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Chapter 7
Similarity and Proportion
• Express a ratio in
simplest form.
• State and apply the
properties of similar
polygons.
• Use the theorems
about similar triangles.
Warm –up
• In complete sentences, explain what a ratio
is. Create a real-life example of a ratio
being used.
• In complete sentences, explain what a
proportion is.
7.1 Ratio and Proportion
Objectives
• Express a ratio in simplest form
• Solve for an unknown in a proportion
Ratio
• A ratio of one number to another is the quotient when the
first number is divided by the second.
• A comparison between numbers
• There are 3 different ways to express a ratio
1
2
3
5
1:2
3:5
a:b
1 to 2
3 to 5
a to b
a
b
Ratio
• Always reduce ratios to the simplest form
• The ratio of 8 to 12 is
8_ =
12
2_
3
O
Z
Find the ratio of OI TO ZD
- Put the answer in your notes.
- make sure to reduce
Find the ratio of LD to LO
Angle ratio on whiteboard
110
14
6b
D
60
70
I
Additional Elements of Ratios
• What is the ratio of 100cm to 10m ?
100 = 10 ?
10
1
• WHY??
– To find the ratio of two lengths, they must
always be measured in the terms of the same
unit.
Unit Conversions
100cm
1m
=
10m
10m
or
100cm
100cm
=
10m
1000cm
**Both ways give you
the same ratio which
is 1 to 10.
Comparing 3 or more numbers
• We use the following form to represent
three or more numbers that are in ratio to
each other…
3:5:7
• Reads as.. “ 3 to 5 to 7"
Comparing 3 or more numbers
• The measures of three angles of a triangle
are in the ratio 2:2:5. Find the measure of
each angle.
• Partners: Set up the problem…
– X represents a “part” of each angle
– 2x + 2x + 5x = 180
• SO WHAT’S EACH MEASURE?
Proportion
• An equation stating to ratios are equal.
5 a

8 b
5:8= a:b
• In both instances you read the proportion as…..
“5 is to 8 as a is to b.”
• WHAT WOULD BE AN EXAMPLE OF A
TRUE PROPORTION?
White Board Practice
• ABCD is a parallelogram. Find the value of
each ratio.
B
A
6
D
10
C
White Board Practice
• AB : BC
– 5:3
• BC : AD
– 1:1
• mA:mC
B
A
– 1:1
• AB : perimeter of
ABCD
6
– 5 : 16
D
10
C
White Board Practice
• Express the ratio in simplest form
• IS : DI : IT
10
D
4
I
12
S
T
White Board Practice
• Express the ratio in simplest form
• IS : DI : IT
• 4 : 10 : 16
• 2 : 5 : 8
----------To reduce, find GCF
10
D
4
I
12
S
T
Whiteboard Practice
• The ratio of the measures of two
complementary angles is 4:5. Find the
measure of each angle.
• 4x + 5x = 90
• 9x = 90
• X = 10
• 40, 50
7.2 Properties of Proportions
Objectives
• Express a given
proportion in an
equivalent form.
Warm - up
1. Come up with an example of a true
proportion
2. How do you solve for a proportion that
has a missing variable?
Means and Extremes
• The extremes of a proportion are the first
and last terms
• The means of a proportion are the middle
terms
a
b
c
=
d
a:b=c:d
Means-Extremes property of
proportions
• The product of the extremes equals the
product of the means.
a
b
c
=
d
ad = cb
Properties of Proportion
[AKA – Different ways to say the same thing]
a c

b d
a. ad  bc
is equivalent to
b.
a

b
c

d
d.

b
d
a b

c d
b
d
c.

a c
Bottom Line: When I cross
multiply any of these, I will
always end up back at
ad=bc.
Rewrite the following in 4 different
ways…
x 5
 As any of these
y 2
x y

1. 2 x  5 y 2.
5 2
x  y 5 2

4.
y
2
y 2

3.
x 5
2(x + y) = y (5+2)
2x + 2y = 7y
2x = 5y
Another Property
call it the “addition property”
a c e
a
a  c  e ...
If   ... then 
b d f
b b  d  f ...
Show whiteboard example
Solving a Proportion
3 a

5 15
5a  45
a 9
First, cross-multiply
Next, divide by 5
White Board Practice
x 4
• If
 , then 2x = _______
7 2
White Board Practice
x 4
• If
 , then 2x = 28
7 2
White Board Practice
2

• If 2x = 3y, then
3
White Board Practice
2 y

• If 2x = 3y, then
3 x
White Board Practice
x 4
x7

• If  , then
7 2
7
White Board Practice
x 4
x7 6
• If  , then

7 2
7
2
White Board Practice
x3
x y2
• If 
, then 3 
3
2
White Board Practice
x3 y
x y2
• If 
, then 3  2
3
2
White Board Practice
Solve for x
X
2
=
9
3
2_
8__
=
x-6
x+3
Brightstorm link
X=6
X=9
Whiteboard practice
• Page 246
– #2
– #11
WARM UP
•
In order for 2 polys to be congruent, 2 rules
must be satisfied…
1. All _______________________________
2. All________________________________
•
In a complete sentence, what do you think the
difference is between 2 polys that are
congruent and 2 polys that are similar?
7.3 Similar Polygons
Objectives
• State and apply
the properties
of similar
polygons.
Similarity
• Coaching Football
– When I need to show my players the
diagram of a play, I am not going to use a
piece of paper that is 50 yards wide and 100
yards long…
– So what do I do???
– Draw the same shape of the field but with a
length and width that is drawn to a smaller
scale.
Similar Polygons
• Same shape
• Not the same size  Why?
Similar Polygons (~)
1. All corresponding angles congruent
A  A’
B  B’
Read as “A prime”,
A
prime, and so on..
C  C’
ORDER
MATTERS!!
Just like
congruent polys
you must make
sure to name the
vertices in the
correct order.
“B
A’
C
B
B’
C’
Similar Polygons (~)
2. All corresponding sides are in proportion
AB = BC = CA
All sides have
equivalent ratios
A’B’ B’C’ C’A’
A
Partners: Come up
A’
with side lengths and
angle measures for the
two triangles that
would make them
similar.
B
C B’
C’
The Scale Factor
• If two polygons are similar, then they have a scale factor
• The reduced ratio between any pair of
corresponding sides or the perimeters.
• 12:3  scale factor of 4:1
12
3
Using the Scale factor to
find Missing Pieces
12 10

3
y
You have to know the
scale factor first to
find missing pieces.
Solve for y by
cross-multiplication
12
3
10
y
What could I do to make the math easier before I try cross-multiplying?
White Board Practice
• Quadrilateral ABCD ~ Quadrilateral
A’B’C’D’. Find their scale factor
A
y
30
D
• 5:3
The first # in the
scale factor will
come from ABCD
B
20
A’
C
12
x
B’
z
50
D’
30
C’
White Board Practice
• Quadrilateral ABCD ~ Quadrilateral
A’B’C’D’. Find the values of x, y, and z
A
y
30
D
• x = 18
• y = 20
• z = 12
B
20
A’
C
12
x
B’
z
50
D’
30
C’
White Board Practice
• Quadrilateral ABCD ~ Quadrilateral
A’B’C’D’. Find the ratio of the perimeters
A
y
30
D
• 5:3
B
20
A’
C
12
x
B’
z
50
D’
30
C’
Additional Problems
• Example
• Page 250
– Classroom ex.
– #1
– #10
Quiz Review
Section 7.1
•
•
Putting ratios into simplest form
Find the measure of each angle based on a ratio
– i.e. Pg. 244 #24 – 29
Section 7.2
•
Properties of proportions ( purple box pg. 245)
– i.e. how can the proportion be changed around and still be equal to the original
# 1-8)
•
Find the value of X ( Cross multiply and solve)
– i.e. Pg. 247 #9 – 20
Section 7.3
•
•
Understand the definition of similar polygons (~)
Finding the scale factor of similar polys
– Compare the lengths of corresponding sides (reduce)
•
Use the scale factor to find unknown lengths
– i.e. Pg. 251 #15 - 26
(i.e. pg. 247
Warm – Up
• Using the book or notes…
• Write down the definitions for the following
–
–
–
–
Ratio
Proportion
Scale factor
Similar Polygons
7.4 A Postulate for Similar
Triangles
Objectives
• Learn to prove triangles are similar.
What we have learned…
• Two polygons are similar
by showing that they
satisfy the definition of
similar polygons (~)
– 3 pair of corresponding
angles are congruent
– 3 pair of corresponding
sides are in proportion
Why does this whole 3 pair thing sound so familiar?
Index Card Experiment
Supplies: Index card, Scissors, Ruler
1. Cut out a triangle using a 3x5 index card
2. Label the vertices A, B, C
3. Take side BC of your triangle
Index Card Experiment
4. Draw a line that is twice the length of BC
and label the endpoints B’ and C’
5. At B’ line up angle B of your triangle and
trace it on the paper. Then do the same
thing for C’.
▲ABC ~ ▲A’B’C’
• Why?
– Corresponding angles are congruent
– The sides are in proportion with a scale factor
of 1:2
• How was ▲A’B’C’ created?
– By using 2 of the corresponding angles from
▲ABC
AA Similarity Postulate
(AA~ Post)
If two angles of one triangle are congruent to two
angles of another triangle, then the triangles are
similar.
A
D
B
C
F
E
Applying AA Similarity in
Proofs
• The key to using this postulate is to first
prove two corresponding angles of two
triangles congruent and then using it
Remote Time
• T – Similar Triangles
• F – Not Similar
T – Similar Triangles
F – Not Similar
T – Similar Triangles
F – Not Similar
T – Similar Triangles
F – Not Similar
Whiteboards
• Page 256
– #11
– #13
brightstorm
• Example
7-5: Theorems for Similar Triangles
Objectives
• Learn about 2 additional ways to prove
triangles are similar.
WARM-UP
What we have learned…
• SAS Congruency – Write down in your own
words what this means.
• SSS Congruency – Write down in your own
words what this means.
SAS Postulate
If two sides and the included angle are
congruent to the corresponding parts of
another triangle, then the triangles are
congruent.
E
B
C
F
D
SAS Similarity Theorem (SAS~)
Partners:
on whatisyou
now know
If an angleBased
of a triangle
congruent
to anabout
angle of
similarity
compared
congruency,
comethose
up with
another triangle
andtothe
sides including
the
wording
for this theorem.
angles
are proportional,
then the triangles are
similar.
A
D
2
1
B
4
C
F
2
E
- What is the scale factor of the ~ triangles
- Name the triangles
C
- Name the postulate or theorem
Included angle
Scale Factor = 2/3
▲CDE ~ ▲CAB by SAS ~
6
D
3
10
E
5
A
B
SSS Postulate
If three sides of one triangle are congruent to
the corresponding parts of another triangle,
then the triangles are congruent.
E
B
A
C
F
D
SSS Similarity Theorem (SSS~)
Partners: Based on what you now know about
If the three sides of one triangle are proportional to
similarity compared to congruency, come up with
thewording
three sides
another triangle, then the
the
thisoftheorem.
triangles are similar.
A
D
2
3
6
B
4
C
F
1
2
E
Example
• The measures of the sides of ▲ABC are 4,
5, 7
• The measures of the sides of ▲XYZ are 16,
20 , 28
• Are the two triangles similar? Why?
• ▲ABC ~ ▲XYZ by SSS ~
4 Ways to Prove Triangles
Similar
1.Definition of similarity
2.AA ~
**PROOFS: Once we have proven
that 2 triangles are similar. We can
then say what about …
3.SAS ~
1. The corresponding angles?
2. The corresponding sides?
4.SSS ~
White Board Practice
• Name the similar triangles and give the
postulate or theorem that justifies your
answer…
A
80◦
D
▲ADE ~ ▲ABC by AA ~
E
80◦
B
C
▲ABC ~ ▲DEF by SSS ~
F
9
A
18
D
3
B
4.5
6
C
9
E
Z
20
S
15
R
25
D
12
T
▲TRS ~ ▲ZRD by SAS ~
• You want to prove ▲RST ~ ▲ XYZ by
SSS ~
– State the ratios that you know have to be equal
to one another
• You want to prove ▲RST ~ ▲ XYZ by
SAS ~
– If you know LR congruent LX, what else do you
need to prove?
7-6: Proportional Lengths
Objectives
• Apply the Triangle Proportionality
Theorem and its corollary
• State and apply the Triangle Angle-bisector
Theorem
Billy and Bob
Billy and Bob want a foot-long sub from
Subway that costs $4
Billy has $1 and Bob has $3
They combine their money and buy the sub
How much of the sub should each person
get based on the amount of money they
paid?
Divided Proportionally
$4
Bob
$3
12in
$2
$1
Billy
Divided Proportionally
If points are placed on segments AB and CD
so that AX  CY , then we say that these
XB
YD
segments are divided proportionally.
B
D
X
Y
A
C
Example
B
D
X
A
2
AX = CY
XB
YD
2 =
4
4
1
2
Y
1
C
AX = XB
CY
YD
2 =
1
4
2
2
AX = CY
AB
CD
2 =
6
1
3
*Partners:
determine
another correct
proportion as
well as one that
wouldn’t work*
Theorem
If a line parallel to one side of a triangle
intersects the other two sides, it divides
them proportionally.
Just think of these 2 sides
Y
A
X
as lines that have been
divided proportionally
B
Z
What can we conclude based on the
diagram?
AY = BY
XY
ZY
▲AYB ~ ▲XYZ by AA ~
The sides are divided into
proportional segments by TH.
7-3
*Find 2 proportions that
can be justified by TH. 7-3
Y
4
2
B
A
1
X
2
Z
White Board Practice
• Are the following proportions possible?
• Answer True or False.
y
c
j
b
d
x
True or False
c
j
d
= x
T
b
y
d
= c
T
y
c
j
y
b
b
j
= x
T
c
b
y = x
F
d
x
Corollary
IfWhat
threetheorem
paralleldoes
linesthis
intersect
tworemind
transversals,
diagram
you of?
then they divide the transversals
proportionally.
R
W
S
T
X
RS = WX
ST
XY
Y
Solve for Y
15
10
y
14
Theorem
If a ray bisects an angle of a triangle, then it
divides the opposite side into segments
proportional to the other two sides.
Y
WX = XY
WZ
ZY
Y has been
bisected into
congruent angles
X
W
Z
White Board Practice
• Find X
12
10
24
X = 20
X
White Board Practice
• Find X
10
5
X
20
X = 15
Ch. 7 Test Review
Section 7.1
• Putting ratios into simplest form
• Find the measure of each angle based on a ratio
– i.e. Pg. 244 #24 – 29
Section 7.2
• Properties of proportions ( purple box pg. 245)
– i.e. how can the proportion be changed around and still be equal to
the original (i.e. pg. 247 # 1-8)
• Find the value of X ( Cross multiply and solve)
– i.e. Pg. 247 #9 - 20
Ch. 7 Test Review
Section 7.3
• Understand the definition of similar polygons (~)
• Finding the scale factor of similar polys
– Compare the lengths of corresponding sides (reduce)
• Use the scale factor to find unknown lengths
– i.e. Pg. 251 #15 - 26
Section 7.4 and 7.5
• **pg. 258 #16**
• Proving 2 triangles similar
– AA ~ , SAS ~ , SSS ~
– i.e. pg. 266 # 1 – 6
– **REMEMBER ORDER MATTERS WHEN
NAMING THE SIMILAR TRIANGLES!!!
Ch. 7 Test Review
Section 7.6
• Understand the 2 theorems and the corollary
– i.e. P. 272 # 3- 9 and 20 – 23
PROOFS
• Study the following – Understand why a certain statement
was given and its reason for it.
• i.e. p 255 proof example
• P. 266 # 11 - 16