7-2
Measuring and Classifying
Angles
7-2
1. Plan
Check Skills You’ll Need
What You’ll Learn
OBJECTIVE
To measure and
describe angles
1
OBJECTIVE
. . . And Why
PowerPoint
1. k + 12 = 15
2. O + 20 = 90
3. m + 5 = 28
4. n + 40 = 180
5. p + 16 = 90
6. s + 22 = 30
Check Skills You’ll Need
Solving Equations Using
Mental Math
New Vocabulary • angle • vertex • acute angle • right angle
To classify angles in
architecture, as in
Example 2
• obtuse angle • straight angle • complementary • supplementary
• adjacent angles • vertical angles • congruent angles
OBJECTIVE
1
Lesson Preview
Use mental math to solve each equation.
To work with pairs
of angles
2
For help, go to Lesson 2-2.
Interactive lesson includes instant self-check, tutorials, and activities.
Lesson 2-2: Example 2. Extra
Practice, p. 685.
Lesson Resources
Optional Materials
• Protractor
Measuring and Describing Angles
Teaching Resources
Practice, Reteaching, Enrichment
An angle (&)is formed by two rays with a
common endpoint. You can call the angle at the
right &DCE, &ECD, &C, or &1.
E
ra
y
Reaching All Students
vertex
1
A vertex is the point of intersection of two sides
of an angle or figure. Point C is the vertex of the
angle at the right. The plural of vertex is vertices.
D
C
ray
You can use a protractor to measure angles. Write
“the measure of &C” as m&C.
1
EXAMPLE
Presentation Assistant Plus!
Measuring Angles
What is the measure of &X at the left?
X
10
0
80 11
70 0
10
170
2
16 0
0
90
0
180
Step 2 Make sure that one side of
the angle passes through zero on
one of the protractor’s scales.
60 70 80
50
0 110 100
40 130 12
0
4
0
3 0 1
15
X
Step 3 Read the same scale where it
intersects the second side of the angle.
150 160 170 180
140 30 20 10
0
130 40
0 50
12 0
6
Step 1 Place your protractor on the
vertex of the angle, as shown.
Practice Workbook 7-2
Spanish Practice Workbook 7-2
Guided Problem Solving 7-2
Hands-On Activities 27, 28
Transparencies
• Check Skills You’ll Need 7-2
• Problem of the Day 7-2
• Additional Examples 7-2
• Student Edition Answers 7-2
• Lesson Quiz 7-2
• Classroom Aid 5
PH Presentation Pro CD-ROM 7-2
Computer Test Generator CD
Technology
Resource Pro® CD-ROM
Computer Test Generator CD
PH Presentation Pro CD-ROM
m&X = 1108
Check Understanding 1 Measure &G using a protractor.
www.PHSchool.com
Student Site
• Teacher Web Code: abk-5500
• Self-grading Lesson Quiz
G
7-2 Measuring and Classifying Angles
M INS
CHECK
TANT
351
Ongoing Assessment and Intervention
PH SuccessNet Teacher Center
• Lesson Planner
• Resources
Plus
SYSTE
Before the Lesson
During the Lesson
After the Lesson
Diagnose prerequisite skills using:
• Check Skills You’ll Need
Monitor progress using:
• Check Understanding
• Additional Examples
• Test Prep
Assess knowledge using:
• Lesson Quiz
• Computer Test Generator CD
351
2. Teach
You can classify angles by their measures.
Reading Math
The ^ symbol indicates a
right angle.
Math Background
An angle is formed when two
lines (or line segments, or rays)
meet at a common point called
the vertex.
formed by
) An angle
)
rays BA and BC can be named in
a variety of ways: ABC, CBA,
or B for short. (When more than
one angle is drawn from a vertex,
the more descriptive names are
needed.) The measure of an angle
is indicated by the lowercase m,
as in mB, in degrees. Angles can
be classified by their measures as
acute, right, obtuse, and straight.
Complementary angles add to 90°
and supplementary angles add
to 180°.
2
straight angle
between
908 and 1808
1808
EXAMPLE
Real
World
Identifying Angles
A
F
B
N
E
C
D
acute angles: &FAN, &NAB, and &DNE
obtuse angles: &F, &C, and &FAB
Check Understanding 2 Classify each angle as acute, right, obtuse, or straight.
Inclusion
Some students may experience
difficulties using a protractor.
Have them use protractors with
the center point clearly marked.
This helps students align the
protractor correctly with
the vertex.
obtuse angle
908
Architecture A geodesic dome, a building system invented by
R. Buckminster Fuller, is pictured below. Identify three acute
angles and three obtuse angles.
Teaching Notes
EXAMPLE
right angle
between
08 and 908
Notice that a straight angle is a line.
When two lines intersect, four
pairs of adjacent angles are
formed. Adjacent angles share
a side and are supplementary
(add to 180°). When two lines
intersect, two pairs of vertical
angles are formed. Vertical angles
are opposite each other and are
congruent (have equal measures).
1
acute angle
a.
right
b.
acute
c.
obtuse
OBJECTIVE
2
Working With Pairs of Angles
Error Prevention!
If the sum of the measures of two angles is 908, the angles are
complementary. If the sum is 1808, the angles are supplementary.
Some students may read the
wrong scale when measuring
with a protractor. Have students
determine whether the angle is
acute or obtuse before measuring.
Students should then read the
corresponding scale.
39
51
complementary
115
65
supplementary
Auditory Learners
Students may confuse the
terms supplementary and
complementary. Invite them to
suggest mnemonic devices to
remember how they differ. For
example, complementary is before
supplementary alphabetically,
and 90 is before 180 numerically.
352
352
Chapter 7 Geometry
Reaching All Students
Below Level Have students fold three
pieces of cardstock in half and turn
them on edge. Students can adjust the
fold to model acute, right, and obtuse
angles.
Advanced Learners Can an angle
English Learners
be larger than 180°? Explain.
Sample: Yes, the exterior angle for
any angle less than 180° is greater
than 180°.
See note on page 353.
Inclusion
See note on page 352.
Suppose you know that two angles are supplementary, and you know the
measure of one of the angles. You can use an equation to find the measure
of the other angle.
3
EXAMPLE
Finding Complements and Supplements Algebra
Let x = the measure of the supplement of &A.
d The angles are supplementary.
d Substitute 37 for ml A.
x + 378 = 1808
x + 378 - 378 = 1808 - 378
d Subtract 37 from each side.
x = 1438
d Simplify.
The measure of the supplement of &A is 1438.
Check Understanding 3 a. Find the measure of the complement of &A in Example 3.
Have students use a protractor
to draw complementary and
supplementary angles. Have them
label their drawings.
13
EXAMPLE
English Learners
Explain to students that it is not
unusual to confuse complementary
and supplementary. Tell them
when someone gives you a
compliment (by saying something
nice about you), you stand up
straighter, making a 90° angle
with the ground. Be sure to tell
them that complement and
compliment are not the same
terms (having different roots).
The measure of &A is 378. Find the measure of its supplement.
x + m&A = 1808
Tactile Learners
53°
PowerPoint
b. Reasoning Is the supplement of an angle always greater than its
complement? Explain. Yes; you have to add a larger angle to make a
180° angle than to make a 90° angle.
Additional Examples
1 What is the measure of V?
Adjacent angles share a vertex and a side but have no interior points in
common. Angles 1 and 2 below are adjacent angles. Adjacent angles
formed by two intersecting lines are supplementary.
45°
V
2 Identify the acute angles,
4
1
3
obtuse angles, and straight
angles in the figure.
2
Angles 1 and 3 above are vertical angles. Vertical angles are formed by two
intersecting lines and are opposite each other. Vertical angles have equal
measures. Angles with equal measures are congruent angles.
Need Help?
Congruent means having
the same size and shape.
4
EXAMPLE
Finding Angle Measures
Algebra
1
4 2
3
m&1 + 1288 - 1288 = 1808 - 1288
m&1 = 528
m&2 = 1288
d l2 and l4 are vertical angles.
d l1 and l3 are vertical angles.
4 Find the measures of 1, 2,
and 4, if m3 = 32°.
ml1 ≠ 32°, ml2 ≠ 148°, and
ml4 ≠ 148°
2
1
3
4
Check Understanding 4 a. Find the measures of &5, &6, and &7,
if m&8 = 728. 108°; 72°; 108°
b. Reasoning For Example 4 and part (a), what is
the sum of the measures of the angles formed
by two intersecting lines? 360°
N
Find the measure of its
complement. 41°
l1 and l4 are
d
supplementary.
m&3 = 528
K L
acute: lMLN, lMNL
obtuse: lKLM, lLMN
straight: lKLN
3 The measure of Q is 49°.
Find the measures of &1, &2, and &3, if m&4 = 1288.
m&1 + 1288 = 1808
M
8
5
7
6
7-2 Measuring and Classifying Angles
353
353
Science Connection
More Than One Way
An astrolabe measures the angles
formed by a star in the sky and
the horizon. Early sailors used the
astrolabe to help them navigate
the seas.
If m&1 = 1408 and m&2 = 408, what is m&3?
2
Carlos’s Method
4
Since &3 and &1 are across from
each other, they are vertical angles.
Since vertical angles have the
same measure, m&3 = m&1.
So m&3 = 1408.
Careers
Surveyors determine the exact
location and measurement
of points, lines and contours of
the Earth’s surface. They use
instruments to measure angles
and distances and they use
geometry to find precise
measurements.
Kayla’s Method
&2 and &3 together form a straight angle, so their measures add up to
1808. This means that they are supplementary angles.
Closure
• What are acute, obtuse, right,
408 + m&3 = 1808
and straight angles? Acute
angles are greater than 0° and
less than 90°; obtuse angles
are greater than 90° and less
than 180°; right angles are
equal to 90°, and straight
angles are equal to 180° or
a straight line.
• What are complementary and
supplementary angles?
Complementary angles are
two angles that add to 90°.
Supplementary angles are
two angles that add to 180°.
• What are vertical and adjacent
angles? When two lines
intersect, vertical angles are
opposite each other; adjacent
angles share a side and a
vertex and are supplementary.
408 - 408 + m&3 = 1808 - 408 d Subtract 408 from each side.
m&3 = 1408
d Simplify.
So m&3 is 1408.
Choose a Method
In the figure at the right, m&BEC = 258 and
m&CED = 1558. Find the m&AEB. Explain
why you chose the method you used.
155°; methods and explanations may vary.
A Practice by Example
B
A
4
A
5
6
A
B
B
E
D
C
Find the measure of each angle using a protractor.
1.
50°
2.
57°
3.
115°
6.
83°
E
D
C
C
D
E
D
C
E
E
D
C
B
A
3
C
B
A
2
C
B
A
1
B
A
For more practice, see Extra Practice.
EXERCISES
Example 1
(page 351)
D
D
E
E
Test-Prep Tip
Estimate the angle
measure first. If the angle
is larger than a corner of
this page, its measure will
be greater than 90º.
354
354
3
1
Chapter 7 Geometry
4.
90°
5.
133°
Example 2
(page 352)
3. Practice
Classify each angle as acute, right, obtuse, or straight.
7.
8.
9.
10.
Assignment Guide
straight
acute
obtuse
1
Objective 1
A B Core 1–11, 29, 34–35
C Extension 38
2
Objective 2
A B Core 12–28, 30 –33, 36
C Extension 37
right
11. City Planning Engineers are planning to put in a new road that will
make an angle of 1208 with an existing road. Classify the angle as acute,
right, obtuse, or straight. obtuse
Example 3
(page 353)
Algebra Find the measures of the complement and supplement of
each angle.
85°, 175°
45°; 135°
14. m&C = 768
15. m&D = 508
16. m&E = 798
17. m&F = 678
18. m&G = 128
19. m&H = 368
23°; 113°
78°; 168°
54°; 144°
Algebra In the diagram at the right,
ml2 ≠ 1238. Find the measure of each
of the following angles.
20. m&1 57°
21. m&3 57°
Test Prep 39 – 42
Mixed Review 43– 47
40°; 130°
13. m&B = 58
11°; 101°
Example 4
(page 353)
14°, 104°
12. m&A = 458
Error Prevention!
Exercises 7–11 If students confuse
obtuse and acute angles, give the
mnemonic “a cute little angle.”
2
1
3
4
22. m&4 123°
Practice 7-2
Measuring and Classifying Angles
Find the measure of each angle. Then classify the angle.
B
32. Answers may vary.
Samples are given.
lJHK and lBHC,
lHCD and lBCL
Algebra Find the measures of the complement and supplement of
each angle.
66.5°; 156.5°
52.4°; 142.4°
1.
90, right
42.1°; 132.1°
24. m&B = 37.68
25. m&C = 47.98
26. m&D = 56.48
27. m&E = 75.18
28. m&F = 82.28
14.9°; 104.9°
3.
135, obtuse
45, acute
Classify each angle as acute, right, obtuse, or straight.
23. m&A = 23.58
33.6°; 123.6°
2.
4. mA = 180
5. mB = 43
6. mC = 127
straight
acute
obtuse
7. mD = 90
right
Use the figure at the right to name the following.
Sample answers are shown.
* )* )* )* )
XU, ST, XP, QY
8. four lines
7.8°; 97.8°
29. Physical Therapy Physical therapists use goniometers to measure the
amount of motion a person has in a joint, like an elbow or a knee.
Estimate the measure of the angle in the photograph. about 65°
M
•
XN, XP, QN
9. three segments
10. a pair of congruent angles
•
U
•
•V
• W
X
S
•
•Q
N T
•
•
P
•
Y•
lVXW and lUXP
11. four right angles
12. two pairs of obtuse vertical angles
lQNT, lTNY, lQNS, and lSNY
13. two pairs of adjacent supplementary angles
lMSW and lUST, lSXP and lVXW
14. two pairs of complementary angles
lMSX and lMSU, lMSX and lXST lQNX and lXNS, lTNP and lPNY
15. Use a protractor to find mA, mB, mC, mD.
Use the figure at the right to name the following.
F
J
mlA ≠ 110
B
K
C
30. two pairs of adjacent supplementary angles
Answers may vary. Sample: lJHK and lKHC;
lBHJ and lJHK
31. four right angles
lABF; lABG; lFBD; lDBG
A
32. two pairs of obtuse vertical angles
D
45˚
L
E
See above left.
lABE and lEBG; lFBH and lHBC
Reteaching 7-2
34. Error Analysis One student measured
&XYZ and said that m&XYZ = 1208.
Explain the student’s error. See below left.
B
A
&A is an acute angle because its measure is less
than 90. If an angle has a measure greater than 90
and less than 180, it is an obtuse angle.
obtuse
greater than 90°
less than 180°
acute
less than 90°
You can measure an angle using a protractor.
Write the measure of &A as m&A.
3
To measure an angle:
1
Can an angle ever have the
same measure as its complement? Explain.
2
Line up one side of the angle with zero on the protractor scale.
3
Read the scale at the second side of the angle. Since &A is an
acute angle, read 80 and not 100.
C
A
B
170 180
160
10 0
20
Writing in Math
80 90 100 11
0 12
70
0
60 110 100 90 80 70
60 130
0
50 12
50
0
13
Place the center point of your protractor on the vertex of the
angle.
30
X
0
Y
15
GPS
C
You can name this angle &A, &BAC, or &CAB.
40
36.
Z
0
35. Data File, p. 345 Classify the loft angle of a
No. 4 driver. acute
Measuring and Classifying Angles
An angle is made up of two rays (the sides of the angle) with a
common endpoint (the vertex of the angle).
14
34. The student used the
wrong scale on the
protractor.
135° 45°
45˚
0 10
20
30
180 170 16
0 15
0 40
14
0
help people regain mobility
after an accident.
17. Use the dot grid
to draw two
supplementary
angles, one of
which is 45.
Do not use a
protractor.
G
33. two pairs of complementary angles
Careers Physical therapists
mlD ≠ 135
16. Use a protractor. Draw two vertical
angles. One angle has a measure of 45.
C
B
mlC ≠ 53
D
A
H
mlB ≠ 62
1
2
m&A = 80
Measure each angle. Then circle acute or obtuse.
A 45° angle has the same measure as its complement.
1.
2.
R
C
7-2 Measuring and Classifying Angles
B
355
120
acute
GPS
Use the Guided Problem
Solving worksheet with
Exercise 36.
A
S
m&B =
3.
T
m&S =
obtuse
acute
obtuse
4.
Z
O
Y
N
X
m&Y =
50
60
acute
m&N =
obtuse
M
110
acute
obtuse
Classify each angle with the given measure as acute or obtuse.
5. 45
acute
6. 148
obtuse
7. 4
9. 65
acute
10. 179
obtuse
11. 23
acute
acute
8. 106
obtuse
12. 115
obtuse
355
4. Assess
PowerPoint
C
37. You know that &B is the complement of &A and m&B = 518, and
m&A = (3x - 12)8. Find x. 17
38. Stretch Your Thinking I am a three-digit number. When you reverse my
digits, a greater number is formed. The product of the new number and
me is 65,125. What number am I? 125
Lesson Quiz 7-2
Use the diagram for questions
1–3.
Test Prep
2
1
50°
5
3
Reading Comprehension
4
1. Find the measures of 2
and 3. ml2 ≠ 40°;
ml3 ≠ 90°
Read the passage and answer Exercises 39 and 40 below.
Identifying Ivory
Ivory smugglers try to slip illegal
ivory from elephant tusks into
the United States by labeling the
elephant tusks as mammoth
tusks. Ed Espinoza of the United
States Fish and Wildlife Service
found that the markings on
2. Name a pair of adjacent,
complementary angles. l1
and l2
3. Name a pair of
supplementary angles. l3
and l4 or l4 and l5
photocopied tusks have different
angles. Markings on mammoth
tusks create angles that measure
908 or less. Markings on elephant
tusks create angles that measure
1158 or more.
39. Tell whether each photocopy comes from a mammoth or an elephant.
mammoth
elephant
a.
b.
Test Prep
Resources
For additional practice with a
variety of test item formats:
• Test Prep, p. 397
• Test-Taking Strategies, p. 393
• Test-Taking Strategies with
Transparencies
Take It to the NET
Online lesson quiz at
www.PHSchool.com
Web Code aba-0702
Multiple Choice
40. Suppose two tusks from the same type of animal have angles that
are complementary. Would the animal be a mammoth or an elephant?
mammoth
41. What is the measure of the complement of a 488 angle? B
A. 328
B. 428
C. 1328
D. 1428
42. The supplement of which angle is NOT acute? F
F. 858
G. 918
H. 1028
Enrichment 7-2
Measuring and Classifying Angles
Patterns in Geometry
Mixed Review
An exterior angle is the angle formed by
one side of the triangle and the extension
of the adjacent side. The exterior angles
are marked in the triangle.
120
1. What is the sum of the exterior angles in the triangle?
Lesson 6-7
360
43. Dining You add a 15% tip to your bill of $37. How much do you pay?
2. For each vertex of the regular polygons below, extend one side
and write the measure of the exterior angle. Then find the sum of
all the exterior angles.
a.
b.
$42.55
c.
360
3. What pattern did you discover?
Lesson 5-7
°
60
72
72°
90°
360
For a scale of 1 cm : 12 km, find the actual length for each drawing length.
44. 0.5 cm 6 km
360
45. 1.2 cm
14.4 km
Sample answer: The sum of the exterior
angles is always 360.
4. Find the sum of the exterior angles for these nonregular polygons.
a. 20
b.
35
30
155
356
Chapter 7 Geometry
145
360
c.
360
d.
150
125
90
Alternative Assessment
60
100
130
130
125
360
135
360
5. Is the pattern the same or different? Why do you think this is true?
Sample answer: The same, since the sum of the exterior angle
measures is the same for all polygons regardless of the number of
sides.
356
I. 1318
Each student in a pair uses a protractor to draw
three adjacent angles whose sum is greater than
180° and less than 360°. Students exchange papers
and find the measure of each other’s angles. Then
they classify each angle as acute, obtuse, right, or
straight.
46. 1.7 cm
20.4 km
47. 2 cm 24 km