Download Section 1.4 PowerPoint File

Put everything away and get out a piece of
paper. Make sure you put your name and
period on the paper.
• Find the length of segment ST. Round to
the nearest tenth.
Definition
Angle: two different rays with the same
endpoint. The rays are the sides of the angle.
The endpoint is the vertex.
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.
Protractor Postulate: The measure of angle AOB
is the absolute value of the difference between
the real numbers for OA and OB.
A
O
B
Classifying Angles: Angles can be classified
as acute, right, obtuse, and straight.
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.
o
o
d. m GHL= 90.
It is a right angle.
GUIDED PRACTICE
1.
for Examples 1and 2
Name all the angles in the diagram. 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
Homework!
HW: P.28: 1-21 odd
Angle Addition Postulate
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 a straight angle, find m
and m NLM.
ANSWER 125°, 55°
KLN
GUIDED PRACTICE
4. Given that
and m HFG.
for Example 3
EFG is a right angle, find m
ANSWER 60°, 30°
EFH
Definition
Congruent Angles: Two angles are contruent
angles if they have the same measure.
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:
DEF ~
JKL and
DEG ~ GKL.
Because  DEG~
GKL,
DEG = m GKL.
So, m GKL = 157°.
GUIDED PRACTICE
for Example 4
Use the diagram shown.
5.
Identify all pairs of congruent angles in the diagram.
ANSWER
T and
S,
P and
R.
GUIDED PRACTICE
for Example 4
Use the diagram shown.
6.
In the diagram, m PQR = 130o, m QRS = 84,o and
m TSR = 121o . Find the other angle measures in
the diagram.
ANSWER
m
PTS = 121°, m
QPT = 84°
Definition
Angle Bisector: A ray that divides an angle into two
angles that are congruent
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,
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°.
m
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.
ANSWER
90°
Homework!
• HW: 29: 23-47 odd, 53, 54