Transcript EXAMPLE 1 Name angles Name the three angles in the diagram. WXY, or YXW YXZ, or ZXY WXZ, or ZXW You should not name any of these angles X because.

EXAMPLE 1
Name angles
Name the three angles in the diagram.
WXY, or
YXW
YXZ, or
ZXY
WXZ, or
ZXW
You should not name any of these angles X
because all three angles have X as their vertex.
EXAMPLE 2
Measure and classify angles
Use the diagram to find the measure of the indicated
angle. Then classify the angle.
a.
KHJ
b.
GHK
c.
SOLUTION
A protractor has an inner and
an outer scale. When you
measure an angle, check to
see which scale to use.
GHJ
d.
GHL
EXAMPLE 2
Measure and classify angles
a.
HJ is lined up with the 0o on the inner scale of the
protractor. HK passes through 55o on the inner
o
scale. So, m KHJ = 55 . It is an acute angle.
b.
HG is lined up with the 0 on the outer scale and
o
HK passes through 125 on the outer scale. So,m
o
GHK = 125 . It is an obtuse angle.
c.
m
GHJ = 180.o It is a straight angle.
d.
m
GHL= 90. It is a right angle.
o
o
GUIDED PRACTICE
1.
for Examples 1and 2
Name all the angles in the diagram at the
right.Which angle is a right angle?
ANSWER
PQR ,
PQS,
RQS .
PQS is a right angle .
GUIDED PRACTICE
2.
for Examples 1and 2
Draw a pair of opposite rays. What type of angle
do the rays form?
ANSWER
Straight Angle
EXAMPLE 3
Find angle measures
ALGEBRA Given that m
and m
MKN.
o
LKN =145 , find m
LKM
SOLUTION
STEP 1
Write and solve an equation to find the value of x.
m
LKN = m LKM + m MKN
o
o
145 = (2x + 10)o + (4x – 3)
145 = 6x + 7
138 = 6x
23 = x
Angle Addition Postulate
Substitute angle measures.
Combine like terms.
Subtract 7 from each side.
Divide each side by 6.
EXAMPLE 3
Find angle measures
STEP 2
Evaluate the given expressions when x = 23.
m
LKM = (2x + 10)° = (2 23 + 10)° = 56°
m
MKN = (4x – 3)° = (4 23 – 3)° = 89°
ANSWER
So, m
LKM = 56° and m
MKN = 89°.
GUIDED PRACTICE
for Example 3
Find the indicated angle measures.
3.
Given that KLM is straight angle, find m
and m NLM.
KLN
SOLUTION
STEP 1
Write and solve an equation to find the value of x.
m
KLM + m NLM = 180°
(10x – 5)° + (4x +3)°= 180°
14x – 2 = 180
14x = 182
x = 13
Straight angle
Substitute angle measures.
Combine like terms.
Subtract 2 from each side.
Divide each side by 14.
GUIDED PRACTICE
for Example 3
STEP 2
Evaluate the given expressions when x = 13.
m
KLM = (10x – 5)° = (10 13 – 5)° = 125°
m
NLM = (4x + 3)° = (4 13 + 3)° = 55°
ANSWER
m
KLM = 125°
m
NLM = 55°
GUIDED PRACTICE
4. Given that
and m HFG.
for Example 3
EFG is a right angle, find m
EFH
SOLUTION
STEP 1
Write and solve an equation to find the value of x.
m
EFG = m
EFG + m HFG = 90°
(2x + 2)° + (x +1)° = 90°
3x + 3 = 90
3x = 87
x = 29
EFG is a right angle
Substitute angle measures.
Combine like terms.
Subtract 3 from each side.
Divide each side by 3.
GUIDED PRACTICE
for Example 3
STEP 2
Evaluate the given expressions when x = 29.
m
EFH = (2x + 2)° = (2 29 +2)° = 60°
m
HFG = (x + 1)° = (29 + 1)° = 30°
ANSWER
m
EFG = 60°
m
HFG = 30°
EXAMPLE 4
Identify congruent angles
Trapeze
The photograph shows some of the angles
formed by the ropes in a trapeze apparatus. Identify the
congruent angles. If m DEG = 157° ,what is m GKL?
SOLUTION
There are two pairs of congruent angles:
JKL and
DEG ~ GKL.
DEF ~
Because  DEG~
GKL,
DEG = m
So, m GKL = 157°.
GKL.
GUIDED PRACTICE
for Example 4
Use the diagram shown below.
5.
Identify all pairs of congruent angles in the
diagram.
SOLUTION
There are two pairs of Congruent angles in the
diagram.
T ~
S and
P~
Q.
GUIDED PRACTICE
for Example 4
Use the diagram shown at the right.
In the diagram, m PQR = 130o, m QRS = 84,o and
m TSR = 121o . Find the other angle measures in
the diagram.
SOLUTION
6.
PTS ~
TSR = 121°
Congruent angles
QRS ~
QPT= 84°
Congruent angles
EXAMPLE 5
Double an angle measure
In the diagram at the right, YW bisects
o
m XYW = 18. Find m XYZ.
XYZ, and
SOLUTION
By the Angle Addition Postulate,
m XYZ = m XYW + m WYZ. Because YW bisects
you know that XYW ~
WYZ.
So, m
M
XYW = m WYZ, and you can write
XYZ = m
XYW + m
WYZ = 18° + 18° = 36°.
XYZ
GUIDED PRACTICE
7.
for Example 5
Angle MNP is a straight angle, and NQ bisects
MNP. Draw MNP And NQ . Use arcs to mark
the congruent angles in your diagram, and give
the angle measures of these congruent angles.
SOLUTION
GUIDED PRACTICE
for Example 5
m
MNQ + m
PNQ
m
MNQ + m
PNQ = 180° Straight angle
m
MNQ + m MNQ = 180° m MNQ = m PNQ
2 m MNQ = 180° Add
m MNQ = 90° Divided each side by 2
The solution is m
Angle addition postulate
MNQ = m
PNQ = 90°