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Ch. 8 Notes
1
Name _________________________________________________________ Hour __________
Warm-up:
Simplify each fraction.
1. 24
36
2.
49
56
Students will be able to solve proportions and ratios.
Ratio
8.1 & 8.2 Ratio and Proportion
________________________________________
________________________________________
Proportions
An equation that sets two ratios equal
a c

b d
Extremes
The numbers a and d
Mean
The numbers b and c
Examples
Simplify the ratios.
1.
12cm
4m
2.
6 ft
18in
Ch. 8 Notes
2
Name _________________________________________________________ Hour __________
Example
The perimeter of rectangle ABCD is 60
centimeters. The ratio of AB:BC is 3:2. Find the
length and width of the rectangle.
D
C
w
A
l
B
Cross
Product
Property
The product of the extremes equals the product
of the means.
Reciprocal
Property
If two ratios are equal, then their reciprocals are
also equal.
If
a c

b d
, then ad = bc.
a
c
If b  d , then
More
Properties
of
Proportions
Geometric
Mean
b d

a c
a
c
a b

c d
a
c
a b c d

b
d
If b  d , then
If b  d , then
If the sides of a triangle are in the following
a x
proportion x  b , then the geometric mean is
x  a b
Ch. 8 Notes
3
Name _________________________________________________________ Hour __________
Examples
Solve the proportions.
1.
4 5

x 7
The measure of the angles in JKL are in the
extended ratio of 1:2:3. Find the measures of the
K
angles.
J
L
The ratios of the side lengths of DEF to the
corresponding side lengths of ABC are 2:1. Find
C
the unknown lengths.
3 in
A
B
F
D
Homework: 8.1 worksheet
8 in
E
Ch. 8 Notes
4
Name _________________________________________________________ Hour __________
Warm-up:
Solve each proportion.
1.
12 40

x 20
2.
7 3

9 x
Students will be able to solve equations using proportions.
Examples
8.2 Problem Solving with Proportions
Tell whether the statement is true.
a.
If
p
r
p 3


,
then
6 10
r 5
b.
If
a c
a 3 c 3
 , then

3 4
3
4
In the diagram
AB AC

BD CE
. Find the length of
BD .
A
16
B
x
D
Homework: 8.2 wkst.
30
C
10
E
Ch. 8 Notes
5
Name _________________________________________________________ Hour __________
Warm-up:
Find the geometric mean of the two numbers.
1.
8 and 18
2.
10 and 15
Students will be able to solve for missing sides of polygons.
Similar
Polygons
8.3 Similar Polygons
________________________________________
________________________________________
Theorem 8.1 If two polygons are similar, then the ratio of their
perimeters is equal to the ratios of their
P
corresponding side lengths.
K
Q
If KLMN ~ PQRS, then
KL  LM  MN  NK KL LM MN NK




PQ  QR  RS  SP
PQ QR
RS
SP
L
S
N
Examples
R
M
Pentagons JKLMN and STUVW are similar. List all
the pairs of congruent angles. Write the ratios of
the corresponding sides in a statement of V
proportionality.
M
W
N
L
J
K
U
S T
Ch. 8 Notes
6
Name _________________________________________________________ Hour __________
Decide whether the figures are similar. If they
are similar, write a similarity statement.
Q
4
R
X
6
Y
6
10
9
S
P
15
8
Z
12
W
Quadrilateral JKLM is similar to quadrilateral
z
Q
PQRS.
J
10
K
6
Find the value of z.
15
M
R
S
L
Parallelogram ABCD is similar to parallelogram
GBEF. Find the value of y. B
E
C
12
G
15
F
A
P
y
24
D
Ch. 8 Notes
7
Name _________________________________________________________ Hour __________
UVW is similar to YXW. Find the value ofU a.
V
a
5
W
6.4
12
X
Y
Homework: 8.3 wkst
Warm-up:
Quiz 8.1-8.3
List all the pairs of congruent angles and write the statement of
proportionality for the figures.
1. ATL~ NYC
Warm-up:
Determine whether the following polygons are similar. If so, write a statement of
similarity.
I
5
L
A
D
O
15
A
1.
2.
4
4
8
6
B
7
C
3
E
4
3.5
F
8
G
15
8
T
B
5
Y
Students will be able to use the properties of similar triangles to find unknown
parts of triangles.
Ch. 8 Notes
8
Name _________________________________________________________ Hour __________
8.4 Similar Triangles
Angle-Angle Postulate 25
If two angles of one triangle are congruent to
Similarity
two angles of another triangle, then the two
Postulate
triangles are similar.
(AA)
If  JKL   XYZ and  KJL K YXZ ,
Y
Then  JKL ~  XYZ
L
Z
X
J
Side-SideSide
Similarity
Postulate
(SSS)
Theorem 8.2
If the corresponding sides of two triangles are
proportional, then the triangles are similar.P
AB
BC
If PQ  QR

CA
,
RP
A
Then ABC ~ PQR
Q
C
B
Side-AngleSide
Similarity
Postulate
(SAS)
R
Theorem 8.3
If an angle of one triangle is congruent to an
angle of a second triangle and the lengths of the
sides including these angles are proportional,
then the triangles are similar.
If X  M and
ZX
XY

PM MN
Then XYZ ~ MNP
,
X
M
N
Y
Z
P
Ch. 8 Notes
9
Name _________________________________________________________ Hour __________
In the diagram,  BTW ~  ETC .
a. Write the statement of proportionality.
Examples:
T
b.
E
Find mTEC
34o
3
20
C
79o
B
c.
12
W
Find ET and BE.
Find the length of the altitude QS .
N
12
M
12
6
Q
R
Homework: 8.4 wkst
Warm-up:
Find the value of the missing sides.
O
24
x
G
A
12
30
T
I
3
B
9
w
L
y
Y
8
S
8
T
P
Ch. 8 Notes
10
Name _________________________________________________________ Hour __________
Students will be able to prove triangles similar.
Examples
8.5 Proving Triangles
Which of the following three triangles are
similar?
A
12
C
E
G
4
6
9
B
14
6
F
J
6
8
10
D
H
Use the given lengths to prove RST ~ PSQ.
Given: SP = 4, PR = 12, SQ = 5, QT = 15
S
Prove: RST ~ PSQ
4
P
12
R
5
Q
15
T
Ch. 8 Notes
11
Name _________________________________________________________ Hour __________
Indirect Measurement: To measure the width of
a river, you use a surveying technique, as shown
in the diagram. Use the given lengths (measured
in feet) to find RQ.
P
63
Q
R
12
T
9
S
Homework: 8.5 wkst
Warm-up:
Determine if the following triangles are congruent. If so, write the
statement of similarity and give the reason.
A
D
1.
4
3
B
5
1.5
2
E
F
C 2.5
Students will be able to use the properties of similar triangles to find
unknown parts of triangles.
8.6 Proportions and Similar Triangles
Triangle
Theorem 8.4
Proportional If a line parallel to one side of a triangle
ity Theorem intersects the other two sides, then it divides the
Q
T
two sides proportionally.
If
TU QS , then
RT RU

TQ US
R
S
U
Ch. 8 Notes
12
Name _________________________________________________________ Hour __________
Converse of Theorem 8.5
If a line divides two sides of a triangle
Triangle
Proportional proportionally, then it is parallel to the third side.
RT RU
Q
ity Theorem
T
If TQ  US , then TU QS .
R
S
Theorem 8.6
U
If three parallel lines intersect two transversals,
then they divide the transversals proportionally.
If r s and s t , and l and m intersect r, s, and
t,
Then
UW VX

WY
XZ
r
U
s
W
X
V
t
Y
Z
l
m
Theorem 8.7 If a ray bisects an angle of a triangle it divides the
opposite side into segments whose lengths are
proportional to the lengths of the other two
sides.
If CD bisects
 ACB , then
AD CA

A
DB CB
C
D
B
Ch. 8 Notes
13
Name _________________________________________________________ Hour __________
In the diagram, AB ED , BD = 8, DC = 4, and AE =
12. What is the length of EC ? C
4
D
8
E
12
B
A
Given the diagram, determine whether
MN GH
M
G
21
56
L
N 16 H
48
In the diagram,  1   2   3 , and PQ = 9, QR =
15, and ST = 11. What is the length of TU ?
P
9
Q
15
R 3
S
1
2
11
T
U
Ch. 8 Notes
14
Name _________________________________________________________ Hour __________
In the diagram, CAD ~ DAB . Use the given
side lengths to find the length of DC .
B
9
D
A
In the diagram
variables.
KL MN .
14
C
15
Find the values of the
J
9
K
L
37.5
7.5
x
13.5
M
Homework: 8.6 wkst
y
N
Ch. 8 Notes
15
Name _________________________________________________________ Hour __________
Warm-up:
Quiz 8.4-8.6
Are the following triangles similar? If so write the similarity statement
and give the reason why.
A
D
1.
o
98
50
o
32
B
o
C
o
E 50
F
Warm-up:
Find the value of the variable.
J
1.
9
32
K
L
12
x
15
M
y
N
Warm-up:
Find the value of the variable.
B
1.
10
D
8
A
21
C
Warm-up:
Ch. 8 Test