1.5 Measuring and Constructing Angles
1.5
COMMON
CORE
Learning Standards
HSG-CO.A.1
HSG-CO.D.12
Measuring and Constructing Angles
Essential Question
How can you measure and classify an angle?
Measuring and Classifying Angles
Work with a partner. Find the degree measure of each of the following angles.
Classify each angle as acute, right, or obtuse.
D
C
0 10
180 170 1 20 3
60
15 0 4
01 0
40
E
a. ¡ÏAOB
e. ¡ÏCOE
B
170 180
60
0 1 20 10 0
15
0 30
14 0
4
80 90 10 0
70 10 0 90 80 110 1
70
2
0
60 0 11
60 0 13
2
0
1
5 0
50 0
13
O
b. ¡ÏAOC
f. ¡ÏCOD
c. ¡ÏBOC
g. ¡ÏBOD
A
d. ¡ÏBOE
h. ¡ÏAOE
Drawing a Regular Polygon
Work with a partner.
a. Use a ruler and protractor to draw the
triangular pattern shown at the right.
b. Cut out the pattern and use it to draw
three regular hexagons, as shown below.
ATTENDING
TO PRECISION
To be proficient in math,
you need to calculate and
measure accurately
and efficiently.
2 in.
2 in.
2 in.
2 in.
120¡ã
2 in.
2 in.
2 in.
2 in.
2 in.
2 in.
2 in.
2 in.
2 in.
2 in.
2 in.
2 in.
2 in.
2 in.
2 in.
2 in.
c. The sum of the angle measures of a polygon with n sides is equal to 180(n ? 2)¡ã.
Do the angle measures of your hexagons agree with this rule? Explain.
d. Partition your hexagons into smaller polygons, as shown below. For each hexagon,
find the sum of the angle measures of the smaller polygons. Does each sum equal
the sum of the angle measures of a hexagon? Explain.
Communicate Your Answer
3. How can you measure and classify an angle?
Section 1.5
Measuring and Constructing Angles
37
1.5 Lesson
What You Will Learn
Name angles.
Core Vocabul
Vocabulary
larry
Measure and classify angles.
angle, p. 38
vertex, p. 38
sides of an angle, p. 38
interior of an angle, p. 38
exterior of an angle, p. 38
measure of an angle, p. 39
acute angle, p. 39
right angle, p. 39
obtuse angle, p. 39
straight angle, p. 39
congruent angles, p. 40
angle bisector, p. 42
Use the Angle Addition Postulate to find angle measures.
Identify congruent angles.
Bisect angles.
Naming Angles
C
An angle is a set of points consisting of two
different rays that have the same endpoint,
called the vertex. The rays are the sides of
the angle.
vertex
1
You can name an angle in several different ways.
A
? Use its vertex, such as ¡ÏA.
? Use a point on each ray and the
vertex, such as ¡ÏBAC or ¡ÏCAB.
? Use a number, such as ¡Ï1.
Previous
protractor
degrees
sides
The region that contains all the points
between the sides of the angle is the
interior of the angle. The region that
contains all the points outside the angle
is the exterior of the angle.
B
exterior
interior
Naming Angles
COMMON ERROR
When a point is the vertex
of more than one angle,
you cannot use the vertex
alone to name the angle.
A lighthouse keeper measures the
angles formed by the lighthouse
at point M and three boats. Name
three angles shown in the diagram.
J
ligh
lighthouse
htho
o
ouse
SOLUTION
¡ÏJMK or ¡ÏKMJ
K
M
¡ÏKML or ¡ÏLMK
L
¡ÏJML or ¡ÏLMJ
Monitoring Progress
Help in English and Spanish at
Write three names for the angle.
1.
2.
Q
38
Chapter 1
Basics of Geometry
E
2
R
P
3.
X
1
Z
Y
F
D
Measuring and Classifying Angles
A protractor helps you approximate the measure of an angle. The measure is usually
given in degrees.
Postulate 1.3 Protractor Postulate
Consider ??
OB and a point A on one
side of ??
OB. The rays of the form ?
OA
can be matched one to one with the
real numbers from 0 to 180.
The measure of ¡ÏAOB, which can
be written as m¡ÏAOB, is equal to
the absolute value of the difference
between the real numbers matched
? on a protractor.
with ?
OA and OB
80 90 10 0
70 10 0 90 80 110 1
70
2
60 0 110
60 0 1
2
3
50 0 1
50 0
13
A
O
B
170 180
60
0 1 20 10 0
15
0 30
14 0
4
You can classify angles according to their measures.
Core Concept
Types of Angles
A
A
A
acute angle
Measures greater
than 0¡ã and less
than 90¡ã
A
right angle
obtuse angle
straight angle
Measures 90¡ã
Measures greater than
90¡ã and less than 180¡ã
Measures 180¡ã
Measuring and Classifying Angles
Find the measure of each angle.
Then classify each angle.
b. ¡ÏJHL
c. ¡ÏLHK
SOLUTION
a. ?
HG lines up with 0¡ã on the outer
scale of the protractor. ?
HK passes
through 125¡ã on the outer scale. So,
m¡ÏGHK = 125¡ã. It is an obtuse angle.
G
L
M
K
H
J
170 180
60
0 1 20 10 0
15
0 30
14 0
4
a. ¡ÏGHK
80 90 10 0
70 10 0 90 80 110 1
70
2
60 0 110
60 0 1
2
3
50 0 1
50 0
13
0 10
180 170 1 20 3
60
15 0 4
01 0
40
Most protractors have an
inner and an outer scale.
When measuring, make
sure you are using the
correct scale.
Postulate
0 10
180 170 1 20 3
60
15 0 4
01 0
40
COMMON ERROR
b. ?
HJ lines up with 0¡ã on the inner scale of the protractor. ?
HL passes through 90¡ã.
So, m¡ÏJHL = 90¡ã. It is a right angle.
c. ?
HL passes through 90¡ã. ?
HK passes through 55¡ã on the inner scale. So,
m¡ÏLHK = ¨O 90 ? 55 ¨O = 35¡ã. It is an acute angle.
Monitoring Progress
Help in English and Spanish at
Use the diagram in Example 2 to find the angle measure. Then classify the angle.
4. ¡ÏJHM
5. ¡ÏMHK
Section 1.5
6. ¡ÏMHL
Measuring and Constructing Angles
39
Identifying Congruent Angles
You can use a compass and straightedge to construct an angle that has the same
measure as a given angle.
Copying an Angle
Use a compass and straightedge to construct an angle that has the same measure as
¡ÏA. In this construction, the center of an arc is the point where the compass point
rests. The radius of an arc is the distance from the center of the arc to a point on the
arc drawn by the compass.
SOLUTION
Step 1
Step 2
Step 3
C
A
A
Step 4
C
A
B
C
A
B
B
F
D
E
D
Draw a segment Draw
an angle such as ¡ÏA,
as shown. Then draw a
segment. Label a point D
on the segment.
F
E
D
Draw an arc Label B, C,
and E. Draw an arc with
radius BC and center E.
Label the intersection F.
Draw arcs Draw an arc
with center A. Using the
same radius, draw an arc
with center D.
E
D
Draw a ray Draw ?
DF.
¡ÏEDF ? ¡ÏBAC.
Two angles are congruent angles when they have the same measure. In the
construction above, ¡ÏA and ¡ÏD are congruent angles. So,
m¡ÏA = m¡ÏD
The measure of angle A is equal to the measure of angle D.
¡ÏA ? ¡ÏD.
Angle A is congruent to angle D.
and
Identifying Congruent Angles
a. Identify the congruent angles labeled
in the quilt design.
b. m¡ÏADC = 140¡ã. What is m¡ÏEFG?
G
H
E
SOLUTION
F
a. There are two pairs of congruent angles:
¡ÏABC ? ¡ÏFGH and
READING
In diagrams, matching arcs
indicate congruent angles.
When there is more than
one pair of congruent
angles, use multiple arcs.
¡ÏADC ? ¡ÏEFG.
b. Because ¡ÏADC ? ¡ÏEFG,
m¡ÏADC = m¡ÏEFG.
C
D
So, m¡ÏEFG = 140¡ã.
A
Monitoring Progress
Help in English and Spanish at
7. Without measuring, is ¡ÏDAB ? ¡ÏFEH in Example 3? Explain your reasoning.
Use a protractor to verify your answer.
40
Chapter 1
Basics of Geometry
B
Using the Angle Addition Postulate
Postulate
Postulate 1.4 Angle Addition Postulate
R
Words
If P is in the interior of
¡ÏRST, then the measure of ¡ÏRST is
equal to the sum of the measures of
¡ÏRSP and ¡ÏPST.
m¡ÏRST
m¡ÏRSP
S
m¡ÏPST
Symbols If P is in the interior of
P
¡ÏRST, then
T
m¡ÏRST = m¡ÏRSP + m¡ÏPST.
Finding Angle Measures
Given that m¡ÏLKN = 145¡ã,
find m¡ÏLKM and m¡ÏMKN.
M
(2x + 10)¡ã
L
(4x ? 3)¡ã
K
SOLUTION
N
Step 1 Write and solve an equation to find the value of x.
m¡ÏLKN = m¡ÏLKM + m¡ÏMKN
Angle Addition Postulate
145¡ã = (2x + 10)¡ã + (4x ? 3)¡ã
Substitute angle measures.
145 = 6x + 7
Combine like terms.
138 = 6x
Subtract 7 from each side.
23 = x
Divide each side by 6.
Step 2 Evaluate the given expressions when x = 23.
?
?
m¡ÏLKM = (2x + 10)¡ã = (2 23 + 10)¡ã = 56¡ã
m¡ÏMKN = (4x ? 3)¡ã = (4 23 ? 3)¡ã = 89¡ã
So, m¡ÏLKM = 56¡ã, and m¡ÏMKN = 89¡ã.
Monitoring Progress
Help in English and Spanish at
Find the indicated angle measures.
8. Given that ¡ÏKLM is a straight angle,
find m¡ÏKLN and m¡ÏNLM.
9. Given that ¡ÏEFG is a right angle,
find m¡ÏEFH and m¡ÏHFG.
N
E
(2x + 2)¡ã
(10x ? 5)¡ã (4x + 3)¡ã
K
L
(x + 1)¡ã
M
F
Section 1.5
H
G
Measuring and Constructing Angles
41
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