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1-6
1-6
Measuring Angles
1. Plan
Objectives
1
To find the measures of
angles
To identify special angle pairs
2
What You’ll Learn
Check Skills You’ll Need
• To find the measures of
Solve each equation.
angles
• To identify special angle
Examples
1
2
Naming Angles
Measuring and Classifying
Angles
Using the Angle Addition
Postulate
Identifying Angle Pairs
Making Conclusions From a
Diagram
3
4
5
pairs
. . . And Why
1. 50 1 a 5 130 80
2. m 2 110 5 20 130
3. 85 2 n 5 40 45
4. x 1 45 5 180 135
5. z 2 20 5 90 110
6. 180 2 y 5 135 45
Finding Angle Measures
Math Background
An angle (&) is formed by two rays with the same endpoint.
The rays are the sides of the angle. The endpoint is the vertex of
)
)
the angle. The sides of the angle shown here are BT and BQ . The
vertex is B. You could name this angle &B, &TBQ, &QBT, or &1.
Vocabulary Tip
Angle measure and segment
measure have important
similarities. Congruent angles can
be moved onto one another so
they match exactly, as with
congruent segments. Congruent
angles are also indicated with tick
marks. Like a ruler, the intervals
on a protractor must be equal.
The most common unit of angle
measure is the degree, which
results when a circle is divided
into 360 equal parts. Finally, the
Angle Addition Postulate
corresponds directly with the
Segment Addition Postulate.
You may also refer to the
angle suggested by the
two segments BT and BQ
as &TBQ.
1
EXAMPLE
B
T 1 Q
Naming Angles
Name &1 in two other ways.
A
C
1
2
E
More Math Background: p. 2D
Lesson Planning and
Resources
D
&AEC and &CEA are other names for &1.
Quick Check
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:
Skills Handbook page 758
New Vocabulary • angle • acute angle • right angle • obtuse angle
• straight angle • congruent angles • vertical angles
• complementary angles • supplementary angles
To find the measures of
angles in flower
arrangements, as in
Exercise 41.
1
GO for Help
36
1 a. Name &CED two other ways. l2, lDEC
b. Critical Thinking Would it be correct to name any of the angles &E? Explain.
No; 3 ' have E for a vertex, so you need more info. in the name to
distinguish them from one another.
One way to measure an angle is in degrees. To indicate the size or
degree measure of an angle, write a lowercase m in front of the
80⬚
angle symbol. The degree measure of angle A is 80. You show this
A
by writing m&A = 80.
Chapter 1 Tools of Geometry
Solving Linear Equations
Skills Handbook, p. 758
Special Needs
Below Level
L1
Students with fine motor difficulties may have
problems using a protractor. You may want to have
students work in pairs so they can assist each other
as needed.
36
learning style: verbal
L2
Demonstrate on an overhead projector how to line up
a protractor with a ray, and how to choose the
appropriate scale. Have students choose the scale for
which one of the sides passes through zero.
learning style: visual
Key Concepts
Postulate 1-7
)
)
Let OA and OB be opposite
) )
rays in a plane. OA , OB , and
all the rays with endpoint O
that can be drawn on one
* )
side of AB can be
paired with the real
numbers from
0 to 180 so that
)
2. Teach
Protractor Postulate
Guided Instruction
1
)
Error Prevention
Discuss as a class why it is
inappropriate to name &1 as &E.
Ask: How could this cause
confusion? There are three
angles whose vertex is E. Explain
also that the measure of an angle
does not need a degree symbol.
a. OA is paired with 0 and OB is paired with 180.
)
EXAMPLE
)
b. If OC is paired with x and OD is paired with y, then m&COD = ux - yu.
Math Tip
You can classify angles according to their measures.
x⬚
x⬚
acute angle
0 < x < 90
right angle
x ⫽ 90
x⬚
x⬚
obtuse angle
90 < x < 180
straight angle
x ⫽ 180
Note the special symbol that’s tucked into the corner of the right angle. When you
see it, you know that the measure of the angle is 90.
2
nline
EXAMPLE
Ask: How is the Protractor
Postulate like the Ruler Postulate
in Lesson 1-5? Both pair numbers
in a one-to-one correspondence
with geometric objects and use
absolute value to determine
measurements.
Auditory Learners
Have students take turns with a
partner explaining the Protractor
Postulate in their own words.
Measuring and Classifying Angles
2
Find the measure of each angle. Classify each as acute, right, obtuse, or straight.
a.
EXAMPLE
Connection to
Algebra
Review the meaning of the
inequality symbol.
b.
PowerPoint
Additional Examples
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Web Code: aue-0775
120, obtuse
Quick Check
90, right
1 Name the angle below in four
ways.
2 Find the measure of each angle. Classify each as acute, right, obtuse, or straight.
a.
30; acute
b.
90; right
c.
140; obtuse
C
G
Angles with the same measure
are congruent angles. In other
words, if m&1 = m&2, then
&1 > &2. You can use these
statements interchangeably.
A
&3, &G, &AGC, &CGA
1
2 Find the measure of each
angle. Classify each as acute, right,
obtuse, or straight.
2
Angles can be marked alike to
show that they are congruent,
as in this photograph of the
Air Force Thunderbirds
precision flying team.
1 2
Lesson 1-6 Measuring Angles
Advanced Learners
3
37
m&1 ≠ 110, obtuse;
m&2 ≠ 80, acute
English Language Learners ELL
L4
Have students write their first names using block
letters. Then have them count the number of acute,
right, obtuse, and congruent angles suggested by
the letters.
learning style: visual
Show how the concept and notation for congruent
angles is closely related to congruent segments.
For example, if AB DE, then AB = DE. Likewise, if
&1 &2, then m&1 = m&2.
learning style: visual
37
3
EXAMPLE
Visual Learners
The Angle Addition Postulate is similar to the Segment Addition Postulate.
Draw the figures for the Angle
Addition Postulate on the board.
Have students place other points
in the interior of &AOC to
reinforce the concept of interior.
Key Concepts
Postulate 1-8
Angle Addition Postulate
If point B is in the interior of &AOC,
then m&AOB + m&BOC = m&AOC.
A
PowerPoint
Additional Examples
If &AOC is a straight angle, then
m&AOB + m&BOC = 180.
B
B
O
A
C
C
O
3 Suppose that m&1 = 42 and
m&ABC = 88. Find m&2.
3
A
What is m&TSW if m&RST = 50
and m&RSW = 125?
1
B
Using the Angle Addition Postulate
EXAMPLE
W
2
m&RST + m&TSW = m&RSW
50 + m&TSW = 125
Quick Check
Teaching Tip
Subtract 50 from each side.
3 If m&DEG = 145, find m&GEF. 35
G
D
.
2
1
Error Prevention!
Students sometimes confuse
complementary and
supplementary angles. One ways
to keep them straight is to
remember that c comes before s
in the alphabet, just as 90 comes
before 180.
Angle Addition Postulate
Substitute.
m&TSW = 75
Guided Instruction
After students read the definition
of vertical angles, ask: What is
another way to define vertical
angles? opposite angles formed
by two intersecting lines
S
R
C
m&2 ≠ 46
E
F
Identifying Angle Pairs
Some angle pairs that have special names.
X
X
X
X
vertical angles
Helvetica
Condensed
Times
Roman
Eurostile
Extended
MarkerFelt
Thin
Chapter 1 Tools of Geometry
2
4
3
1
two angles whose sides are
opposite rays
complementary angles
2
4
two coplanar angles with a
common side, a common vertex,
and no common interior points
supplementary angles
105⬚
50⬚
3
2
1
38
adjacent angles
1
3
In each font, a capital X
suggests vertical angles.
38
T
A B
40⬚
4
75⬚
two angles whose measures
have sum 90
two angles whose measures
have sum 180
Each angle is called the
complement of the other.
Each angle is called the
supplement of the other.
4
EXAMPLE
4
Identifying Angle Pairs
a. complementary
&2 and &3
c. vertical
&3 and &5
2
1
5
Connection to
Language Arts
Students are familiar with the
word compliment. Point out that
the word in this lesson has an e
instead of an i. Ask students to
find non mathematical contexts
where complement and
supplement are used.
In the diagram identify pairs of numbered angles that are related as follows:
b. supplementary
&4 and &5; &3 and &4
EXAMPLE
3
4
PowerPoint
Quick Check
4 a. Name two pairs of adjacent angles in the photo below. Answers may vary.
Sample: lAFB and
b. If m&EFD = 27, find m&AFD. 153
lBFC; lBFD and lDFE
Additional Examples
4
2 3
1 4
When entering the roadway,
turn and look for oncoming
traffic regardless of what you
see in the rear-view mirror.
Name all pairs of angles in the
diagram that are
a. vertical
l1 and l3; l2 and l4
b. supplementary l1 and l2;
l2 and l3; l3 and l4;
l4 and l1
c. complementary none
Whether you draw a diagram or use a given diagram, you can make some
conclusions directly from the diagrams. You can conclude that angles are
• adjacent angles
5a. Yes; the congruent
segments are
marked.
b. No; there are no
markings.
c. No; there are no
markings.
d. No; there are no
markings.
nline
• adjacent supplementary angles
5 Use the diagram from
Example 2. Which of the
following can you conclude:
&3 is a right angle, &1 and &5
are adjacent, &3 &5? l1 and
l5 are adjacent.
• vertical angles
Unless there are marks that give this information, you cannot assume
• angles or segments are congruent
• an angle is a right angle
• lines are parallel or perpendicular
5
EXAMPLE
Resources
• Daily Notetaking Guide 1-6 L3
• Daily Notetaking Guide 1-6—
L1
Adapted Instruction
Making Conclusions From a Diagram
What can you conclude from the information
in the diagram?
• &1 > &2, by the markings.
4
2
1
5
• &2 and &3, for example, are adjacent angles.
Visit: PHSchool.com
Web Code: aue-0775
3
Closure
• &4 and &5, for example, are adjacent supplementary
angles,
or m&4 + m&5 = 180 by the Angle Addition Postulate.
Ask: How are angles classified?
By their angle measure:
acute (R 90), right (≠ 90), obtuse
(S 90), and straight (≠ 180) What
are the special angle pairs?
vertical, adjacent,
complementary, and
supplementary
• &1 and &4, for example, are vertical angles.
Quick Check
5 Can you make each conclusion from the information in the diagram? Explain.
a. TW > WV
b. PW > WQ a–d. See left.
c. TV ' PQ
d. TV bisects PQ.
e. W is the midpoint of TV.Yes; the congruent
segments are marked.
T
P
W
Q
V
Lesson 1-6 Measuring Angles
39
39
EXERCISES
3. Practice
For more exercises, see Extra Skill, Word Problem, and Proof Practice.
Practice and Problem Solving
Assignment Guide
A
1 A B 1-14, 33-47
2 A B
Practice by Example
Example 1
15-32
C Challenge
48-49
Test Prep
Mixed Review
50-54
55-59
GO for
Help
Name each angle in three ways.
1.
Example 2
(page 37)
Exercises 9–12 Using a corner of
paper to model a 90° angle makes
classifying acute and obtuse
angles visually apparent.
GPS Guided Problem Solving
7. a straight angle, &EFG
8. a right angle, &GHI
about 42°
10. the angle formed by the skis
11.
L3
90; right
B
Example 4
L3
Date
(page 39)
The Coordinate Plane
Graph each point in the coordinate plane.
7. N(1, 0), P(3, 8)
9. S(0, 5), T(0, ⫺3)
14. E(14, ⫺2), F(7, ⫺8)
15. O(0, 0), G(⫺5, 12)
16. H(2.8, 1.1), I(⫺3.4, 5.7)
17. J(2 12 , - 14 ), K(3 14 , -1)
24. What is the perimeter of PQSR?
25. What is the midpoint of QR?
40
© Pearson Education, Inc. All rights reserved.
23. Graph quadrilateral PQSR.
E
B
60⬚ O
D
C
In the diagram above, find the measure of each of the following angles.
20. The midpoint of EF is (⫺3, 7). The coordinates of E are (⫺3, 10).
Find the coordinates of F.
Quadrilateral PQSR has coordinates as follows:
P(0, 0), Q(–1, 4), R(8, 2), and S(7, 6).
A
19. a pair of vertical angles lAOB and lDOC or
lBOC and lAOD
19. The midpoint of CD is (4, 11). The coordinates of D are (4, 12).
Find the coordinates of C.
22. A crow flies to a point that is 1 mile east and 20 miles south of its
starting point. How far does the crow fly?
15. supplementary to &AOD lAOB or lDOC
18. complementary to &EOD lDOC or lAOB
18. The midpoint of AB is (1, 2). The coordinates of A are (⫺3, 6).
Find the coordinates of B.
21. Graph the points A(2, 1), B(2, ⫺5), C(⫺4, ⫺5), and D(⫺4, 1).
Draw the segments connecting A, B, C, and D in order.
Are the lengths of the sides of ABCD the same? Explain.
E
Name an angle or angles in the diagram described by each of the following.
17. supplementary to &EOA lEOC
11. W(2, 7), X(1, 2)
13. C(⫺1, 5), D(2, ⫺3)
F
16. adjacent and congruent to &AOE lEOC
Find the coordinates of the midpoint of each segment. The coordinates of
the endpoints are given.
12. A(6, 7), B(4, 3)
J
A
5. E(⫺4, ⫺2)
Find the distance between the points to the nearest tenth.
8. Q(10, 10), R(10, ⫺2)
14. Find m&GFJ if m&EFG = 110. 70
G
L1
Adapted Practice
10. U(11, 0), V(⫺1, 0)
13. Find m&CBD if m&ABC = 45
and m&ABD = 79. 34
C
L2
Reteaching
135; obtuse
D
L4
Enrichment
6. L(⫺4, 11), M(⫺3, 4)
D
60; acute
(page 38)
4. D(⫺4, 0)
1 2
B
A
5–8. See margin.
6. an acute acute, &BCD
9.
Example 3
3. C(0, 6)
C
5. an obtuse angle, &RST
12.
2. B(5, ⫺2)
lMCP, lPCM, lC, or l1
Use a protractor. Measure and classify each angle.
Exercise 16 Point out to students
that two conditions must be met
in this exercise.
1. A(⫺2, 5)
Z
4. &2
lCBD, lDBC
Draw and label a figure to fit each description.
Error Prevention!
Class
P
M
3. &1 lABC, lCBA
Tactile Learners
Practice 1-6
C
1
Use the figure at the right. Name the
indicated angle in two different ways.
To check students’ understanding
of key skills and concepts, go over
Exercises 12, 18, 33, 41, 47.
Name
lXYZ, lZYX, lY
Y
Homework Quick Check
Practice
2.
X
(page 36)
20. &EOC 90
40
Chapter 1 Tools of Geometry
21. &DOC 30
22. &BOC 150
23. &AOB 30
(page 39)
24. &J > &D Yes; the markings show
they are congruent.
25. &JAC > &DAC
No; there are no markings.
26. &JAE and &EAF are adjacent
and supplementary. See left.
26. Yes; you can
conclude that the
angles are adjacent
and supplementary
from the diagram.
28. Yes; you can conclude
that angles are
supplementary from
the diagram.
B
Exercises 24, 25 Ask: Why are
three letterers used to name the
angles in Exercise 25 but only one
letter is used in Exercise 24?
Vertex J and vertex D each apply
to only one angle, but many
angles share vertex A.
Can you make each conclusion from the information in the diagram? Explain.
Example 5
E
F
A
27. m&JCA = m&DCA
No; there are no markings.
28. m&JCA + m&ACD = 180 See left.
C
J
D
Error Prevention!
No; there are no markings.
29. AJ > AD
30. C is the midpoint of JD.
Yes; there are markings.
)
31. &EAF and &JAD are vertical angles. 32. AC bisects &JAD.
No; there are no markings.
See left.
A
In the diagram, mlACB ≠ 65. Find each of the following.
Apply Your Skills
33. m&BCD 115
31. Yes; you can
conclude that '
are vertical from
the diagram.
34. m&ECD 65
E
35. 6:00 180
36. 7:00 150
37. 11:00 30
38. 4:40 100
39. 5:20 40
40. 10:40 80
B
C
Estimation Estimate the measure of the angle
formed by the hands of a clock at each time.
D
Exercises 33–34
41. Flower Arranging In Japanese flower
arranging, you match a stem that is
vertical with 0. You match other
stems with numbers from 0 to
90, in both directions from the
vertical. What numbers would
the flowers shown be paired
with on a standard protractor?
45, 75, and 165, or 135, 105, and 15
Real-World
42. 12; mlAOC ≠ 82,
mlAOB ≠ 32;
mlBOC ≠ 50
43. 8; mlAOB ≠ 30,
mlBOC ≠ 50;
mlCOD ≠ 30
44. 18; mlAOB ≠ 28,
ml BOC ≠ 52;
mlAOD ≠ 108
x 2 Algebra Use the diagram, below right, for Exercises 42–45. Solve for x. Find the
Connection
angle measures to check your work. 42–45. See margin.
Japanese flower arranging
makes precise use of angles
to create a mood.
Exercise 46 Students may be
misled or confused because the
drawing is not drawn to scale
with m&MQV = 90. Students can
redraw the figure, but by
examining the answer choices
students should be able to
identify the correct answer.
42. m&AOC = 7x - 2, m&AOB = 2x + 8,
m&BOC = 3x + 14
45. 7; mlAOB ≠ 31,
mlBOC ≠ 49;
mlAOD ≠ 111
B
A
C
43. m&AOB = 4x - 2, m&BOC = 5x + 10,
m&COD = 2x + 14
O
D
44. m&AOB = 28, m&BOC = 3x - 2, m&AOD = 6x
45. m&AOB = 4x + 3, m&BOC = 7x, m&AOD = 16x - 1
46. Multiple Choice If m&MQV = 90, which
expression can you use to find m&VQP? A
m/MQP 2 90
90 2 m/MQV
m/MQP 1 90
90 1 m/VQP
47c. Answers may vary.
Sample: The sum of
the l measures
should be 180.
GO
Homework Help
GPS
Visit: PHSchool.com
Web Code: aua-0106
and m&TQS = 6x + 20. 19.5
b. What is m&RQS? m&TQS? 43; 137
c. Show how you can check your answer.
See left.
M
N
6.
R
8.
B
Q
T
41
G
C
5.
D
T
S
S
Lesson 1-6 Measuring Angles
lesson quiz, PHSchool.com, Web Code: aua-0106
5–8. Drawings may vary.
Samples are given.
P
V
x 2 47. a. Algebra Solve for x if m&RQS = 2x + 4
nline
R
Q
7.
E
F
H
I
G
41
4. Assess & Reteach
C
Challenge
PowerPoint
)
)
)
)
48. XC bisects &AXB,
XD bisects &AXC,
)
) XE bisects &AXD, XF bisects
&EXD, XG bisects &EXF, and XH bisects &DXB. If m&DXC = 16, find
m&GXH. 30
49. Technology Leon constructed an angle. Then he constructed a ray from the
vertex of the angle to a point in the interior of the angle. He measured all the
angles formed. Then he moved the interior ray. What postulate do the two
pictures support? Angle Add. Post.
Lesson Quiz
Use the figure below for Exercises
1–2.
C
D
105⬚
105⬚
1
A
2
B
24⬚
1. Name &2 two different ways.
&DAB, &BAD
2. Measure and classify &1, &2,
and &BAC. 90, right; 30,
acute; 120, obtuse
2
42⬚
Test Prep
Use the figure below for Exercises
3–4.
1
63⬚
81⬚
Multiple Choice
3
4
50. Two angles are congruent, adjacent, and supplementary. What is the
measure of each? B
A. 45
B. 90
C. 180
D. cannot be determined
51. Two angles are congruent and complementary. What is the measure
of each? F
F. 45
G. 90
H. 180
J. cannot be determined
52. Two angles are adjacent and supplementary. What is the measure of each? D
A. 45
B. 90
C. 180
D. cannot be determined
3. Name a pair of supplementary
angles. Samples: l1 and l3,
l2 and l4
53. When 15 is subtracted from the measure of an angle, the result is the
measure of a right angle. What is the measure of the original angle? H
F. 75
G. 85
H. 105
J. 115
4. Can you conclude that there
are vertical angles in the
diagram? Explain. No; no
angle pairs are formed by
opposite rays.
Short Response
54. You are given that m&ABD + m&DBC = m&ABC.
a. Draw a diagram to show the above. a–b. See margin.
b. If m&ABD = 12 and &ABC is obtuse, what are the least and greatest
whole number measures possible for &DBC? Explain.
Alternative Assessment
Have students draw diagrams to
illustrate the Angle Addition
Postulate. Then have them write
examples that use each postulate
to find a missing measurement
when two of the three
measurements are known.
Mixed Review
Lesson 1-5
GO for
Help
Test Prep
55. If EG 5 75 and EF 5 28, what is FG? 47
E
56. If EG 5 49, EF 5 2x 1 3, and FG 5 4x 2 2,
find x. Then find EF and FG. x = 8; EF = 19; FG = 30
42
57. Writing Explain the difference between an orthographic drawing and an
isometric drawing. See back of book.
Lesson 1-1
Find one counterexample to show that each conjecture is false.
58. The quotient of two integers is
not an integer.
452
2
Chapter 1 Tools of Geometry
54. [2] a.
D
A
C
B
42
F
Lesson 1-2
Resources
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 75
• Test-Taking Strategies, p.70
• Test-Taking Strategies with
Transparencies
Use the figure at the right for Exercises 55–56.
b. An obtuse l
measures between
90 and 180 degrees;
the least and
greatest whole
number values are 91
and 179 degrees.
Part of lABC is 12°.
So the least and
greatest l measures
for lDBC are 79 and
167.
59. An even number cannot have
5 as a factor.
10 = 2 ⫻ 5, so 10 has 5 as a
factor and 10 is even.
[1] one part correct
G