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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