Chapter 7: Proportions and Similarity

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Chapter 7 Proportions and Similarity
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7.1- Proportions

• Make a Frayer foldable

7.1 Ratio and Proportion
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Ratio

• A comparison of two quantities using division
• 3 ways to write a ratio
• a to b
• a b

Ex There are 480 sophomores and 520 juniors in a
high school. Find the ratio of juniors to
sophomores. Then find the ratio of sophomores to
students.
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Proportion

• An equation stating that two ratios are equal
• Example
• Cross products means and extremes
• Example

a and d extremes b and c means
ad bc
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Your Turn solve these examples

• Ex

Ex
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Your Turn solve this example

• The ratios of the measures of three angles of a
  triangle are 578. Find the angle measures.

A strip of wood molding that is 33 inches long is
cut into two pieces whose lengths are in the
ratio of 74. What are the lengths of the two
pieces?
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7.2 Similar Polygons

• Similar polygons have
• Congruent corresponding angles
• Proportional corresponding sides
• Scale factor the ratio of corresponding sides

A
Polygon ABCDE Polygon LMNOP
L
B
E
M
P
Ex
N
O
C
D
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If ?ABC ?RST, list all pairs of congruent
angles and write a proportion that relates the
corresponding sides.
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Determine whether the triangles are similar.
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A. The two polygons are similar. Find x and y.
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If ABCDE RSTUV, find the scale factor of ABCDE
to RSTUV and the perimeter of each polygon.
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If LMNOP VWXYZ, find the perimeter of each
polygon.
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7.3 Similar Triangles

• Similar triangles have congruent corresponding
  angles and proportional corresponding sides

Z
Y
A
C
X
B
angle A angle X angle B angle Y angle C
angle Z
ABC XYZ
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7.3 Similar Triangles

• Triangles are similar if you show
• Any 2 pairs of corresponding sides are
  proportional and the included angles are
  congruent (SAS Similarity)

R
B
12
6
18
C
T
A
4
S
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7.3 Similar Triangles

• Triangles are similar if you show
• All 3 pairs of corresponding sides are
  proportional (SSS Similarity)

R
B
6
5
10
C
7
T
14
A
3
S
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7.3 Similar Triangles

• Triangles are similar if you show
• Any 2 pairs of corresponding angles are congruent
  (AA Similarity)

R
B
C
T
A
S
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A. Determine whether the triangles are similar.
If so, write a similarity statement. Explain your
reasoning.
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B. Determine whether the triangles are similar.
If so, write a similarity statement. Explain your
reasoning.
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A. Determine whether the triangles are similar.
If so, write a similarity statement. Explain your
reasoning.
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B. Determine whether the triangles are similar.
If so, write a similarity statement. Explain your
reasoning.
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A. Determine whether the triangles are similar.
If so, choose the correct similarity statement to
match the given data.
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B. Determine whether the triangles are similar.
If so, choose the correct similarity statement to
match the given data.
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SKYSCRAPERS Josh wanted to measure the height of
the Sears Tower in Chicago. He used a 12-foot
light pole and measured its shadow at 1 p.m. The
length of the shadow was 2 feet. Then he measured
the length of the Sears Towers shadow and it
was 242 feet at the same time. What is the
height of the Sears Tower?
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7.4 Parallel Lines and Proportional Parts

• If a line is parallel to one side of a triangle
  and intersects the other two sides of the
  triangle, then it separates those sides into
  proportional parts.

A
X
Y
B
C
If XY ll CB, then
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7.4 Parallel Lines and Proportional Parts

• Triangle Midsegment Theorem
• A midsegment of a triangle is parallel to one
  side of a triangle, and its length is half of the
  side that it is parallel to

A
E
B
If E and B are the midpoints of AD and AC
respectively, then EB DC
C
D
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7.4 Parallel Lines and Proportional Parts

• If 3 or more lines are parallel and intersect two
  transversals, then they cut the transversals into
  proportional parts

C
B
A
D
E
F
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7.4 Parallel Lines and Proportional Parts

• If 3 or more parallel lines cut off congruent
  segments on one transversal, then they cut off
  congruent segments on every transversal

C
B
A
D
E
If , then
F
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MAPS In the figure, Larch, Maple, and Nuthatch
Streets are all parallel. The figure shows the
distances in between city blocks. Find x.
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ALGEBRA Find x and y.
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7.5 Parts of Similar Triangles

• If two triangles are similar, then the perimeters
  are proportional to the measures of corresponding
  sides

X
A
B
C
Y
Z
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7.5 Parts of Similar Triangles
If two triangles are similar

• the measures of the corresponding altitudes are
  proportional to the corresponding sides

• the measures of the corresponding angle bisectors
  are proportional to the corresponding sides

X
A
S
M
C
B
D
Y
Z
W
R
L
N
T
U
O
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7.5 Parts of Similar Triangles

• If 2 triangles are similar, then the measures of
  the corresponding medians are proportional to the
  corresponding sides.

• An angle bisector in a triangle cuts the opposite
  side into segments that are proportional to the
  other sides

E
A
G
T
D
B
C
J
H
I
F
H
G
U
W
V
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In the figure, ?LJK ?SQR. Find the value of x.
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In the figure, ?ABC ?FGH. Find the value of x.
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Find x.
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Find n.