Download 1-4 Study Guide and Intervention

NAME ______________________________________________ DATE
1-4
____________ PERIOD _____
Study Guide and Intervention
Angle Measure
Measure Angles If two noncollinear rays have a common
endpoint, they form an angle. The rays are the sides of the angle.
The common endpoint is the vertex. The angle at the right can be
named as /A, /BAC, /CAB, or /1.
B
1
A
A right angle is an angle whose measure is 90. An acute angle
has measure less than 90. An obtuse angle has measure greater
than 90 but less than 180.
Example 1
S
R
1 2
C
Example 2
Measure each angle and
classify it as right, acute, or obtuse.
T
3
Q
P
E
D
a. Name all angles that have R as a
vertex.
Three angles are /1, /2, and /3. For
other angles, use three letters to name
them: /SRQ, /PRT, and /SRT.
A
B
C
a. /ABD
Using a protractor, m/ABD 5 50.
50 , 90, so /ABD is an acute angle.
b. Name the sides of /1.
##$, RP
##$
RS
b. /DBC
Using a protractor, m/DBC 5 115.
180 . 115 . 90, so /DBC is an obtuse
angle.
c. /EBC
Using a protractor, m/EBC 5 90.
/EBC is a right angle.
Exercises
A
B
4
1. Name the vertex of /4.
1
D
2. Name the sides of /BDC.
3
2
C
3. Write another name for /DBC.
Measure each angle in the figure and classify it as right,
acute, or obtuse.
N
M
S
4. /MPR
P
5. /RPN
R
6. /NPS
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Glencoe/McGraw-Hill
19
Glencoe Geometry
Lesson 1-4
Refer to the figure.
NAME ______________________________________________ DATE
1-4
____________ PERIOD _____
Study Guide and Intervention
(continued)
Angle Measure
Congruent Angles
Angles that have the same measure are
congruent angles. A ray that divides an angle into two congruent
angles is called an angle bisector. In the figure, ##$
PN is the angle
bisector of /MPR. Point N lies in the interior of /MPR and
/MPN > /NPR.
M
N
P
R
Q
R
Example
Refer to the figure above. If m/MPN 5 2x 1 14 and
m/NPR 5 x 1 34, find x and find m/MPR.
Since ##$
PN bisects /MPR, /MPN > /NPR, or m/MPN 5 m/NPR.
2x 1 14 5 x 1 34
2x 1 14 2 x 5 x 1 34 2 x
x 1 14 5 34
x 1 14 2 14 5 34 2 14
x 5 20
m/NPR 5 (2x 1 14) 1 (x 1 34)
5 54 1 54
5 108
Exercises
##$ bisects /PQT, and QP
##$ and QR
##$ are opposite rays.
QS
1. If m/PQT 5 60 and m/PQS 5 4x 1 14, find the value of x.
S
T
P
2. If m/PQS 5 3x 1 13 and m/SQT 5 6x 2 2, find m/PQT.
##$ and BC
##$ are opposite rays, BF
##$ bisects /CBE, and
BA
#BD
#$ bisects /ABE.
E
D
3. If m/EBF 5 6x 1 4 and m/CBF 5 7x 2 2, find m/EBC.
F
1
A
2 3
B
4
C
4. If m/1 5 4x 1 10 and m/2 5 5x, find m/2.
5. If m/2 5 6y 1 2 and m/1 5 8y 2 14, find m/ABE.
6. Is /DBF a right angle? Explain.
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Glencoe/McGraw-Hill
20
Glencoe Geometry
NAME ______________________________________________ DATE
1-5
____________ PERIOD _____
Study Guide and Intervention
Angle Relationships
Pairs of Angles
Adjacent angles are angles in the same plane that have a common
vertex and a common side, but no common interior points. Vertical angles are two
nonadjacent angles formed by two intersecting lines. A pair of adjacent angles whose
noncommon sides are opposite rays is called a linear pair.
Example
Identify each pair of angles as adjacent angles, vertical angles,
and/or as a linear pair.
a.
b.
S
T
U
R
M
4
R
/SRT and /TRU have a common
vertex and a common side, but no
common interior points. They are
adjacent angles.
c.
A
S
/1 and /3 are nonadjacent angles formed
by two intersecting lines. They are vertical
angles. /2 and /4 are also vertical angles.
608
6
B
3N
2
d.
D
5
P
1
C
308
/6 and /5 are adjacent angles whose
noncommon sides are opposite rays.
The angles form a linear pair.
B
A
1208
F
608
G
/A and /B are two angles whose measures
have a sum of 90. They are complementary.
/F and /G are two angles whose measures
have a sum of 180. They are supplementary.
Exercises
Identify each pair of angles as adjacent, vertical, and/or
as a linear pair.
2. /1 and /6
V
2
1
3. /1 and /5
4. /3 and /2
3 4
6Q
R
R
S
P
For Exercises 5–7, refer to the figure at the right.
5. Identify two obtuse vertical angles.
S
5
V
N
U
6. Identify two acute adjacent angles.
T
7. Identify an angle supplementary to /TNU.
8. Find the measures of two complementary angles if the difference in their measures is 18.
©
Glencoe/McGraw-Hill
25
Glencoe Geometry
Lesson 1-5
1. /1 and /2
T
U
NAME ______________________________________________ DATE
1-5
____________ PERIOD _____
Study Guide and Intervention
(continued)
Angle Relationships
Perpendicular Lines
Lines, rays, and segments that form four right
angles are perpendicular. The right angle symbol indicates that the lines
#$ is perpendicular to @#$
are perpendicular. In the figure at the right, @AC
BD ,
@
#$
@#$
or AC ⊥ BD .
A
B
C
D
Example
Find x so that D
wZ
w⊥w
PZ
w.
If D
wZ
w⊥P
wZ
w, then m/DZP 5 90.
m/DZQ 1 m/QZP
(9x 1 5) 1 (3x 1 1)
12x 1 6
12x
x
5
5
5
5
5
m/DZP
90
90
84
7
D
Q
(9x 1 5)8
(3x 1 1)8
Sum of parts 5 whole
Substitution
Z
Simplify.
P
Subtract 6 from each side.
Divide each side by 12.
Exercises
#$ ⊥ @MQ
#$.
1. Find x and y so that @NR
N
P
2. Find m/MSN.
5x 8
M
x8
(9y 1 18)8 S
Q
R
#$ ⊥ #BF
#$. Find x.
3. m/EBF 5 3x 1 10, m/DBE 5 x, and #BD
E
D
4. If m/EBF 5 7y 2 3 and m/FBC 5 3y 1 3, find y so
#$ ⊥ ##$
that #EB
BC .
F
B
A
C
5. Find x, m/PQS, and m/SQR.
P
S
3x 8
(8x 1 2)8
Q
R
6. Find y, m/RPT, and m/TPW.
T
(4y 2 5)8
(2y 1 5)8
R
P
W
V
S
©
Glencoe/McGraw-Hill
26
Glencoe Geometry