A
B
Endpoint
AB
Endpoint
Ray YX
X
Y
YX Endpoint
Opposite rays are two collinear rays with the same endpoint. Opposite rays always form a line.
Q)
)R
S
RQ and RS are opposite rays.
Real-World Connection
A sunbeam models a ray. The sun is its endpoint.
Quick Check
1 EXAMPLE Naming Segments and Rays
Name the segments and rays in the figure at the right.
? The three segments a)re LP, )PQ, a) nd L) Q. ? The) four ra) ys are LP or LQ , PQ , PL , and
QP or QL .
Q LP
))
1 Critical Thinking LP and PL form a line. Are they opposite rays? Explain.
No, they do not have the same endpoint.
Lesson 1-4 Segments, Rays, Parallel Lines and Planes
23
1-4
1. Plan
Objectives
1 To identify segments and rays 2 To recognize parallel lines
Examples
1 Naming Segments and Rays 2 Identifying Parallel and Skew
Segments 3 Identifying Parallel Planes
Math Background
The undefined terms point, line, and plane form the basis for the definitions of ray, segment, and parallel planes. Together these terms form the beginning vocabulary for the study of geometry. Euclid used this approach in Book 1 of The Elements.
More Math Background: p. 2C
Lesson Planning and Resources
See p. 2E for a list of the resources that support this lesson.
PowerPoint
Bell Ringer Practice
Check Skills You'll Need For intervention, direct students to: Basic Postulates of Geometry Lesson 1-3: Examples 3, 4 Extra Skills, Word Problems,
Proof Practice, Ch. 1
) ) Special Needs
Draw line AB on
L1 the board
and
ask:
Are
AB)
and
BA)
opposite rays? Why or why not? No, AB and BA are
not opposite rays because they do not have only
one endpoint in common, they share many points.
Below Level L2 Remind students that the different notations for line, line segment, and ray readily identify and distinguish them.
learning style: visual
learning style: verbal
23
2. Teach
Guided Instruction
Teaching Tip Point out that the first letter naming a ray is always its endpoint. The second letter is any other point on the ray. Emphasize that opposite rays are two distinct collinear rays with only their endpoints in common.
1 EXAMPLE Visual Learners Remind students to associate the notations for line, segment, and ray with the actual figures.
PowerPoint
Additional Examples
1 Name the segments and rays in the figure.
A
B C
BA, BC, BA), BC)
Guided Instruction
2 EXAMPLE Tactile Learners Some students may have trouble visualizing skew lines from the figure shown. Provide physical models for these students.
12 Recognizing Parallel Figures
Vocabulary Tip
You
read
*)
AB
6
*)
EF
as
"line
AB is parallel to line EF."
Lines that do not intersect may or may not be coplanar.
Parallel lines are coplanar lines that do not intersect. Skew lines are noncoplanar; therefore, they are not parallel and do not intersect.
D A
H E
C B
G F
* )* ) *AB) 6 EF* )
AB and CG are skew.
nline
Visit: Web Code: aue-0775
Segments or rays are parallel if they lie in p* ara)llel lin* es.)They are skew if they lie in
skew lines. AB and CG are skew because AB and CG are skew.
2 EXAMPLE Identifying Parallel and Skew Segments
a. Name all labeled segments that are parallel to DC.
AB, GH, and JI are parallel to DC.
A D
B C
b. Name all labeled segments that are skew to DC.
NJ, GJ, and HI are skew to DC.
G
H
N
J
I
Quick Check
2 Use the diagram in Example 2. a. Name all labeled segments that are parallel to GJ. HI, DN b. Name all labeled segments that are skew to GJ. AB, CD, CH c. Name another pair of parallel segments; of skew segments. DN, HI; DN, HC
Parallel planes are planes that do not intersect. A line and a plane that do not intersect are also parallel.
G
A J
B
H
Plane ABCD 6 *Plan)e GHIJ.
Plane ABCD 6 GH.
I
D C
24 Chapter 1 Tools of Geometry
Advanced Learners L4 Have students justify the statement, "Skew lines are noncoplanar; therefore they are not parallel and do not intersect." They may need to reason indirectly.
English Language Learners ELL Use Exercises 25-33 to reinforce the meaning of the new vocabulary in the lesson as well as the terms always, sometimes, and never in the context of mathematical reasoning.
24
learning style: verbal
learning style: verbal
3 EXAMPLE Identifying Parallel Planes
Use the diagram at the right to name
G
the figures.
a. two pairs of parallel planes
A
J
plane ABHG 6 plane DCIJ
plane ADJ 6 plane BCI
D
b. *a lin) e that is parallel to plane GHIJ AB is parallel to GHIJ.
Quick Check 3 Name the figures.
S
3a. PSWT n RQVU, PRUT n SQVW, PSQR n TWVU
a. three pairs of parallel planes
b. a line that is parallel to plane QRUV
P
4
Answers may vary. Sample: PS
W
T
B C
R U
H I
Q V
EXERCISES
For more exercises, see Extra Skill, Word Problem, and Proof Practice.
Practice and Problem Solving
A Practice by Example
GO
for Help
Example 1 (page 23)
Example 2 (page 24)
Example 3 (page 25)
13. Answers may vary. 44
Sample: CD, AB 44 4 14. BG, DH , CL
Use the figure at the right for Exercises 1?3.
RS, RT, RW, ST, SW, TW 1. Name all the labeled segments.
RS
SSS SSS 2. Name all the labeled rays. RS, ST, TW, WT, TS, SR
3. a. Name a pair of opposite rays with T
S SS
as an endpoint. TS or TR, TW
SS
b. Name another pair of opposite rays. SR, ST
TW
Name all segments shown in the diagram that are
parallel to the given segment.
B
4. AC DF
5. EF BC
6. AD BE, CF
E D
F
Name all segments shown in the diagram that are skew to the given segment.
AC
Exercises 4?11
7. AC DE, EF, BE 8. EF AD, AB, AC 9. AD BC, EF
10 and 11. Answers may vary. Samples are given. Use the diagram above and name a pair of figures to match each description.
ABC n DEF
4
10. parallel planes
11. a line and a plane that are parallel BC, DEF
Use the figure at the right to na*me)th4e following.
C
12. all lines that are parallel to* A)B FG
13. two lines that are skew to EJ
J
D
L
FG
H
14. all lines that are parallel to plane JFAE
E
4 15. the intersection of plane FAB and plane FAE AF
AB
PowerPoint
Additional Examples
2 Use the figure from Example 3. Name all segments that are parallel to AD. Name all that are skew to AD. parallel: GJ, HI, BC; skew: GH, JI, BH, CI
3 Identify a pair of parallel planes in your classroom. Sample: floor and ceiling
Resources
? Daily Notetaking Guide 1-4 L3
? Daily Notetaking Guide 1-4--
Adapted Instruction
L1
Closure
How are parallel and skew lines alike? How are they different? Both parallel and skew lines never intersect; parallel lines are coplanar, whereas skew lines are not.
Lesson 1-4 Segments, Rays, Parallel Lines and Planes
25
25
3. Practice
Assignment Guide
1 A B 1-3, 34-36
2 A B 4-33, 37-39
C Challenge
40-45
Test Prep Mixed Review
46-50 51-66
Homework Quick Check
To check students' understanding of key skills and concepts, go over Exercises 2, 6, 22, 35, 38.
Error Prevention!
TERxe) arcnisdeS2WS) taurdeeonptps omsaityetrhaiynsk.
that Ask:
How many points do opposite rays
have in common? exactly 1 What
is it? the endpoint of both rays
? Pearson Education, Inc. All rights reserved.
GPS Guided Problem Solving
L3
Enrichment
L4
Reteaching
L2
Adapted Practice
L1
PNamreactice
Class
Date
L3
Practice 1-4
If GJ 32, find the value of each of the following.
1. x 2. GH 3. HJ
3x
x 16
G
H
J
Measuring Segments and Angles
4. Find PD if the coordinate of P is -7 and the coordinate of D is -1. 5. Find SK if the coordinate of S is 17 and the coordinate of K is -5. 6. Find the coordinate of B if AB = 8 and the coordinate of A is -2. 7. Find the coordinate of X if XY = 1 and the coordinate of Y is 0.
8. Name the angle at the right in three different ways.
M
O
1
P
If AX 45, find the value of each of the following.
9. y 10. AQ 11. QX
2y 1
y1
A
Q
X
Find the measure of each angle. 12. &EBF 13. &EBA 14. &DBE 15. &DBC 16. &ABF 17. &DBF
D
E
F
18. Name all acute angles in the figure. 19. Name all obtuse angles in the figure. 20. Name all right angles in the figure.
73
39
A
B
C
21. If AC = 62, find the value of x. Then find AB and BC.
22. If AC = 206, find the value of x. Then find AB and BC.
3x 4
3x 4
A
B
C
24.
J
F E
D
26
B Apply Your Skills
21. False; they intersect above pt. A.
In Exercises 16?23, describe the statement as
true* or)fal*se. I)f false, explain. 16. *CB ) 6 H* G ) true
17. ED 6 HG False; they are skew.
18. plane AED 6 plane FGH true
H
B
I
G
A
CJ
F
E
D
19. *plan) e AB* H 6) plane CDF
False; they intersect above CG.
*) *)
20. *AB) and *HG) are skew lines. true
21. *AE) and*BC) are skew lines. See left.
22. CG and AI are skew lines.
23. CF and AJ are skew lines.
False; they are n.
False; they are n.
*)
24. The following steps show how to draw planes A and B intersecting in FG.
Step 1
Step 2
Step 3
AG B
F
Vocabulary Tip
Always, sometimes, and never refer to all possible cases, not to intervals of time.
GO nline
Homework Help
Visit: Web Code: aue-0104
*)
Use similar steps to draw plane DFE and plane DFJ intersecting in DF. See margin.
Complete Exercises 25?33 with always, sometimes, or never to make
a true statement.
always
never
25. Two parallel lines are 9 coplanar.
always
27.
Two)
opposite)
rays 9 never
form
a
line.
29. GH and HG are 9 the same ray.
26. *Two) skew* lin) esaalwrea9 ys coplanar.
28.
TQ)
and
QT)
are 9 the same sometimes
line.
30. JK and JL are 9 the same ray.
31. Two planes that do not intersect are 9 parallel. always
32. Two lines that lie in parallel planes are 9 parallel. sometimes
33. Two lines in intersecting planes are 9 skew. sometimes
34. Multiple Choice FG has endpoints F(-3, 3) and G(3, 1). Which point is also
on FG? C
(-6, 4)
(-1, 2)
(0, 2)
(6, 0)
)
GPS 35. Coordinate Geometry AB has endpoint A(2, 3) )and cont)ains B(4, 6).
Give possible coordinates for point C so that AB and AC are opposite rays.
Graph your answer.
Answers may vary. Sample: (0, 0); check students' graphs.
36. Directional Compass On a directional compass, the directions
north and south can be represented by opposite rays.
a. Name two other compass directions that can be
represented by opposite rays.
b. What other pairs of opposite directions, if any,
can you find? a?b. See margin.
37. Open-Ended Summarize the three ways in which two lines may be related. Give examples from the real world that illustrate the relationships. See margin.
26 Chapter 1 Tools of Geometry
36. a. Answers may vary. Sample: northeast and southwest
b. Answers may vary. Sample: northwest and southeast, east and west
37. Two lines can be parallel, skew, or intersecting in one point. Samples: Train tracks ? parallel; vapor trail of a northbound jet and an eastbound jet at different altitudes ?
skew; streets that cross ? intersecting
39. b. Examples may vary: Sample: The floor and ceiling are parallel. A wall intersects both. The lines of intersection are parallel.
38. Answers may vary. Sample: Skew lines cannot be contained in one plane. Therefore, they have "escaped" a plane.
C Challenge
43. Answers may vary. 444
Sample: VR, QR, SR
38. Writing The term skew is a Middle English word meaning "to escape." Explain how this meaning might be appropriate for skew lines. See left.
39. Critical Thinking Suppose two parallel planes A and B are each intersected by a third plane C. a. Make a conjecture about the intersection of planes A and C and the intersection of planes B and C. The lines of intersection are parallel. b. Find examples in your classroom. See margin.
40. a. Draw a line. Draw points E and F on the line. How many different segments do points E and F determine? Name the segments. See margin.
b. Draw another line. Draw points E, F, and G on the line. How many segments do points E, F, and G determine? Name them. See margin.
c. Continue to draw lines, labeling one more point each time. Make a table showing the number of points and the number of segments determined. Look for and describe a pattern in the data. See margin.
d. Use your pattern to find how many segments are determined if you label 10 points on a line. 45 segments
e. If you label n points on a line, how many segments can you name? n(n 2 1)
2
Use the figure at the right for Exercises 41 and 42.
41. Do planes A and B have oth*er li)nes in common that are parallel to CD ?
A B
Explain. See margin.
42. Visualization Are there planes tha* t in)tersect planes A and B in lines parallel to CD ?
CD
Draw a sketch to support your answer. See margin.
The figure at the right is a pyramid.
V
43.
Name
three
lines
that
intersect*
at)
one point. 4
See
left.
44. What line could be parallel to PS ? QR
Q
P
45. Visualization Consider a plane through V that is
parallel to pla*ne )PQRS. Can a line*in t)hat plane be parallel *to S) R ? Can it intersect SR? Can it
S
R
be skew to SR? Explain each answer.
Yes; no; yes; explanations may vary.
Test Prep
Multiple Choice
Use the figure at the right for Exercises 46?49.
46. How many labeled segments are in the figure? D
A. 1
B. 4
)
47. Which) ray is opposite BC ? ) H
F. BE
G. BD
)
48. What )is another name for)CA ? B
A. AC
B. CB
C. 6
)
H. BA
)
C. CE
A B C DE
D. 10
)
J. AB
)
D. DC
49. Which figure could be the intersection of two planes? F
F. line
G. ray
H. point
J. segment
lesson quiz, , Web Code: aua-0104
Lesson 1-4 Segments, Rays, Parallel Lines and Planes
27
4. Assess & Reteach
PowerPoint
Lesson Quiz
Use the figure below for Exercises 1?3.
Q R
O
S
TP
1. Name the segments that form the triangle. RS, TR, ST
2. Name the rays that have point
TTOa)s, TthPe),irTeRn),dTpSo) int.
3. Explain how you can tell that no lines in the figure are parallel or skew. The three pairs of lines intersect, so they cannot be parallel or skew.
Use the figure below for Exercises 4 and 5.
CD A B
X
W Q
4. Name a pair of parallel planes. plane ABCD n plane XWQ
5. *NXWam).e*AaCli) noer *tBhDa)t is skew to
Alternative Assessment
Provide each student with a model of a rectangular solid, such as an empty cereal box. Have students describe how to find each of the following on the model: intersecting lines, parallel lines, skew lines, parallel planes, and intersecting planes.
40. a.
F
E
one segment, EF b.
EF G
3 segments, EF, EG, FG
c. Number Number of
of points segments
2
1
3
3
4
6
5
10
6
15
Answers may vary.
Sample: For each "new" point, the number of new segments equals the number of "old" points.
41. No; two different planes cannot intersect in more than one line.
42. yes; plane P, for example
P A B
CD
27
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