Download 1.4 Measure and Classify Angles

1.4
Measure and Classify Angles
Goal
Your Notes
p Name, measure, and classify angles.
VOCABULARY
Angle
Sides of an angle
Vertex of an angle
Measure of an angle
Acute angle
Right angle
Obtuse angle
Straight angle
Congruent angles
Angle bisector
Example 1
You should not
name any of these
angles B because
all three angles
have B as their
.
12
Name angles
Name the three angles in the diagram.
Lesson 1.4 • Geometry Notetaking Guide
, or
, or
, or
A
C
B
D
Copyright © Holt McDougal. All rights reserved.
1.4
Measure and Classify Angles
Goal
Your Notes
p Name, measure, and classify angles.
VOCABULARY
Angle An angle consists of two different rays with
the same endpoint.
Sides of an angle In an angle, the rays are called
the sides of the angle.
Vertex of an angle In an angle, the endpoint is the
vertex of the angle.
####$ and ###$
Measure of an angle In ∠ AOB, OA
OB can be
matched one to one with real numbers from 0 to
180. The measure of ∠ AOB is equal to the absolute
value of the difference between the real
OA and ###$
OB.
numbers for ####$
Acute angle An angle that measures between 08
and 908
Right angle An angle that measures 908
Obtuse angle An angle that measures between 908
and 1808
Straight angle An angle that measures 1808
Congruent angles Angles with the same measure
Angle bisector A ray that divides an angle into two
angles that are congruent
Example 1
You should not
name any of these
angles B because
all three angles
have B as their
vertex .
12
Name angles
Name the three angles in the diagram.
∠ ABC , or ∠CBA
∠CBD , or ∠ DBC
∠ ABD , or ∠ DBA
Lesson 1.4 • Geometry Notetaking Guide
A
C
B
D
Copyright © Holt McDougal. All rights reserved.
POSTULATE 3: PROTRACTOR POSTULATE
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
170 180
60
0
0 1 20 10
15
0 30
14 0
4
Consider ###$
OB and 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
0 10
180 170 1 20 3
60
15 0 4
01 0
40
Your Notes
O
B
is equal to
between the real numbers
The measure of
###$.
for ###$
OA and OB
Measure and classify angles
A protractor is
a drawing tool
that allows you to
measure angles.
You can also use a
protractor to draw
angles of given
measures.
a. ∠ WSR
b. ∠ TSW
c. ∠ RST
d. ∠ VST
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
R
T
scale of the
on the
angle.
ST is lined up with the 08 on the
b. ###$
SW passes through
protractor. ###$
scale. So, m∠ TSW 5
. It is
d. m∠ VST 5
V
S
a. ###$
SR is lined up with the 08 on the
SW passes through
protractor. ###$
scale. So, m∠ WSR 5
. It is
c. m∠ RST 5
W
170 180
60
0
0 1 20 10
15
0 30
14 0
4
Use the diagram to find the
measure of the indicated angle.
Then classify the angle.
0 10
180 170 1 20 3
60
15 0 4
01 0
40
Example 2
scale of the
on the
angle.
. It is
angle.
. It is
angle.
Checkpoint Complete the following exercises.
1. Name all the angles in
the diagram at the right.
J
F
G
H
2. What type of angles do the x-axis and y-axis form in
a coordinate plane?
Copyright © Holt McDougal. All rights reserved.
Lesson 1.4 • Geometry Notetaking Guide
13
POSTULATE 3: PROTRACTOR POSTULATE
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
170 180
60
0
0 1 20 10
15
0 30
14 0
4
Consider ###$
OB and 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 .
0 10
180 170 1 20 3
60
15 0 4
01 0
40
Your Notes
O
B
The measure of ∠ AOB is equal to the absolute
value of the difference between the real numbers
###$.
for ###$
OA and OB
Measure and classify angles
A protractor is
a drawing tool
that allows you to
measure angles.
You can also use a
protractor to draw
angles of given
measures.
a. ∠ WSR
b. ∠ TSW
c. ∠ RST
d. ∠ VST
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
W
R
V
170 180
60
0
0 1 20 10
15
0 30
14 0
4
Use the diagram to find the
measure of the indicated angle.
Then classify the angle.
0 10
180 170 1 20 3
60
15 0 4
01 0
40
Example 2
S
T
a. ###$
SR is lined up with the 08 on the outer scale of the
SW passes through 658 on the outer
protractor. ###$
scale. So, m∠ WSR 5 658 . It is an acute angle.
ST is lined up with the 08 on the inner scale of the
b. ###$
SW passes through 1158 on the inner
protractor. ###$
scale. So, m∠ TSW 5 1158 . It is an obtuse angle.
c. m∠ RST 5 1808 . It is a straight angle.
d. m∠ VST 5 908 . It is a right angle.
Checkpoint Complete the following exercises.
1. Name all the angles in
the diagram at the right.
J
F
G
H
∠ FGH or ∠ HGF, ∠ FGJ or ∠ JGF, ∠ JGH or ∠ HGJ
2. What type of angles do the x-axis and y-axis form in
a coordinate plane?
right angles
Copyright © Holt McDougal. All rights reserved.
Lesson 1.4 • Geometry Notetaking Guide
13
Your Notes
A point is in the
interior of an angle
if it is between
points that lie on
each side of the
angle.
interior
POSTULATE 4: ANGLE ADDITION POSTULATE
Words If P is in the interior of
∠ RST, then the measure of
∠ RST is equal to the sum of
the measures of ∠
and ∠
.
R
m/RST
S
m/PST
T
Symbols If P is in the interior of ∠ RST,
then m∠ RST 5 m∠
1 m∠
Example 3
m/RSP P
.
Find angle measures
Given that m∠ GFJ 5 1558,
find m∠ GFH and m∠ HFJ.
H
(4x 2 1)8
(4x 1 4)8
Solution
G
J
F
Step 1 Write and solve an equation to find the value of x.
m∠ GFJ 5 m∠
5(
1 m∠
)8 1 (
Angle Addition
Postulate
)8
Substitute.
5
Combine like terms.
5
Subtract
each side.
5x
Divide each side
by
.
from
Step 2 Evaluate the given expressions when x 5
m∠ GFH 5 (
)8 5 (
)8 5
m∠ HFJ 5 (
)8 5 (
)8 5
So, m∠ GFH 5
and m∠ HFJ 5
.
.
.
.
Checkpoint Complete the following exercise.
3. Given that ∠ VRS is a right angle,
find m∠ VRT and ∠ TRS.
(x 2 4)8
V
T
(3x 1 2)8
R
14
Lesson 1.4 • Geometry Notetaking Guide
S
Copyright © Holt McDougal. All rights reserved.
Your Notes
A point is in the
interior of an angle
if it is between
points that lie on
each side of the
angle.
interior
POSTULATE 4: ANGLE ADDITION POSTULATE
Words If P is in the interior of
∠ RST, then the measure of
∠ RST is equal to the sum of
the measures of ∠ RSP
and ∠ PST .
R
m/RST
m/RSP P
S
m/PST
T
Symbols If P is in the interior of ∠ RST,
then m∠ RST 5 m∠ RSP 1 m∠ PST .
Example 3
Find angle measures
Given that m∠ GFJ 5 1558,
find m∠ GFH and m∠ HFJ.
H
(4x 2 1)8
(4x 1 4)8
Solution
G
J
F
Step 1 Write and solve an equation to find the value of x.
m∠ GFJ 5 m∠ GFH 1 m∠ HFJ
1558 5 ( 4x 1 4 )8 1 ( 4x 2 1 )8
Angle Addition
Postulate
Substitute.
155 5 8x 1 3
Combine like terms.
152 5 8x
Subtract 3 from
each side.
19 5 x
Divide each side
by 8 .
Step 2 Evaluate the given expressions when x 5 19 .
m∠ GFH 5 ( 4x 1 4 )8 5 ( 4 p 19 1 4 )8 5 808 .
m∠ HFJ 5 ( 4x 2 1 )8 5 ( 4 p 19 2 1 )8 5 758 .
So, m∠ GFH 5 808 and m∠ HFJ 5 758 .
Checkpoint Complete the following exercise.
3. Given that ∠ VRS is a right angle,
find m∠ VRT and ∠ TRS.
(x 2 4)8
V
T
m∠VRT 5 198, m∠TRS 5 718
(3x 1 2)8
R
14
Lesson 1.4 • Geometry Notetaking Guide
S
Copyright © Holt McDougal. All rights reserved.
Your Notes
Example 4
Identify congruent angles
L
Identify all pairs of congruent angles
in the diagram. If m∠ P 5 1208, what
is m∠ N?
M
P
Solution
There are two pairs of congruent angles:
∠P >
and ∠ L >
Because ∠ P >
So, m∠ N 5
Example 5
N
, m∠ P 5
.
.
Double an angle measure
In the diagram at the right, ###$
WY
bisects ∠ XWZ, and m∠ XWY 5 298.
Find m∠ XWZ.
Z
Y
X
W
Solution
By the Angle Addition Postulate,
1
m∠ XWZ 5
.
Because ###$
WY bisects ∠ XWZ, you know
>
.
So,
m∠ XWZ 5
5
5
1
, and you can write
1
5
.
Checkpoint Complete the following exercises.
4. Identify all pairs of congruent
angles in the diagram. If
m∠ B 5 1358, what is m∠ D?
B
C
A
D
Homework
5. In the diagram below, ###$
KM bisects
∠ LKN and m∠ LKM 5 788. Find
m∠ LKN.
M
N
L
K
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Lesson 1.4 • Geometry Notetaking Guide
15
Your Notes
Example 4
Identify congruent angles
L
Identify all pairs of congruent angles
in the diagram. If m∠ P 5 1208, what
is m∠ N?
M
P
Solution
There are two pairs of congruent angles:
N
∠ P > ∠ N and ∠ L > ∠ M
Because ∠ P > ∠ N , m∠ P 5 m∠ N .
So, m∠ N 5 1208 .
Example 5
Double an angle measure
In the diagram at the right, ###$
WY
bisects ∠ XWZ, and m∠ XWY 5 298.
Find m∠ XWZ.
Z
Y
X
W
Solution
By the Angle Addition Postulate,
m∠ XWZ 5 m∠ XWY 1 m∠ YWZ .
Because ###$
WY bisects ∠ XWZ, you know
∠ XWY > ∠ YWZ .
So, m∠ XWY 5 m∠ YWZ , and you can write
m∠ XWZ 5 m∠ XWY 1 m∠ YWZ
5 298 1 298 5 588 .
Checkpoint Complete the following exercises.
4. Identify all pairs of congruent
angles in the diagram. If
m∠ B 5 1358, what is m∠ D?
Homework
B
C
A
D
∠ B > ∠ D and ∠ A > ∠ C; 1358
5. In the diagram below, ###$
KM bisects
∠ LKN and m∠ LKM 5 788. Find
m∠ LKN.
1568
Copyright © Holt McDougal. All rights reserved.
M
N
L
K
Lesson 1.4 • Geometry Notetaking Guide
15