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Similarity
Similar Polygons
1
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• Dilating a figure produces a figure that is the same ________________ as the
original figure, but a different _________________ . Like ____________ motions,
dilations preserve ______________ measures. Unlike rigid motions, dilations do
not preserve the __________________ of line segments. Instead, they produce a
figure with sides that are ____________________ to the sides of the pre-image.
So, the original figure and its __________________ image are
_________________ figures.
• A _________________ polygon is a polygon in which all sides have the
________________ length and all angles have the same ____________________.
Any two regular polygons of the same type – having the same number of sides -
are __________________ similar to each other.
• The symbol for similar is _______________ .
2
MAKING CONNECTIONS
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• Dilating a figure produces a figure that is the same ________________ as the
original figure, but a different _________________ . Like ____________ motions,
dilations preserve ______________ measures. Unlike rigid motions, dilations do
not preserve the __________________ of line segments. Instead, they produce a
figure with sides that are ____________________ to the sides of the pre-image.
So, the original figure and its __________________ image are
_________________ figures.
• A _________________ polygon is a polygon in which all sides have the
________________ length and all angles have the same ____________________.
Any two regular polygons of the same type – having the same number of sides -
are __________________ similar to each other.
• The symbol for similar is _______________ .
3
MAKING CONNECTIONS
length
size rigidangle
shape
proportional
resultingsimilar
regular
same measure
always
~
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Characteristics of Similar Polygons
1. Corresponding angles are _____________ .
2. Corresponding sides are _______________ .
Similarity Statement
ABC ~ DEF
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EXS: Are the pairs of figures similar? EXPLAIN.
2)1)
3) 4)
Corresponding Sides : Corresponding Sides
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EXS: Are the pairs of figures similar? EXPLAIN.
2)1)
3) 4)
Corresponding Sides : Corresponding Sides
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Setting Up the Ratio of Similar Polygons
1) Find the Scale Factor.
2) Set the scale factor equal to a ratio containing the missing side to form a proportion.
3) Solve for the variable.
8/24/2018
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EXS: Each pair of polygons is similar. Write a proportion to find each missing side. Solve for x.
2)1)
3)
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EXS: Each pair of polygons is similar. Write a proportion to find each missing side. Solve for x.
2)1)
3)
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Angle Relationships Side Relationships
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EX 1: Solve for x and y.
~ABC SLT A
B C
S
L
T
x
5 cm
24 cm
10 cm
13 cm
y
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EX 1: Solve for x and y.
~ABC SLT A
B C
S
L
T
x
5 cm
24 cm
10 cm
13 cm
y
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A B
CD
6
x
E F
GH
18
27
EX 2: ABCD ~ EFGH. Solve for x.
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A B
CD
6
x
E F
GH
18
27
EX 2: ABCD ~ EFGH. Solve for x.
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EX 3: A tree cast a shadow 18 feet long. At the same time a person who is 6 feet tall cast a shadow 4 feet long. How tall is the tree?
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EX 3: A tree cast a shadow 18 feet long. At the same time a person who is 6 feet tall cast a shadow 4 feet long. How tall is the tree?
tree's shadow tree's height
person's shadow person's height
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Corresponding Sides : Corresponding Sides
Or
Perimeter : Perimeter
A : B8/24/2018
Setting Up the Ratio of Similar Polygons
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Area : Area
A2 : B2
8/24/2018
Volume: Volume
A3 : B3
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REMEMBER! The ratio of the perimeters of
two similar polygons equals the ratio of
any pair of corresponding sides.
A
C T
O
D G
6 4
10
y
EX 4: The ratio of the perimeters of CAT to
DOG is 3:2. Find the value of y.
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A
C T
O
D G
6 4
10
y
EX 4: The ratio of the perimeters of CAT to
DOG is 3:2. Find the value of y.
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12 cm 4 cm
Perimeter = 60 cmPerimeter = x
EX 5: Find the perimeter of the smaller triangle.
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12 cm 4 cm
Perimeter = 60 cmPerimeter = x
EX 5: Find the perimeter of the smaller triangle.
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U
W
V X
Z
Y
7.5
5
12
61. ~ ____
2.
3. ?
4. ?
5. 50 30 , ?
UVW
What is the scale factor of UVW to XYZ
What is VW
What is XZ
If m U and m Y what is m Z
Warm-Up
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U
W
V X
Z
Y
7.5
5
12
61. ~ ____
2.
3. ?
4. ?
5. 50 30 , ?
UVW
What is the scale factor of UVW to XYZ
What is VW
What is XZ
If m U and m Y what is m Z
XYZ
5/6
10
9
100
Warm-Up
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Similarity
Proving Triangles Similar
25
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Shared angles are congruent in each triangle by the REFLEXIVE property.
∠𝑨 ≅ ∠𝑨
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Shared angles are congruent in each triangle by the REFLEXIVE property.
𝐒𝐢𝐧𝐜𝐞 ∠𝑨 ≅ ∠𝑨,
t𝐡𝐞𝐧, ∠𝑪𝑨𝑩 ≅ ∠𝑫𝑨𝑬
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VERTICAL ANGLES are the opposite angles created when two lines intersect one another. VERTICAL ANGLES ARE ALWAYS CONGRUENT.
∠𝟏 ≅ ∠𝟑 ∠𝟐 ≅ ∠𝟒
14
32
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Before we start…let’s get a few things straight
A B
C
X Z
Y
INCLUDED ANGLEIt’s stuck in between!
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• Two triangles are similar if all of their
corresponding angles have ___________
measures and all of their corresponding sides
have _______________________ lengths.
However, you do not need to know every one of
those angle measures and side lengths to
______________ that two triangles are
________________ . 30
KEY IDEAS
equal
proportional
prove
similar
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• For instance, if two angles of one triangle are
__________________ to two angles of another triangle, then the
triangles are ___________________ . Also, if the three sides of
one triangle are _______________________ to the three sides
of another triangle, then the triangles are ___________________
. Finally, if two sides of one triangle are ____________________
to two sides of another triangle and the ____________________
angles of those sides are _____________________, then the
triangles are ____________ .31
KEY IDEAS (cont.)
congruentsimilar
proportionalsimilar
proportional
includedcongruent
similar
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Angle-Angle Similarity (AA~)
Postulate
If two angles of one triangle
are congruent to two angles
of another triangle, then the
triangles are similar.
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Side-Side-Side Similarity (SSS~)
Theorem
If the three sides of one
triangle are proportional in
length to the three sides of
another triangle, then the
triangles are similar.
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Side-Angle-Side Similarity (SAS~)
Theorem
If two sides of one triangle have
lengths that are proportional to
two sides of another triangle
and the included angles of
those sides are congruent, then the triangles are similar.
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Ex. Determine whether the triangles are similar. If so, tell which similarity test is used and complete the statement.
F
G
H
KL
M
43°
43°68°
68°
W
V
U
7
11X
Y
Z5
3
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Prove that RST~ PSQ
R
S
T
P Q
12
4 5
15SS
reflexive
5
20
4
16
1
4
1
4
1. Two sides are proportional
2. Included angle is congruentSAS~
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37
Using only the information given, can it be shown that CDE ~ FGE?
A) Yes, by the AA~ Postulate
B) Yes, by the SAS~ Theorem
C) Yes, by the SSS~ Theorem
D) No, not enough information is given.