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```Similar Figures
M.C. Escher
Some artists use
mathematics to
help them design
their creations.
In M.C. Escher’s
Square Limit, the
fish are arranged
so that there are
no gaps or
overlapping
pieces.
Square Limit by M.C. Escher
How are the
fish in the
middle of the
design and the
surrounding
fish alike?
How are they
different?
Square Limit by M.C. Escher
Escher used a pattern
of squares and
triangles to create
Square Limit.
These two triangles
are similar.
Similar figures have
the same shape but
not necessarily the
same size.
Similar Figures
For each part of one
similar figure there
is a corresponding
part on the other
figure.
Segment AB
corresponds to
segment DE.
Name another pair of
corresponding
segments.
B
A
C
D
E
F
Similar Figures
Angle A
corresponds to
angle D.
B
Name another
pair of
corresponding
angles.
A
C
D
E
F
Similar Figures
•Corresponding sides have
lengths that are proportional.
• Corresponding angles are
congruent.
Congruent Figures
In order to be congruent, two
figures must be the same size
and same shape.

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.

Similar Figures
A
3 cm
2 cm
B
D
2 cm
3 cm
W
9 cm
6 cm
Z
6 cm
C
X
Corresponding sides:
AB corresponds to WX.
BC corresponds to XY.
CD corresponds to YZ.
9 cm
Y
Similar Figures
A
3 cm
2 cm
B
D
2 cm
3 cm
W
9 cm
6 cm
Z
6 cm
C
Corresponding angles:
A corresponds to
B corresponds to
W.
X.
C corresponds to
D corresponds to
Y.
Z.
X
9 cm
Y
Similar Figures
A
3 cm
2 cm
B
D
2 cm
3 cm
W
9 cm
6 cm
Z
6 cm
C
X
9 cm
In the rectangles above, one proportion is
AB
2
3
=
, or
= .
WX WZ
6
9
Y
If you cannot use corresponding side lengths to
write a proportion, or if corresponding angles are
not congruent, then the figures are not similar.
Missing Measures in Similar Figures
The two triangles are similar. Find the
missing length y and the measure of D.
100
111
Write a proportion using
____
= ___
200
y
corresponding side lengths.
200 • 111 = 100 • y The cross products are equal.
The two triangles are similar. Find the
missing length y.
22,200 = 100y
22,200
100y
______
= ____
100
100
y is multiplied by 100.
Divide both sides by 100
to undo the multiplication.
222 mm = y
The two triangles are similar. Find the
measure of angle D.
Angle D is congruent to angle C.
If angle C = 70°, then angle D = 70° .
Try This
The two triangles are similar. Find the
missing length y and the measure of B.
A 60 m
65°
50 m
45° 52 m
B
120 m
100 m
y
50
52
____
___
=
100
y
5,200 = 50y
5,200 = 50y
_____
___
50
50
104 m = y
Write a proportion using
corresponding side lengths.
Divide both sides by 50 to
undo the multiplication.
Try This
The two triangles are similar. Find the
missing length y and the measure of B.
A 60 m
65°
50 m
45° 52 m
B
120 m
100 m
y
Angle B is congruent to angle A.
If angle A = 65°, then angle B = 65°
Using Proportions with Similar Figures
This reduction is similar to a
picture that Katie painted. The
height of the actual painting is
54 centimeters. What is the
width of the actual painting?
Actual
Reduced
2
54
3
w
Using Proportions with Similar Figures
Actual
Reduced
2
54
3
w
2 cm
3 cm
_____
=
Write a proportion.
w
cm
54 cm
54 • 3 = 2 • w The cross products are equal.
162 = 2w
81 = w
w is multiplied by 2.
Divide both sides by 2 to
undo the multiplication.
Try these 5 problems.
These two triangles are similar.
1. Find the missing length x. 30 in.
2. Find the measure of
J.
36.9°
3. Find the missing length y. 4 in.
4. Find the measure of
P.
90°
5. Susan is making a wood deck from plans for
an 8 ft by 10 ft deck. However, she is going to
increase its size proportionally. If the length is
12 ft
to be 15 ft, what will the width be?
Joan used a mirror to estimate the
height of a flagpole. What is the
height of the flagpole?
5 ft
flagpole
2 ft---|-----------------9 ft-----------------------
2 5

9 x
2 x  95
x  22.5 ft
2 x  45
```
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