1. Proportions and Similarity
§ 9.1 Using Ratios and Proportions
§ 9.2 Similar Polygons
§ 9.3 Similar Triangles
§ 9.4 Proportional Parts and Triangles
§ 9.5 Triangles and Parallel Lines
§ 9.6 Proportional Parts and Parallel Lines
§ 9.7 Perimeters and Similarity

2. Using Ratios and Proportions
What You'll Learn
You will learn to use ratios and proportions to solve problems.
Vocabulary
1) ratio
2) proportion
3) cross products
4) extremes
5) means

3. Using Ratios and Proportions
In 2000, about 180 million tons of solid waste was created in the United States.
The paper made up about 72 million tons of this waste.
The ratio of paper waste to total waste is 72 to 180.
This ratio can be written in the following ways.
72 to 180
72:180
72 ÷ 180
Definition of
Ratio
A ratio is a comparison of two numbers by division.
a to b
a:b
a ÷ b
where b  0

4. Using Ratios and Proportions
A __________ is an equation that shows two equivalent ratios.
proportion
Every proportion has two cross products.
In the proportion to the right, the terms 20 and 3 are called the extremes,
and the terms 30 and 2 are called the means.
30(2) =
20(3)
The cross products are 20(3) and 30(2).
60 = 60
The cross products are always _____
in a proportion.
equal

5. Using Ratios and Proportions
Theorem 9-1
Property of
Proportions
For any numbers a and c and any nonzero numbers b and d,
Likewise,

6. Using Ratios and Proportions
Solve each proportion:
15(2x) =
30(6)
3(x) =
(30 – x)2
30x = 180
3x = 60 – 2x
x = 6
5x = 60
x = 12

7. Using Ratios and Proportions
Driving gear
Driven gear
The gear ratio is the number of teeth on
the driving gear to the number of teeth on
the driven gear.
If the gear ratio is 5:2 and the driving gear
has 35 teeth, how many teeth does the
driven gear have?
given ratio
equivalent
ratio =
driving gear
driven gear 5 35
driving gear = 2 x
driven gear
5x = 70
x = 14
The driven gear has 14 teeth.

8. Using Ratios and Proportions
End of Section 9.1

9. Vocabulary What You'll Learn
Similar Polygons
What You'll Learn
You will learn to identify similar polygons.
Vocabulary
1) polygons
2) sides
3) similar polygons
4) scale drawing

10. Polygons that are the same shape but not necessarily the same size are
Similar Polygons
A polygon is a ______ figure in a plane formed by segments called sides.
closed
It is a general term used to describe a geometric figure with at least three sides.
Polygons that are the same shape but not necessarily the same size are
called ______________.
similar polygons
The symbol ~ is used to show that two figures are similar.
ΔABC is similar to ΔDEF A B C D F E
ΔABC ~ ΔDEF

11. Polygon ABCD ~ polygon EFGH
Similar Polygons
Definition of
Similar
Polygons
Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are ___________.
proportional C D A B F E G H
and
Polygon ABCD ~ polygon EFGH

12. Determine if the polygons are similar. Justify your answer.
Similar Polygons
Determine if the polygons are similar. Justify your answer. 7 6 6 4 5 4 5 7
1) Are corresponding angles are _________.
congruent
2) Are corresponding sides ___________.
proportional =
0.66 = 0.71
The polygons are NOT similar!

13. Find the values of x and y if ΔRST ~ ΔJKL
Similar Polygons
Find the values of x and y if ΔRST ~ ΔJKL R T S J L K 4 5 6 7 x
y + 2
Write the proportion that can be solved for y. 4 6 =
y + 2 7
Write the proportion that can be solved for x.
4(y + 2) = 42
4y + 8 = 42 4 5 =
4y = 34 7 x
4x = 35

14. Contractors use scale drawings to represent the floorplan of a house.
Similar Polygons
Scale drawings are often used to represent something that is too large or too small to be drawn at actual size.
Contractors use scale drawings to represent the floorplan of a house.
Dining
Room
Kitchen
Living
Garage
Utility
1.25 in.
.75 in.
1 in.
.5 in.
Scale: 1 in. = 16 ft.
Use proportions to find the actual
dimensions of the kitchen.
width
length
1 in
1.25 in.
1 in
.75 in. = =
16 ft
w ft.
16 ft
L ft.
(16)(1.25) = w
(16)(.75) = L
20 = w
12 = L
width is 20 ft.
length is 12 ft.

15. Similar Polygons
End of Section 9.2

16. Vocabulary What You'll Learn
Similar Triangles
What You'll Learn
You will learn to use AA, SSS, and SAS similarity tests for
triangles.
Vocabulary
Nothing New!

17. Some of the triangles are similar, as shown below.
Similar Triangles
Some of the triangles are similar, as shown below.
The Bank of China building in Hong Kong is one of the ten tallest buildings in
the world.
Designed by American architect I.M. Pei, the outside of the 70-story building
is sectioned into triangles which are meant to resemble the trunk of a bamboo
plant.

18. three corresponding parts of each triangle.
Similar Triangles
In previous lessons, you learned several basic tests for determining whether
two triangles are congruent. Recall that each congruence test involves only
three corresponding parts of each triangle.
Likewise, there are tests for similarity that will not involve all the parts of
each triangle.
Postulate
9-1
AA Similarity
If two angles of one triangle are congruent to two corresponding angles of another triangle, then the triangles are ______.
similar C F A D E B
If A ≈ D and B ≈ E, then ΔABC ~ ΔDEF

19. If the measures of the sides of a triangle are ___________
Similar Triangles
Two other tests are used to determine whether two triangles are similar.
Theorem 9-2
SSS Similarity
If the measures of the sides of a triangle are ___________
to the measures of the corresponding sides of another triangle, then the triangles are similar.
proportional C 6 F 2 3 1 A B D 4 E 8
then the triangles are similar
then ΔABC ~ ΔDEF

20. If the measures of two sides of a triangle are ___________
Similar Triangles
Theorem 9-3
SAS Similarity
If the measures of two sides of a triangle are ___________
to the measures of two corresponding sides of another triangle and their included angles are congruent, then the triangles are similar.
proportional C F 2 1 A B D 4 E 8
then ΔABC ~ ΔDEF

21. is used and complete the statement.
Similar Triangles
Determine whether the triangles are similar. If so, tell which similarity test
is used and complete the statement. G H K M P J 6 14 10 9 15 21 6 10 14
Since
, the triangles are similar by SSS similarity. = = 9 15 21
Therefore, ΔGHK ~ Δ
JMP

22. If Fransisco is 6 feet tall, how tall is the tree?
Similar Triangles
Fransisco needs to know the tree’s height. The tree’s shadow is 18 feet long at the same time that his shadow is 4 feet long.
If Fransisco is 6 feet tall, how tall is the tree?
1) The sun’s rays form congruent angles with the ground.
2) Both Fransisco and the tree form right angles with the ground. 4 6 = t 18
4t = 108
t = 27
6 ft.
The tree is 27 feet tall!
4 ft.
18 ft.

23. Similar Triangles
Slade is a surveyor.
To find the distance across Muddy Pond, he forms similar triangles and measures distances as shown.
8 m
10 m
45 m x
What is the distance
across Muddy Pond? 10 8 =
It is 36 meters across Muddy Pond! 45 x
10x = 360
x = 36

24. Similar Triangles
End of Section 9.3

25. Proportional Parts and Triangles
What You'll Learn
You will learn to identify and use the relationships between
proportional parts of triangles.
Vocabulary
Nothing New!

26. Proportional Parts and Triangles
In ΔPQR,
and
intersects the other two sides of ΔPQR.
Are ΔPQR and ΔPST, similar? R Q P
PST  PQR
corresponding angles
P  P
ΔPQR ~ ΔPST.
Why? (What theorem / postulate?) S T
AA Similarity (Postulate 9-1)

27. Proportional Parts and Triangles
Theorem 9-4
If a line is _______ to one side of a triangle, and intersects the other two sides, then the triangle formed is _______ to the original triangle.
parallel
similar A B C D E If
ΔABC ~ ΔADE.

28. Proportional Parts and Triangles
Since
, ΔSVW ~ ΔSRT.
Complete the proportion: T W S V R SV

29. Proportional Parts and Triangles
Theorem 9-5
If a line is parallel to one side of a triangle and intersects the other two sides, then it separates the sides into segments of
__________________.
proportional lengths E D C B A

30. Proportional Parts and TrianglesC B H G A 5 3
x + 5 x 4

31. Proportional Parts and Triangles
Jacob is a carpenter.
Needing to reinforce this roof rafter, he must
find the length of the brace.
10 ft
6 ft
4 ft
Brace 4 x
4 ft x = 10 4
10x = 16
x = 1 3 5 ft

32. Proportional Parts and Triangles
End of Section 9.4

33. Triangles and Parallel Lines
What You'll Learn
You will learn to use proportions to determine whether lines
are parallel to sides of triangles.
Vocabulary
Nothing New!

34. Triangles and Parallel Lines
You know that if a line is parallel to one side of a triangle and intersects the
other two sides, then it separates the sides into segments of proportional
lengths (Theorem 9-5).
The converse of this theorem is also true.
Theorem 9-6
If a line intersects two sides of a triangle and separates the sides into corresponding segments of proportional lengths, then the line is parallel to the third side. E D C B A 9 6 4

35. Triangles and Parallel Lines
Theorem 9-7
If a segment joins the midpoints of two sides of a triangle, then it is parallel to the third side, and its measure equals ________ the measure of the third side.
one-half E D C B A 2x x

36. Triangles and Parallel Lines
Use theorem 9 – 7 to find the length of segment DE. A 5 8 11 x D E 22 B C

37. Triangles and Parallel Lines
A, B, and C are midpoints of the sides of ΔMNP. C B A N P M
Complete each statement.
1) MP || ____ AC
2) If BC = 14, then MN = ____ 28
3) If mMNP = s, then mBCP = ___ s
4) If MP = 18x, then AC = __ 9x

38. Triangles and Parallel Lines
A, B, and C are midpoints of the sides of ΔDEF. C B A D F E 8 7 5
1) Find DE, EF, and FD.
14; 10; 16
2) Find the perimeter of ΔABC 20
3) Find the perimeter of ΔDEF 40
4) Find the ratio of the perimeter of ΔABC to the perimeter of ΔDEF.
20:40 =
1:2

39. Triangles and Parallel Lines
ABCD is a quadrilateral. A D C B
E is the midpoint of AD F
F is the midpoint of DC E H
H is the midpoint of CB
G is the midpoint of BA G
Q1) What can you say about EF and GH ?
They are parallel
(Hint: Draw diagonal AC .)
Q2) What kind of figure is EFHG ?
Parallelogram

40. Triangles and Parallel Lines
End of Section 9.5

41. Proportional Parts and Parallel Lines
What You'll Learn
You will learn to identify and use the relationships between parallel lines and proportional parts.
Vocabulary
Nothing New!

42. Proportional Parts and Parallel Lines
Hands-On
On your given paper, draw two (transversals) lines intersecting the parallel lines. A D
Label the intersections of the
transversals and the parallel lines,
as shown here. B E F C
Measure AB, BC, DE, and EF. ,
Calculate each set of ratios: BC AB EF DE AC AB DF DE ,
Do the parallel lines divide the transversals proportionally?
Yes

43. Proportional Parts and Parallel Lines
Theorem 9-8
If three or more parallel lines intersect two transversals, the lines divide the transversals proportionally. l m n B A C F E D
If l || m || n
Then BC AB EF DE = , AC AB DF DE =
and , AC BC DF EF =

44. Proportional Parts and Parallel Lines
Find the value of x. a b c H G J W V U 18 12 x 15 GH UV = HJ VW 12 15 = 18 x
12x = 18(15)
12x = 270
x = 22 2 1

45. Proportional Parts and Parallel Lines
Theorem 9-9
If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. l m n B A C F E D
If l || m || n and
AB  BC,
Then
DE  EF.

46. Proportional Parts and Parallel Lines
Find the value of x. A 10
Since AB  BC, B 10
DE  EF
Theorem 9 - 9 C
(x + 3) = (2x – 2)
x + 3 = 2x – 2
5 = x
(2x – 2)
(x +3) 8 8 F D E

47. Proportional Parts and Parallel Lines
End of Section 9.6

48. Perimeters and Similarity
What You'll Learn
You will learn to identify and use proportional relationships of
similar triangles.
Vocabulary
1) Scale Factor

49. Perimeters and Similarity
These right triangles are similar! Therefore, the measures of their corresponding sides are ___________.
proportional
Use the ____________ theorem to calculate the length of the
hypotenuse.
Pythagorean 10 6 15 9 8 12 6 8 10 2
We know that = = = 9 12 15 3
Is there a relationship between the measures of the perimeters of the two
triangles?
perimeter of small Δ
perimeter of large Δ = = 36 24 = 3 2

50. Perimeters and Similarity
Theorem
9-10
If two triangles are similar, then
the measures of the
corresponding perimeters are proportional to
the measures
of the corresponding sides. A C B F E D
If ΔABC ~ ΔDEF, then
perimeter of ΔABC
perimeter of ΔDEF = DE AB = EF BC = FD CA

51. Perimeters and Similarity
The perimeter of ΔRST is 9 units, and ΔRST ~ ΔMNP.
Find the value of each variable. M N
4.5 P R S T 3 6 z Y x
perimeter of ΔMNP
perimeter of ΔRST RS MN = RS MN = ST NP RS MN = TR PM
Theorem 9-10 2 3 = y 6 2 3 = z
4.5
13.5 9 x 3 =
The perimeter of ΔMNP is
3y = 12
3z = 9
27 = 13.5x
Cross Products
x = 2
y = 4
z = 3

52. Perimeters and Similarity
The ratio found by comparing the measures of corresponding sides of
similar triangles is called the constant of proportionality or the ___________.
scale factor D E F 14 10 6 A B C 7 5 3 DE AB = EF BC FD CA
If ΔABC ~ ΔDEF, then or 6 3 = 14 7 10 5 2 1
The scale factor of ΔABC to ΔDEF is 2 1
Each ratio is equivalent to 1 2
The scale factor of ΔDEF to ΔABC is

53. Perimeters and Similarity
End of Section 9.7

54. Proportional Parts and Triangles
Hands-On A
Step 1)
On a piece of lined paper, pick a point on one of the lines and label it A.
Use a straightedge and protractor to draw A so that mA < 90 and only the vertex lies on the line.
Step 3)
Label the points where the horizontal lines intersect segment AG (B through D).
Label the points where the horizontal lines intersect segment AI (F through H).
Step 2)
Extend one side of A down four lines. Label this point E.
Do the same for the other side of A. Label this point I.
Now connect points E and I to form ΔAEI. B F C G D H E I
What can you conclude about the lines through the sides of ΔAEI and
parallel to segment EI?
This activity suggests Theorem 9-5.