Download Similar Polygons

Similar Polygons
Two polygons are similar polygons iff the
corresponding angles are congruent and the
corresponding sides are proportional.
Similarity Statement:
N
C
M
CORN ~ MAIZ
Corresponding Angles:
C  M
C
R  I
O  NA
N  Z
O
R
A
M
StatementOof Proportionality:
R
CO  OR  RN  NC
MA AI IZ ZM
A
I
Z
Example 1
Triangles ABC
and ADE are
similar. Find the
value of x.
D
B
A
6 cm
9 cm
8 cm
C
x
E
Example 1
Triangles ABC and ADE
are similar. Find the
value of x.
8 𝑥+8
=
6
9
D
B
A
6 cm
9 cm
8 cm
6𝑥 + 48 = 72
C
6𝑥 = 24
𝑥=4
x
E
Example 2
Are the triangles below similar?
8
4
6
3
37
53
5
10
Do you REALLY have to check all the sides and angles?
Investigation 1
In this Investigation we will check the first
similarity shortcut. If the angles in two
triangles are congruent, are the triangles
necessarily similar?
F
C
A
50
40
B
D
50
40
E
Investigation 1
Step 1: Draw ΔABC where m<A and m<B
equal sensible values of your choosing.
C
A
50
40
B
Investigation 1
Step 1: Draw ΔABC where m<A and m<B
equal sensible values of your choosing.
Step 2: Draw ΔDEF where m<D = m<A and
m<E = m<B and AB ≠ DE.
F
C
A
50
40
B
D
50
40
E
Investigation 1
Now, are your triangles similar? What would
you have to check to determine if they are
similar?
F
C
A
50
40
B
D
50
40
E
6.4-6.5: Similarity Shortcuts
Objectives:
1. To find missing measures in similar
polygons
2. To discover shortcuts for determining that
two triangles are similar
Angle-Angle Similarity
AA Similarity
Postulate
If two angles of one
triangle are
congruent to two
angles of another
triangle, then the two
triangles are similar.
Example 3
Determine whether the triangles are similar.
Write a similarity statement for each set of
similar figures.
Example 3
Yes, from AA.
∆𝐶𝐷𝐸~∆𝐾𝐺𝐻
Yes, from AA.
(They share angle A)
∆𝐴𝐵𝐸~∆𝐴𝐶𝐷
Thales
The Greek mathematician
Thales was the first to
measure the height of a
pyramid by using
geometry. He showed
that the ratio of a
pyramid to a staff was
equal to the ratio of one
shadow to another.
Example 4
If the shadow of the pyramid is 576 feet, the
shadow of the staff is 6 feet, and the height
of the staff is 5 feet, find the height of the
pyramid.
Example 5
Explain why Thales’ method worked to find
the height of the pyramid?
Example 6
If a person 5 feet tall casts a 6-foot shadow
at the same time that a lamppost casts an
18-foot shadow, what is the height of the
lamppost?
Example 6
5
𝑥
=
6 18
6𝑥 = 90
𝑥 = 15′
Side-Side-Side Similarity
SSS Similarity Theorem:
If the corresponding side lengths of two
triangles are proportional, then the two
triangles are similar.
Side-Angle-Side Similarity
SAS Similarity Theorem:
If two sides of one triangle are proportional to two
sides of another triangle and the included angles
are congruent, then the two triangles are similar.
Example 8
Are the triangles below similar? Why or why
not?
Example 8
Are the triangles below similar? Why or why not?
Yes, from SSS since
all the corresponding
sides have the
same ratio 2/3.
Yes, from SAS since the two pairs of corresponding
sides have the ratio 6/7 and they
have a congruent angle
between them.
Example 9
Use your new conjectures to find the missing
measure.
28
24
24
x
18
y
Example 9
Use your new conjectures to find the missing measure.
24
𝑥
=
18 24
𝑥 = 32
24 28
=
18
𝑦
𝑦 = 21
28
24
24
x
18
y
Example 10
Find the value of x that makes ΔABC ~
ΔDEF.
Example 10
Find the value of x that makes ΔABC ~ ΔDEF.
4
𝑥−1
=
12
18
12𝑥 − 12 = 72
12𝑥 = 84
𝑥=7