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Similar Polygons
(page 240)
Ratio and Proportion
(page 241)
Chapter 7:
7-1:
RATIO: the quotient of
where b ≠ 0.
numbers, a÷ b, usually written as
or a : b,
examples:
A ratio is simply a
.
A ratio is usually expressed in
form.
Ratios can be used to
two numbers, ie. a : b .
The quantities being compared must be in the same
.
Note: ratios have no
of numbers.
, it is a
Ratios can be used to
more than two numbers.
To compare more than two numbers, use the notation a : b : c : d .
PROPORTION: an
stating that two ratios are
a c
=
b d
The 1st term is
and
, the 2nd term is
a:b =
a c e
= =
b d f
and
________: ________
, the 3rd term is
When 3 or more ratios are equal, you can write an
a:b =
.
, & the 4th term is
.
proportion.
________: ________ = ________: ________
examples:
(1)
What is the ratio of 250 mL to 0.5 L?
:
(2)
Two complementary angles have measures in the ratio 2 : 7.
Find the measure of each angle.
m ∠1 =
m ∠2 =
(3)
Five angles of a pentagon have their measures in the ratio 9 : 8 : 6 : 4 : 3.
Find the measure of each angle.
m ∠1 =
m ∠2 =
m ∠3 =
m ∠4 =
m ∠5 =
Assignment: Written Exercises, pages 243 & 244: 1-31 odd #’s
7-2:
Properties of Proportions
In the proportion,
a:b = c:d
The first & last terms,
The middle terms,
&
or
a c
=
b d
(pages 245)
,
, are called the
&
.
, are called the
.
The product of the means is equal to the product of the
.
Properties of Proportions:
(1)
This is called the
-
property of proportions.
example:
(2)
If you interchange the means, you get an equivalent
.
example:
(3)
If you take the reciprocals of the
, the result is an equivalent proportion.
example:
(4)
If you add
to both sides, you still have an equivalent proportion.
example:
(5)
In an extended proportion, the ratio of the sum of all the numerators to the
all the denominators is equivalent to each of the
example:
of
ratios.
examples:
Use the proportion
(1)
5a=
(3)
a+b
b =
examples:
(5)
In the figure,
a 3
=
b 5
to complete each statement.
(2)
5
b=
(4)
5
3=
AD CE
=
DB EB .
If CE = 2 , EB = 6 , and AD = 3, then DB =
.
A
D
C
(6)
E
B
If AB = 10 , DB = 8 , and CB = 7.5, then EB =
.
A
D
C
E
B
Assignment: Written Exercises, pages 247 & 248: 1-27 odd #’s, 33, 35, 37
Similar Polygons
7-3:
(page 248)
SIMILAR POLYGONS: Two polygons are similar if their vertices can be paired so that
corresponding angles are
and
corresponding sides are in
.
Similar figures have the same
A
, but not necessarily the same
B
G
.
F
J
E
H
D
I
C
If polygon ABCDE ~ polygon FGHIJ , then:
!A " ____; !B " ____; !C " ____; !D " ____; !E " ____
AB
and
=
BC
=
CD
DE
=
=
AE
SCALE FACTOR: the ratio of the lengths of two
sides.
Quad. ABCD ~ Quad. A’B’C’D’
example:
A
24
(A’ is read “A prime”)
B
A’
28
32
D
y
Scale Factor = SF = _____ ; x =
18
B’
z
C
x
D’
; y =
27
; z =
C’
Quad. ABCD ~ Quad. WXYZ , find the unknown angle measures.
example:
D
C
Z
130º
Y
60º
B
A
W
X
m!B = ____; m!D = ____; m!Y = ____; m!Z = ____
Quad. GEOM ~ Quad. G’E’O’M’.
example:
Find the scale factor, the perimeter of GEOM, and the perimeter of G’E’O’M’.
O’
24
O
M’
M
18
G
21
27
E
24
G’
E’
scale factor = S.F. =
perimeter of GEOM =
perimeter of G’E’O’M’ =
Compare the ratio of the perimeters. The ratio of perimeters =
.
Assignment: Written Exercises, pages 250 to 252: 1-21 odd #’s, 24, 25, 27, 34, 35
Prepare for Quiz on Lessons 7-1 to 7-3: Ratio, Proportion, and Similarity
A Postulate for Similar Triangles
7-4:
(page 254)
To prove two triangles congruent, you didn’t need to compare all
pairs of
corresponding parts.
Postulates and theorems gave easier ways to
congruence.
There are also postulates and theorems for proving triangles
comparing all of the
without
parts.
AA Similarity Postulate
Postulate 15
If two angles of one triangle are congruent to
the triangles are similar.
example:
angles of another triangle, then
Tell whether the triangles are similar or not similar.
If you can’t reach a conclusion, write no conclusion is possible.
(1)
(2)
62º
54º
60º
60º
1.
2.
(3)
(4)
75º
40º
45º
3.
45º
4.
example:
Find the values of “x” and “y”.
(5)
8
10
x
x = ________
4
y
y = ________
15
(6)
AC ! DE
C
3
D
12
6
y
A
x
E
4
B
x = ________
y = ________
Assignment: Written Exercises, pages 257 to 259: 1-15 ALL #’s
7-5:
Theorem 7-1
Theorems for Similar Triangles
(page 263)
SAS Similarity Theorem
If an angle of one triangle is congruent to an angle of another triangle and the sides
including those angles are in
Given:
, then the triangles are similar.
∠A ≅ ∠D
A
D
AB AC
=
DE DF
E
Prove: ∆ ABC ~ ∆
Theorem 7-2
B
C
SSS Similarity Theorem
If the sides of two triangles are in
, then the triangles are similar.
A
Given:
F
D
AB BC AC
=
=
DE EF DF
Prove: ∆ ABC ~ ∆
E
B
F
C
To prove two polygons similar, you may need to compare
sides.
A useful technique is to compare the
the
sides,
sides, and so on.
Perimeters of similar polygons are in the same
as the corresponding sides.
By using similar triangles, you can prove that corresponding segments,
such as diagonals of similar polygons, also have this
.
example:
(1)
Can the information given be used to prove ∆ PQR ~ ∆ TSP? If so, how?
SP = 8, TS = 6, PT = 12, QR = 12, PQ = 9, RP = 18
Q
P
T
R
S
(2)
SP = 7, TS = 6, PQ = 9, QR = 10.5, m∠S = 80º, m∠QPR + m∠QRP = 100º.
Q
P
T
R
S
Assignment: Written Exercises, pages 266 & 267: 1-10 ALL #’s, 14 [Bonus]
BONUS Quiz on Lessons 7-4 & 7-5
Proportional Lengths
7-6:
(page 269)
If AB : BC = XY : YZ , then AC and XZ are said to be divided
A
B
C
X
.
Y
Z
Triangle Proportionality Theorem
Theorem 7-3
If a line parallel to one side of a triangle intersects the other two sides, then it divides
those sides
.
T
Given: ∆ RST ;
Prove:
PQ ! RS
RP SQ
=
PT QT
P
Q
R
NOTE:
Since ∆ RST ~ ∆
S
, many equivalent proportions can be justified.
Here are many equivalent proportions along with informal statements describing them.
RP SQ
=
PT QT
smaller piece smaller piece
=
larger piece
larger piece
RP SQ
=
RT ST
smaller piece smaller piece
=
whole side
whole side
PT QT
=
RT ST
larger piece larger piece
=
whole side
whole side
RP PT RT
=
=
SQ QT ST
smaller piece larger piece whole side
=
=
smaller piece larger piece whole side
PQ
Name some other equivalent proportions:
=
PT
=
QT
example:
Find the value of “x”.
(1)
20
15
8
x
x=
REMEMBER: If 3 parallel lines cut off congruent segments on one transversal,
then they cut off congruent segments on every transversal.
Corollary
If three parallel lines intersect two transversals, then they divide the transversals
.
Given:
R
RX ! SY ! TZ
S
Prove:
example:
RS XY
=
ST YZ
X
Y
T
Z
Find the value of “x”.
(2)
25 - x
x
16
4
x=
Triangle Angle-Bisector Theorem
Theorem 7-4
If a ray bisects an angle of a triangle, then it divides the
side into
segments proportional to the other 2
Given:
Prove:
.
!DEF ; DG bisects !FDE
F
GF DF
=
GE DE
G
E
D
example:
Find the value of “x”.
(3)
(4)
12
27
45
18
x
x
|<---------------------40---------------------->|
x=
24
x=
Assignment: Written Exercises, pages 272 & 273: 1-11 ALL #’s, 13, 15, 17, 21, 23
Prepare for Quiz on Lessons 7-4 to 7-6: Working with Similar Triangles
Prepare for Test on Chapter 7: Similar Polygons