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

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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
3 6 x 21 18 21x x 18/21 x 6/7
5(x 2) 2 4 5x 10 8 5x 18 x 18/5 x
3 3/5
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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.

5x 7x 8x 180 20x 180 x 9
45, 63, 72
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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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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)
• All 3 pairs of corresponding sides are
  proportional (SSS Similarity)
• Any 2 pairs of corresponding angles are congruent
  (AA Similarity)

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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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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 2 triangles are similar, the measures of the
  corresponding altitudes are proportional to the
  corresponding sides

• If 2 triangles are similar, 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