Similar Polygons
You will learn to identify similar polygons.
1) polygons
2) sides
3) similar polygons
4) scale drawing
Similar Polygons
closed figure in a plane formed by segments called sides.
A polygon is a ______
It is a general term used to describe a geometric figure with at least three sides.
Polygons that are the same shape but not necessarily the same size are
similar polygons
called ______________.
The symbol ~ is used to show that two figures are similar.
D
ΔABC is similar to ΔDEF
A
C
B
ΔABC ~ ΔDEF
F
E
Similar Polygons
Two polygons are similar if and only if their corresponding
angles are congruent and the measures of their corresponding
proportional
sides are ___________.
D
C
G
H
Definition of
Similar
Polygons
A
B
A  E,
C  G,
E
B  F
D  H
F
and
AB BC CD DA



EF FG GH HE
Polygon ABCD ~ polygon EFGH
Guided Practice pg. 373
#1 List all pairs of congruent angles. Write the ratios of the
corresponding side lengths in a statement of proportionality.
Similar Polygons
Determine if the polygons are similar. Justify your answer.
6
4
4
5
7
5
7
4
6
congruent
1) Are corresponding angles are _________.
proportional
2) Are corresponding sides ___________.
=
0.66 = 0.71
The polygons are NOT similar!
Similar Polygons
Find the values of x and y if ΔRST ~ ΔJKL
R
4
7
T
6
J
S
6
=
7
y + 2
4(y + 2) = 42
4y + 8 = 42
4y = 34
y 8
5
4
Write the proportion that
can be solved for y.
1
2
x
Write the proportion that
can be solved for x.
L
4
7
5
=
x
4x = 35
x8
3
4
y + 2
K
Similarity
The ratio found by comparing the measures of corresponding sides of
similar triangles is called the constant of proportionality or the ___________.
scale factor
D
A
5
3
B
7
C
E
If ΔABC ~ ΔDEF, then
or
10
6
F
14
CA
BC
AB
=
=
FD
EF
DE
7
5
3
=
=
6
14
10
Each ratio is equivalent to
1
2
The scale factor of ΔABC to ΔDEF is
1
2
The scale factor of ΔDEF to ΔABC is
2
1
Guided Practice
#2, and 3 pg. 373
2. What is the scale factor of QRST to ABCD?
3. Find the value of x.
Perimeters and Similarity
If two triangles are similar, then the measures of the
corresponding perimeters are proportional to the measures
of the corresponding sides.
D
A
Theorem
9-10
C
B
F
E
If ΔABC ~ ΔDEF, then
perimeter of ΔABC
perimeter of ΔDEF
=
CA
BC
AB
=
=
FD
EF
DE
Perimeters and Similarity
The perimeter of ΔRST is 9 units, and ΔRST ~ ΔMNP.
Find the value of each variable.
perimeter of ΔMNP
MN
=
RS
perimeter of ΔRST
3
x
=
13.5
9
27 = 13.5x
x = 2
NP
MN
=
Theorem 9-10 ST
RS
6
3
=
The perimeter of ΔMNP 2is 3 +y 6 + 4.5
Cross Products
3y = 12
y = 4
PM
MN
=
TR
RS
4.5
3
=
z
2
3z = 9
z = 3
Similar Triangles
You will learn to use AA similarity tests for
triangles.
Nothing New!
Similar Triangles
Somebuilding
of the
triangles
asofshown
below.
The
Designed
Bank of
by China
American
architect
in Hong
I.M.are
Kong
Pei,similar,
the
is one
outside
the
often
the
tallest
70-story
buildings
buildingin
the
is sectioned
world. into triangles which are meant to resemble the trunk of a bamboo
plant.
Similar Triangles
In previous lessons, you learned several basic tests for determining whether
two triangles are congruent. Recall that each congruence test involves only
three corresponding parts of each triangle.
Likewise, there are tests for similarity that will not involve all the parts of
each triangle.
If two angles of one triangle are congruent to two
corresponding angles of another triangle, then the triangles
similar
are ______.
Postulate
9-1
AA Similarity
C
F
A
B
D
E
If A ≈ D and B ≈ E, then ΔABC ~ ΔDEF
Similar Triangles
Fransisco needs to know the tree’s height. The tree’s shadow is 18 feet long
at the same time that his shadow is 4 feet long.
If Fransisco is 6 feet tall, how tall is the tree?
1) The sun’s rays form congruent angles with the
ground.
2) Both Fransisco and the tree form right
angles with the ground.
4
=
18
6
t
4t = 108
t
= 27
The tree is
27 feet tall!
6 ft.
4 ft.
18 ft.
Similar Triangles
Slade is a surveyor.
To find the distance across Muddy Pond,
he forms similar triangles and measures
distances as shown.
What is the distance
across Muddy Pond?
10
45
=
8
x
10x = 360
x = 36
45 m
x
8 m
10 m
It is 36 meters across Muddy Pond!
Guided Practice
# 2 Read the directions and draw the diagram before
you determine if the triangles are similar
Solution:
Angle CDF is 58 degrees ( congruent to angle DEF)
Angle DFC is congruent to Angle 90 degrees
Angle DFE is congruent to Angle 90 degrees
Since two corresponding angles are congruent then
the triangles are similar.
Similar Triangles
Two other tests are used to determine whether two triangles are similar.
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.
C
2
Theorem 9-2
SAS
Similarity
A
1
B
8
If
8 2

and
4 1
D
F
4
E
Angle A is congruent to Angle D
then the triangles are similar
Similar Triangles
proportional
If the measures of two sides of a triangle are ___________
to the measures of two corresponding sides of another triangle
and their included angles are congruent, then the triangles are
similar.
C
Theorem 9-3
SAS
Similarity
2
A
1
8
B
D
F
4
AB8 AC
2
and A  D
If If 
DE4 DF
1
then ΔABC ~ ΔDEF
E
Similar Triangles
Determine whether the triangles are similar.
is used and complete the statement.
14
G
6
K
If so, tell which similarity test
J
9
21
10
H
M
15
P
Since
6
9
=
10
15
=
14
21
Therefore, ΔGHK ~ Δ JMP
, the triangles are similar by SSS similarity.
Guided Practice #3,4
Please read the directions and draw the triangles on your
paper to help you out.
#3 Solution:
<R  < N and
SR RT 4


PN NQ 3
therefore the triangles are similar by
the SAS Similarity Theorem.
#4 Solution:
<WZX

<XZY and
WZ XZ WX 4



XZ YZ
XY 3
therefore the triangles
are similar by either SSS or SAS Similarity Theorem.