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Similar Figures
(Not exactly the same,
but pretty close!)
Let’s do a little
review work
before discussing
similar figures.
Congruent Figures
• In order to be congruent, two
figures must be the same size
and same shape.

Similar shapes
• Are Enlargements of each other
• Corresponding angles are equal
• Sides are related by the same scale factor
Similar Figures
• Similar figures must be the
same shape, but their sizes may
be different.

Similar Figures
This is the symbol that
means “similar.”
These figures are the same
shape but different sizes.


SIZES
• Although the size of the two
shapes can be different, the
sizes of the two shapes4must
differ by a factor.
2
3
1
3

6
6
2
SIZES
• In this case, the factor is x 2.
4
2
3
3
1

6
6
2
SIZES
• Or you can think of the factor
as
2.
4
2
3
3
1

6
6
2
Enlargements
• When you have a photograph
enlarged, you make a similar
photograph.

X3
Reductions
• A photograph can also be
shrunk to produce a slide.
4

Determine the length of the
unknown side.
15
12

?
4
3
9
These triangles differ by a factor
of 3.
15
15
12

3= 5
?
4
3
9
Determine the length of the
unknown side.
?
2
4

24
These dodecagons differ by a
factor of 6.
?
2
4

24
Sometimes the factor between 2
figures is not obvious and some
calculations are necessary.
15
12
18

?=
8
10
12
To find this missing factor,
divide 18 by 12.
15
12
18

?=
8
10
12
18 divided by 12
= 1.5
The value of the missing
factor is 1.5.
15
12
18

1.5 =
8
10
12
This is termed as SSS
similarity criterion
To calculate a length
15
5x
5
3
4
x3
6
Scale factor 3
15
1
3
18
Scale factor 1/3
12
Find the length of the missing
side.
50
30
?
6
40
8
This looks messy. Let’s
translate the two triangles.
50
30
?
6
40
8
Now “things” are easier to see.
50
30
?
6
40
8
The common factor between
these triangles
is 5.
50
30
?
6
40
8
So the length of
the missing side
is…?
That’s right! It’s ten!
50
30
10
6
40
8
Similar Triangles
Triangles are similar if matching
angles remain the same size.
100º
30º
50º
100º
30º
50º
When changing the size of a
figure, will the angles of the
figure also change?
?
40
70
70
?
?
Nope! Remember, the sum of all 3
angles in a triangle MUST add to 180
degrees.
If the size of the
angles were
40
increased,
the sum
40
would exceed
180
degrees.
70
70
70
70
We can verify this fact by placing
the smaller triangle inside the
larger triangle.
40
40
70
70
70
70
The 40 degree angles
are congruent.
40
70
70
70
70
The 70 degree angles
are congruent.
40
40
70
70
70 70
The other 70 degree
angles are congruent.
4
40
70
7070
70
70
•This is termed as
AAA similarity
criterion
Similarity is used to answer real
life questions.
• Suppose that you
wanted to find the
height of this tree.
Unfortunately all that
you have is a tape
measure, and you are
too short to reach the
top of the tree.
You can measure the length of
the tree’s shadow.
10 feet
Then, measure the length of your
shadow.
10 feet
2 feet
If you know how tall you are,
then you can determine how tall
the tree is.
10 feet
6 ft
2 feet
The tree must be 30 ft tall. Boy,
that’s a tall tree!
10 feet
6 ft
2 feet
Similar figures “work” just like
equivalent fractions.
30
66
5
11
These numerators and
denominators differ by a factor of
6.
30
6
66
6
5
11
Two equivalent fractions are
called a proportion.
30
66
5
11
Similar Figures
• So, similar figures are
two figures that are the
same shape and whose
sides are proportional.
Practice Time!
1) Determine the missing side of
the triangle.
?
3
5
4
9
12
1) Determine the missing side of
the triangle.
15
3
5
4
9
12
2) Determine the missing side of
the triangle.
6
6
36
36
4
?
2) Determine the missing side of
the triangle.
6
6
36
36
4
24
3) Determine the missing sides of
the triangle.
39
33
?
?
8
24
3) Determine the missing sides of
the triangle.
39
33
13
11
8
24
4) Determine the height of the
lighthouse.
?
8
2.5
10
4) Determine the height of the
lighthouse.
32
8
2.5
10
5) Determine the height of the
car.
?
3
5
12
Show that these triangles are
similar
10º
50º
120º
10º
50º
120º
…and then…
AB & DE are parallel
Explain why ABC is similar to CDE
5
A
B
<CED = <BAC Alternate Angles
<EDC = <ABC Alternate Angles
<ECD = <ACB Vert Opp Angles
3
C
6
E
D
?
Triangle ABC is similar to Triangle CDE
…and then…
Calculate the length of DE
AC corresponds to CE
Scale Factor = 2
5
A
B
AB corresponds to DE
DE = 2 x AB
3
C
DE = 10cm
6
E
D
?
Summary – Similar shapes
• To calculate missing sides, we first of all need the
scale factor
• We then either multiply or divide by the scale
factor
• To show that 2 shapes are similar we can either
show that all of the sides are connected by the
scale factor or show that matching angles are the
same
THE END!