Chapter 7 Similarity and Proportion

1
Chapter 7Similarity and Proportion

• Express a ratio in simplest form.
• State and apply the properties of similar
  polygons.
• Use the theorems about similar triangles.

2
7.1 Ratio and Proportion

• Objectives
• Express a ratio in simplest form
• Solve for an unknown in a proportion

3
Ratio

• A comparison between numbers

5 7
5 7 s 5 t
4
Ratio

• Always reduce ratios to the simplest form

5
Proportion

• An equation containing ratios

6
Solving a Proportion
First, cross-multiply
Next, divide by 5
7
White Board Practice

• ABCD is a parallelogram. Find the value of each
  ratio.

8
White Board Practice

• AB BC

9
White Board Practice

• 5 3

10
White Board Practice

• BC AD

11
White Board Practice

• 1 1

12
White Board Practice

• m ? A m ? C

13
White Board Practice

• 1 1

14
White Board Practice

• AB perimeter of ABCD

15
White Board Practice

• 5 16

16
White Board Practice

• x 2 and y 3. Write each ratio in simplest
  form.
• x to y

17
White Board Practice

• x 2 and y 3. Write each ratio in simplest
  form.
• 2 to 3

18
White Board Practice

• x 2 and y 3. Write each ratio in simplest
  form.
• 6x2 to 12xy

19
White Board Practice

• x 2 and y 3. Write each ratio in simplest
  form.
• 1 to 3

20
White Board Practice

• x 2 and y 3. Write each ratio in simplest
  form.
• y x
• x

21
White Board Practice

• x 2 and y 3. Write each ratio in simplest
  form.
• 1
• 2

22
7.2 Properties of Proportions

• Objectives
• Express a given proportion in an equivalent form.

23
Means and Extremes

• The extremes of a proportion are the first and
  last terms
• The means of a proportion are the middle terms

a b c d
24
Properties of Proportion
is equivalent to
1.
3.
2.
4.
25
That just means that you can rewrite
As any of these
1.
3.
2.
4.
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Another Property
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White Board Practice

• If , then 2x _______

28
White Board Practice

• If , then 2x 28

29
White Board Practice

• If 2x 3y, then

30
White Board Practice

• If 2x 3y, then

31
White Board Practice

• If , then

32
White Board Practice

• If , then

33
White Board Practice

• If , then

34
White Board Practice

• If , then

35
7.3 Similar Polygons

• Objectives
• State and apply the properties of similar
  polygons.

36
Similar Polygons

• Same shape
• Not the same size ? Why?

37

• Not the same size ? Why?

Because then they would be congruent !
38
Similar Polygons ()

• All corresponding angles congruent
• ?A ? ?A
• ?B ? ?B
• ?C ? ?C

A
A
C
B
C
B
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Similar Polygons ()

• All corresponding sides in proportion
• AB BC CA
• AB BC CA

A
A
B
C
C
B
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The Scale Factor

• The reduced ratio between any pair of
  corresponding sides or the perimeters.
• 123

12
3
41
Finding Missing Pieces

• You have to know the scale factor first to find
  missing pieces.

12
3
10
y
42
White Board Practice

• Quadrilateral ABCD Quadrilateral ABCD.
  Find their scale factor

43
White Board Practice

• 53

44
White Board Practice

• Quadrilateral ABCD Quadrilateral ABCD.
  Find the values of x, y, and z

45
White Board Practice

• x 18
• y 20
• z 13.2

46
White Board Practice

• Quadrilateral ABCD Quadrilateral ABCD.
  Find the ratio of the perimeters

47
White Board Practice

• 53

48
7.4 A Postulate for Similar Triangles

• Objectives
• Learn to prove triangles are similar.

49
AA Simliarity Postulate(AA Post)

• If two angles of one triangle are congruent to
  two angles of another triangle, then the
  triangles are similar.

A
D
F
E
B
C
50
Remote Time

• T Similar Triangles
• F Not Similar

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T Similar TrianglesF Not Similar
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T Similar TrianglesF Not Similar
53
T Similar TrianglesF Not Similar
54
T Similar TrianglesF Not Similar
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7-5 Theorems for Similar Triangles

• Objectives
• More ways to prove triangles are similar.

56
SAS Similarity Theorem (SAS)

• If an angle of a triangle is congruent to an
  angle of another triangle and the sides including
  those angles are proportional, then the triangles
  are similar.

A
D
F
E
B
C
57
SSS Similarity Theorem (SSS)

• If the three sides of one triangle are
  proportional to the three sides of another
  triangle, then the triangles are similar.

A
D
F
E
B
C
58
Homework Set 7.5

• 7-5 1-19 odd
• WS PM 40

59
7-6 Proportional Lengths

• Objectives
• Apply the Triangle Proportionality Theorem and
  its corollary
• State and apply the Triangle Angle-bisector
  Theorem

60
Divided Proportionally

• If points are placed on segments AB and CD so
  that , then we say that these
• segments are divided proportionally.

B
D
X
Y
A
C
See It!
61
Theorem 7-3

• If a line parallel to one side of a triangle
  intersects the other two sides, it divides them
  proportionally.

Y
See It!
B
A
X
Z
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Corollary

• If three parallel lines intersect two
  transversals, they they divide the transversal
  proportionally.

R
W
S
X
T
Y
See It!
63
Theorem 7-4

• If a ray bisects an angle of a triangle, then it
  divides the opposite side into segments
  proportional to the other two sides.

Y
See It!
W
X
Z
64
Construction 12
Given a segment, divide the segment into a given
number of congruent segments. Given Construct S
teps
65
Construction 13
Given three segments, construct a fourth segment
so that the four segments are proportional. Given
Construct Steps
66
Homework Set 7.6

• WS PM 41
• WS Constructions 12 and 13
• 7-6 1-23 odd
• Quiz next class day