Download Parallel Lines and Angle Relationships

Parallel Lines and
Angle Relationships
Objective To explore and apply angle relationships.
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Calculating Total Price
• Add and subtract multidigit
whole numbers. Math Journal 1, p. 197
Students calculate percents of
numbers to find total prices that
include sales tax or tips.
• Determine angle measures by
applying definitions and properties of
angles, triangles, and quadrangles. [Geometry Goal 1]
• Identify congruent figures. Math Boxes 5 9
Math Journal 1, p. 198
Students practice and maintain skills
through Math Box problems.
[Geometry Goal 2]
Study Link 5 9
Key Activities
Students explore and apply the special
relationships between angles that are formed
when parallel lines are cut by a transversal.
Math Masters, p. 172
Geometry Template/protractor
Students practice and maintain skills
through Study Link activities.
Ongoing Assessment:
Recognizing Student Achievement
Use journal page 194. [Geometry Goal 1]
Ongoing Assessment:
Informing Instruction See page 383.
ENRICHMENT
Applying Angle Relationships
and Algebra
Math Masters, p. 173
compass straightedge
Students apply knowledge of angle
relationships and algebraic expressions
to find missing angle measures.
ENRICHMENT
Finding Linear Perspective
Students find examples of parallel lines
that are drawn in perspective.
ELL SUPPORT
Supporting Language with Drawings
straightedge
Students draw and label parallel lines
cut by a transversal and identify the
resulting angles.
Key Vocabulary
parallel lines parallel line segments skew transversal adjacent angles supplementary angles vertical angles
Materials
Math Journal 1, pp. 194 –196
Student Reference Book, pp. 163 and 233
Study Link 58
compass straightedge ruler
Advance Preparation
Ask the school’s art instructor to discuss the second optional Enrichment activity with your students.
Teacher’s Reference Manual, Grades 4–6 pp. 192–195
380
Unit 5
Geometry: Congruence, Constructions, and Parallel Lines
Interactive
Teacher’s
Lesson Guide
Differentiation Options
Ongoing Learning & Practice
Key Concepts and Skills
[Operations and Computation Goal 1]
Curriculum
Focal Points
Mathematical Practices
SMP1, SMP3, SMP5, SMP6, SMP7, SMP8
Getting Started
Mental Math and Reflexes
Students mentally solve problems like the following. If
time permits, have a volunteer share her or his strategy.
Suggestions and possible strategies:
133 + 55 + 47 235; 140 + 40 + 55
35 ∗ 12 420; (35 ∗ 10) + (35 ∗ 2)
360 - 155 205; 360 - 160 + 5
240 ÷ 5 48; Divide by 10, then multiply by 2
Content Standards
6.NS.6b
Math Message
Complete the problems on journal page 194.
Study Link 5 8 Follow-Up
Ask students to compare the coordinates of the
preimage to the coordinates of the reflected image
in Problem 3. Ask how the image coordinates would have been
different if the reflection had been over the x-axis. When a figure
is reflected over the y-axis, the y-coordinates remain the same
and the x-coordinates change signs. If the reflection had been
over the x-axis, the x-coordinates would have remained the same
and the y-coordinates would have changed signs.
1 Teaching the Lesson
► Math Message Follow-Up
WHOLE-CLASS
ACTIVITY
(Math Journal 1, p. 194)
Ongoing Assessment:
Recognizing Student Achievement
Journal
Page 194
Use journal page 194, Problems 1–6 to assess students’ ability to determine
angle measures by applying the properties of adjacent angles, supplementary
angles, and sums of angle measures in triangles and quadrangles. Students are
making adequate progress if they are able to solve Problems 1–6.
NOTE While discussing the angle measures
on journal page 194, you might want to
ask students if they can find a pair of
complementary angles. In Problem 2, the
vertical angles x and z are complementary
because the sum of their measures equals 90°.
[Geometry Goal 1]
Student Page
Algebraic Thinking When students have completed the page,
discuss their answers and reasoning. To be successful with the
content of this and the following lessons, it is important that
students understand the solutions to the problems on journal
page 194. Some students may benefit from reviewing pages 163
and 233 of the Student Reference Book and completing the Check
Your Understanding problems at the bottom of each page.
► Exploring Angle Relationships
Date
LESSON
5 9
Time
Angle Measures
Math Message
Write the measures of the angles indicated in Problems 1– 6.
Do not use a protractor.
1.
ma ⫽
120°
2.
mx ⫽
45°
3.
mp ⫽
20°
4.
mr ⫽
90°
60°
a
my ⫽
135°
mz ⫽
163 233
z
45°
135°
y x
p
PARTNER
ACTIVITY
70°
(Math Journal 1, p. 195)
SOLVING
ms ⫽
60°
mt ⫽
t
30°
r
Algebraic Thinking This is a discovery activity. No introduction is
necessary. Students who complete journal page 195 before the rest
of the class may go on to page 196.
s
120°
h
5.
mh ⫽
65°
80°
125°
75°
6.
md ⫽
mf ⫽
75°
75°
105°
105°
mg ⫽
me ⫽
75° g
f
d
e
75°
Math Journal 1, p. 194
Lesson 5 9
381
When most students have completed journal page 195, bring the
class together to discuss what they have discovered. Ask them to
define parallel lines and parallel line segments. Expect them
to mention the following properties:
Parallel lines never meet. They are always the same distance
apart; that is, the shortest distance between two parallel lines
is always the same, no matter where you measure the distance.
The lines on notebook paper and the rails on a long stretch of
straight railroad track are examples of parallel line segments.
Lines, line segments, or rays in the same plane (a flat surface
that extends forever) are parallel if they never cross or meet, no
matter how far they are extended.
Adjusting the Activity
Help students remember the definition of parallel by pointing out that the
three l’s in parallel are, in fact, parallel. Some students may also be interested in
the mathematical shorthand used for parallel lines or parallel line segments: Line
⎯⎯⎯. Consider
⎯⎯⎯ CD
segment AB is parallel to line segment CD is written as AB
reviewing the mnemonic devices for adjacent and supplementary angles from
Lesson 5-2 as well.
A U D I T O R Y
K I N E S T H E T I C
T A C T I L E
V I S U A L
Mention that if two lines (or line segments or rays) are in different
planes, they may be neither intersecting nor parallel. For example,
an east-west line on the floor and a north-south line on the ceiling
never meet, but they are not parallel either. These lines are
called skew.
Ask the class which pairs of lines on journal page 195 appear to
be parallel. The lines in Problems 1, 3, and 6
Students should observe the following:
When two parallel lines are cut by a transversal (a line that
intersects both parallel lines), the pattern of the angle
measures is the same at both intersections. There is no such
pattern if the lines are not parallel.
Student Page
Date
Time
LESSON
Parallel Lines and Angle Relationships
5 9
Without using a protractor, find the degree measure of each angle in
Problems 1–6 below. Write the measure inside the angle. Then circle
the figures in which 2 of the lines appear to be parallel.
1.
2.
70° b
d c
110°
3.
97°
83°
110°
70°
4.
i 83°
l k
97°
90° m
70° e 110°
110° h g 70°
n
90° p
d
c
95°
h g
85° 95° 95° 85°
r
90°w 90°
u
s
v
90° 90°
90°
6.
90° i
85°
85°e
90°q
90°
90°
5.
95° b
163
90°
l
90°
48°m n132°
k
132°
70°
r
t s
110°
110° 70°
p
48°
70°
x
110°
v
w
110° 70°
7.
A line that intersects 2 parallel lines is called a transversal. The angles
formed by 2 parallel lines and a transversal have special properties.
Refer to the picture of parallel lines below to describe one of these properties.
Example: Angles such as b and f, which lie on the same side of the transversal,
have the same measure.
Sample answers: Angles such as c and
f, which lie between the parallel lines
on the same side of the transversal, are
supplementary. Therefore, the sum of
their measures is 180°.
transversal
a
d
b
c
parallel lines
h
e f
g
Math Journal 1, p. 195
382
Unit 5
Geometry: Congruence, Constructions, and Parallel Lines
Student Page
Ask students to discuss their answers to Problem 7. Help them
summarize the patterns they found. Some possible conclusions
follow:
Date
LESSON
5 9
Time
Working with Parallel Lines
1.
Using only a compass and a straightedge, construct 2 parallel lines. Do this
construction without referring to the Student Reference Book. (Hint: This
construction involves copying an angle.)
2.
Draw 2 parallel lines using only a ruler and a pencil.
3.
Draw a parallelogram that is not a rectangle, using only a ruler and a pencil.
transversal
a
d
b
c
parallel lines
e
g
f
(From Problem 7)
Adjacent angles above or below parallel lines (such as angles
a and b in the diagram) are supplementary angles. The sum
of their measures is 180°.
There are two pairs of vertical angles at each intersection.
Angles in each pair have the same measure.
Angles between the parallel lines on the same side of
the transversal (such as angles c and f in the diagram)
are supplementary.
Math Journal 1, p. 196
Pairs of angles between the parallel lines and on opposite
sides of the transversal (such as angles d and f in the
diagram) have the same measure.
Pairs of angles outside the parallel lines on opposite sides of
the transversal (such as angles b and h in the diagram) have
the same measure.
Any two angles in the figure are either supplementary
or congruent.
Ongoing Assessment: Informing Instruction
It may be easier for students to recognize lines as parallel if they extend the lines
so they are about the same length (as students could do in Problems 3 and 6).
► Working with Parallel Lines
PARTNER
ACTIVITY
(Math Journal 1, p. 196)
Students construct two parallel lines using only a compass and a
straightedge. They draw two parallel lines and a parallelogram
using only a ruler and a pencil. Students should only use the
longer edges of the ruler to draw the parallel lines needed in
Problems 2 and 3. If needed, refer students to the illustration at
the top of journal page 199.
NOTE Discussion of this exercise will introduce the investigation of
parallelograms in Lesson 5-10.
Lesson 5 9
383
Student Page
Date
Time
LESSON
2 Ongoing Learning & Practice
Calculating Total Price
5 9
A total price is the sum of the price of an item (or subtotal) and the sales tax or tip
that is a percentage of that item:
Total Price ⫽ Subtotal ⫹ Sales Tax (or Tip)
Two-Step Method
► Calculating Total Price
One-Step Method
Subtotal: $49.75
Sales Tax: 8%
Find the total price.
Subtotal: $49.75
Sales Tax: 8%
Find the total price.
Step 1: Find the sales tax in
dollars: 8% of $49.75.
0.08 $49.75 ⫽ $3.98
The total price equals 100%
of the subtotal plus 8%
of the subtotal, so
Step 2: Add the sales tax amount
to the subtotal.
Total ⫽ 100% subtotal ⫹ 8% subtotal
⫽ 1.08 subtotal
Subtotal ⫹ Sales Tax ⫽ Total Price
$49.75 ⫹ $3.98 ⫽ $53.73
Find 108% of $49.75.
108% of $49.75 ⫽ 1.08 $49.75
⫽ $53.73
The total price is $53.73.
The total price is $53.73.
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 197)
The problems on journal page 197 provide practice using a
one- or two-step method for computing a total price that includes
sales tax or tips.
Use either method to find the total price. Round your answer to the nearest cent,
if necessary.
Subtotal: $89.00
Sales Tax: 6%
Total Price:
1.
3.
5.
Subtotal: $25.20
Tip: 15%
Total Price:
2.
Subtotal: $325.00
Sales Tax: 7%
Total Price:
4.
Subtotal: $103.50
Tip: 20%
Total Price:
$94.34
$347.75
► Math Boxes 5 9
Subtotal: $448.40
Sales Tax: 4.5%
Total Price:
6.
$468.58
INDEPENDENT
ACTIVITY
$124.20
$28.98
Subtotal: $876.00
Sales Tax: 6.25%
Total Price:
(Math Journal 1, p. 198)
$930.75
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 5-6. The skill in Problem 4
previews Unit 6 content.
Math Journal 1, p. 197
► Study Link 5 9
INDEPENDENT
ACTIVITY
(Math Masters, p. 172)
Home Connection Students explore relationships between
angles formed by parallel lines cut by a transversal.
Remind students to take home a protractor or their
Geometry Templates.
Study Link Master
Student Page
Date
LESSON
STUDY LINK
Math Boxes
5 9
1.
59
1.
Reflect figure PQST over the x-axis. Then plot and label
the vertices of the image that results from that reflection.
P⬘ ⫽ ( ⫺4 , 1
S⬘ ⫽ ( ⫺2 , 4
y
S'
T'
5
4
)
)
Q⬘ ⫽ ( ⫺2 , 2
T⬘ ⫽ ( ⫺4 , 3
Q'
P
Q
S
)
122°
)
58°
2
3
4
x
5
58°
122°
122° 58°
1
1
163
230 231
Sample answers:
58°
–2
–4
122°
180 234
Use the partial-quotients algorithm to
divide the numerator by the denominator.
Round the result to the nearest hundredth
and rename the result as a percent.
3.
Choose the best estimate for
2
1
the product 11ᎏ3ᎏ ᎏ4ᎏ.
2.
12
3.
6
⫽ 0.
92
⫽
92 %
3
55–57
90
Write a number sentence for each word sentence. Then tell whether the number
sentence is true or false.
Word Sentence
Number Sentence
True or
False?
If 19 is subtracted from 55, the result is 36.
55 ⫺ 19 ⫽ 36
true
78 added to 62 is less than 160.
45 is 5 times as great as 9.
What patterns do you notice in your angle measures?
Practice
Remember: 1,000 milliliters (mL) ⫽ 1 liter (L)
62 ⫹ 78 ⬍ 160 true
true
45 ⫽ 5 9
241–243
Math Journal 1, p. 198
Unit 5
Measure the 8 angles in your figure.
Write each measure inside the angle.
All of the vertical angles have the same
measure; all of the angles along the
transversal and on the same side are
supplementary; opposite angles along the
transversal are equal in measure.
11
384
Use a ruler and a straightedge to draw 2 parallel lines. Then draw another
line that crosses both parallel lines.
–3
T
4.
Time
2
–5 –4 –3 –2 –1 0
–1
11
ᎏᎏ
12
Date
Parallel Lines and a Transversal
3
P'
2.
Name
Time
Geometry: Congruence, Constructions, and Parallel Lines
4.
500 mL ⫽
6.
1,300 mL ⫽
8.
3,250 mL ⫽
0.5 L
1.3 L
3.25 L
2,500 mL
950 mL
45 mL
0.045 L ⫽
5.
2.5 L ⫽
7.
0.95 L ⫽
9.
Math Masters, p. 172
Teaching Master
Name
3 Differentiation Options
LESSON
59
Date
Time
Angle Relationships and Algebra
Apply your knowledge of angle relationships to find the missing values
and angle measures. Do not use a protractor.
1.
150⬚
ENRICHMENT
► Applying Angle Relationships
INDEPENDENT
ACTIVITY
x⫽
2x ⫹ 10⬚
10
°
2.
5–15 Min
and Algebra
100°
y ⫹ 40°
2y °
n
p
q
s
r
Lines l and m are parallel.
l
40 °
100 °
°
mq ⫽ 80
y⫽
m
mp ⫽
mr ⫽
ms ⫽
100 °
80 °
(Math Masters, p. 173)
3.
Algebraic Thinking To apply their knowledge of adjacent,
supplementary, and vertical angles, students find the missing
values in algebraic expressions that represent angle measures.
Encourage students to describe the strategies they used to find
the missing measurements.
50° a 70°
x⫽
x ⫹ 30°
c
2 x ⫹ 70°
20 °
60
ma ⫽
°
mc ⫽
70
°
70°
4.
Turn this page over. Using only a straightedge and a compass, design a
problem that uses angle relationships. Create an answer key for your problem.
Then ask a classmate to solve your problem.
Answers vary.
ENRICHMENT
► Finding Linear Perspective
INDEPENDENT
ACTIVITY
30+ Min
Math Masters, p. 173
Art Link Professional illustrators use linear perspective to
represent special relationships as they appear to the human
observer. All parallel lines that seem to go back into a picture (like
rails of a railroad track) are angled so that the lines converge at a
point in the distance, called a vanishing point. The vanishing point
is always set on a real or imaginary horizontal line in a picture.
Have students look for examples of linear perspective in comic
strips and sketches. Encourage them to talk to an artist,
illustrator, or architect about the various techniques used to
create the illusion of depth and distance.
ELL SUPPORT
► Supporting Language
SMALL-GROUP
ACTIVITY
5–15 Min
with Drawings
To provide language support for angle relationships, ask students
to draw a pair of parallel lines cut by a transversal. Have them
name the resulting angles and then use these names to identify
pairs of vertical, supplementary, and adjacent angles.
Planning Ahead
For the optional Readiness activity in Lesson 5-10, use the cut-out
quadrangles from Part 3 of Lesson 5-6, or make copies of Math
Masters, page 164. Also set aside sheets of 11 in. by 17 in. paper.
Lesson 5 9
385