1. Chapter 7 Similarity and Proportion
Express a ratio in simplest form.
State and apply the properties of similar polygons.
Use the theorems about similar triangles.

2. Warm –up
In complete sentences, explain what a ratio is. Create a real-life example of a ratio being used.
In complete sentences, explain what a proportion is.

3. 7.1 Ratio and Proportion Objectives Express a ratio in simplest form
Solve for an unknown in a proportion

4. Ratio
A ratio of one number to another is the quotient when the first number is divided by the second.
A comparison between numbers
There are 3 different ways to express a ratio
This just defines a ratio. Can any of these ratios be simplified? What could the extended ratio be used to do? 3 5 1 2 a b
3 : 5
a : b
1 : 2
1 to 2
3 to 5
a to b

5. Ratio Always reduce ratios to the simplest form
The ratio of 8 to 12 is = 8_ 12 2_ 3 O Z
Find the ratio of OI TO ZD
- Put the answer in your notes.
- make sure to reduce
110
This just defines a ratio. Can any of these ratios be simplified? What could the extended ratio be used to do? 14 6b 60 70 D I
Find the ratio of LD to LO
Angle ratio on whiteboard

6. Additional Elements of Ratios
What is the ratio of 100cm to 10m ?
NO! = ?
WHY??
To find the ratio of two lengths, they must always be measured in the terms of the same unit.

7. 100cm 1m = 10m 10m 100cm 100cm = 10m 1000cm Unit Conversions or
**Both ways give you the same ratio which is 1 to 10. or
100cm cm
10m cm =

8. Comparing 3 or more numbers
We use the following form to represent three or more numbers that are in ratio to each other…
Reads as.. “ 3 to 5 to 7"
3 : 5 : 7

9. Comparing 3 or more numbers
The measures of three angles of a triangle are in the ratio 2:2:5. Find the measure of each angle.
Partners: Set up the problem…
X represents a “part” of each angle
2x + 2x + 5x = 180
SO WHAT’S EACH MEASURE?

10. Proportion 5 : 8 = a : b An equation stating to ratios are equal.
Again, just a definition. This does not show how to solve the proportion. Which of these proportions can be solved? Which cannot? Why?
In both instances you read the proportion as…..
“5 is to 8 as a is to b.”
WHAT WOULD BE AN EXAMPLE OF A TRUE PROPORTION?

11. White Board Practice
ABCD is a parallelogram. Find the value of each ratio. A D C B 10 6

12. White Board Practice AB : BC BC : AD m  A : m  C
5 : 3
BC : AD
1 : 1
m  A : m  C
AB : perimeter of ABCD
5 : 16 A D C B 10 6

13. White Board Practice Express the ratio in simplest form IS : DI : IT10 4 12 D I S T

14. White Board Practice Express the ratio in simplest form IS : DI : IT
4 : 10 :  To reduce, find GCF
2 : 5 : 8 10 4 12 D I S T

15. Whiteboard Practice
The ratio of the measures of two complementary angles is 4:5. Find the measure of each angle.
4x + 5x = 90
9x = 90
X = 10
40, 50

16. 7.2 Properties of Proportions
Objectives
Express a given proportion in an equivalent form.

17. Warm - up Come up with an example of a true proportion
How do you solve for a proportion that has a missing variable?

18. Means and Extremes a b c d a : b = c : d
The extremes of a proportion are the first and last terms
The means of a proportion are the middle terms
The role of the extremes and means is to make the manipulations learned today easier to describe. = a b c d
a : b = c : d

19. Means-Extremes property of proportions
The product of the extremes equals the product of the means. = a b c d
ad = cb

20. Properties of Proportion [AKA – Different ways to say the same thing]
is equivalent to a. b. c.
Each of these is a different property. d.
Bottom Line: When I cross multiply any of these, I will always end up back at ad=bc.

21. Rewrite the following in 4 different ways…
As any of these 1. 2. 3.
Each of these is a different property. +y
2(x + y) = y (5+2)
2x + 2y = 7y
2x = 5y 4.

22. Another Property call it the “addition property”
Select and work several example problems off of the classroom exercises.
Show whiteboard example

23. Solving a Proportion First, cross-multiply Next, divide by 5
This shows how to solve a proportion. A proportion can be solved if it is expressed with a single variable.

24. White Board Practice
If , then 2x = _______

25. White Board Practice
If , then 2x = 28

26. White Board Practice
If 2x = 3y, then

27. White Board Practice
If 2x = 3y, then

28. White Board Practice
If , then

29. White Board Practice
If , then

30. White Board Practice
If , then

31. White Board Practice
If , then

32. White Board Practice Solve for x X = 6 X = 9 X 9 2 3 = 2_ x-6 8__ x+3
Brightstorm link

33. Whiteboard practice
Page 246 #2
#11

34. WARM UP
In order for 2 polys to be congruent, 2 rules must be satisfied…
All _______________________________
All________________________________
In a complete sentence, what do you think the difference is between 2 polys that are congruent and 2 polys that are similar?

35. 7.3 Similar Polygons Objectives
State and apply the properties of similar polygons.

36. Similarity Coaching Football
When I need to show my players the diagram of a play, I am not going to use a piece of paper that is 50 yards wide and 100 yards long…
So what do I do???
Draw the same shape of the field but with a length and width that is drawn to a smaller scale.

37. Because they would be congruent!!
Similar Polygons
Same shape
Not the same size  Why?
Because they would
be congruent!!

38. Similar Polygons (~) All corresponding angles congruent A  A’
B  B’
C  C’
Read as “A prime”, “B prime, and so on.. A
ORDER MATTERS!!
Just like congruent polys you must make sure to name the vertices in the correct order. A’ A’
Write the extended proportion and the congruencies on the diagram. C B B’ C’

39. Similar Polygons (~) 2. All corresponding sides are in proportion
AB = BC = CA
A’B’ B’C’ C’A’
All sides have equivalent ratios A
Partners: Come up with side lengths and angle measures for the two triangles that would make them similar. A’
Write the extended proportion and the congruencies on the diagram. B C B’ C’

40. The Scale Factor
If two polygons are similar, then they have a scale factor
The reduced ratio between any pair of corresponding sides or the perimeters.
12:3  scale factor of 4:1 12
Work several examples of how to find the scale factor, and how to use it to find the unknown parts. 3

41. Using the Scale factor to find Missing Pieces
You have to know the scale factor first to find missing pieces.
Solve for y by cross-multiplication 12
Work several examples of how to find the scale factor, and how to use it to find the unknown parts. 3 10 y
What could I do to make the math easier before I try cross-multiplying?

42. White Board Practice
Quadrilateral ABCD ~ Quadrilateral A’B’C’D’. Find their scale factor
5:3
The first # in the scale factor will come from ABCD A D C B A’ D’ C’ B’ 50 y 30 20 12 x z

43. White Board Practice
Quadrilateral ABCD ~ Quadrilateral A’B’C’D’. Find the values of x, y, and z
x = 18
y = 20
z = 12 A D C B A’ D’ C’ B’ 50 y 30 20 12 x z

44. White Board Practice
Quadrilateral ABCD ~ Quadrilateral A’B’C’D’. Find the ratio of the perimeters
5:3 A D C B A’ D’ C’ B’ 50 y 30 20 12 x z

45. Additional Problems
Example
Page 250
Classroom ex. #1
#10

46. Quiz Review Section 7.1 Section 7.2 Section 7.3
Putting ratios into simplest form
Find the measure of each angle based on a ratio
i.e. Pg #24 – 29
Section 7.2
Properties of proportions ( purple box pg. 245)
i.e. how can the proportion be changed around and still be equal to the original (i.e. pg. 247 # 1-8)
Find the value of X ( Cross multiply and solve)
i.e. Pg #9 – 20
Section 7.3
Understand the definition of similar polygons (~)
Finding the scale factor of similar polys
Compare the lengths of corresponding sides (reduce)
Use the scale factor to find unknown lengths
i.e. Pg #

47. Warm – Up Using the book or notes…
Write down the definitions for the following
Ratio
Proportion
Scale factor
Similar Polygons

48. 7.4 A Postulate for Similar Triangles
Objectives
Learn to prove triangles are similar.

49. Why does this whole 3 pair thing sound so familiar?
What we have learned…
Two polygons are similar by showing that they satisfy the definition of similar polygons (~)
3 pair of corresponding angles are congruent
3 pair of corresponding sides are in proportion
Why does this whole 3 pair thing sound so familiar?

50. Index Card Experiment Supplies: Index card, Scissors, Ruler
Cut out a triangle using a 3x5 index card
Label the vertices A, B, C
Take side BC of your triangle

51. Index Card Experiment
Draw a line that is twice the length of BC and label the endpoints B’ and C’
At B’ line up angle B of your triangle and trace it on the paper. Then do the same thing for C’.

52. ▲ABC ~ ▲A’B’C’ Why? How was ▲A’B’C’ created?
Corresponding angles are congruent
The sides are in proportion with a scale factor of 1:2
How was ▲A’B’C’ created?
By using 2 of the corresponding angles from ▲ABC

53. AA Similarity Postulate (AA~ Post)
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. A D
This can be shown to prove triangles similar, although there is no proof of it. Why are the sides of a triangle in proportional if the angles are congruent? Do a proof using this postulate. F E B C

54. Applying AA Similarity in Proofs
The key to using this postulate is to first prove two corresponding angles of two triangles congruent and then using it

55. Remote Time
T – Similar Triangles
F – Not Similar

56. T – Similar Triangles F – Not Similar

57. T – Similar Triangles F – Not Similar

58. T – Similar Triangles F – Not Similar

59. Whiteboards
Page 256
#11
#13

60. brightstorm
Example

61. 7-5: Theorems for Similar Triangles
Objectives
Learn about 2 additional ways to prove triangles are similar.

62. WARM-UP What we have learned…
SAS Congruency – Write down in your own words what this means.
SSS Congruency – Write down in your own words what this means.

63. SAS Postulate
If two sides and the included angle are congruent to the corresponding parts of another triangle, then the triangles are congruent. B E
Select a proof that uses SAS and do it with them. C D F

64. SAS Similarity Theorem (SAS~)
Partners: Based on what you now know about similarity compared to congruency, come up with the wording for this theorem.
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
This is a difficult theorem to prove, so it is not wise to prove it in class, unless it is an honors group. Do a proof that uses it, however. Also talk about how this makes the triangles “almost” congruent. Compare the SAS, the SAS and the SAS. 2 1 F 2 E B C 4

65. - What is the scale factor of the ~ triangles
- Name the triangles
- Name the postulate or theorem
Included angle C
Scale Factor = 2/3
▲CDE ~ ▲CAB by SAS ~ 10 6 D E 3 5 A B

66. SSS Postulate
If three sides of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. B E
Select a proof that uses SSS and do it with them. C D A F

67. SSS Similarity Theorem (SSS~)
Partners: Based on what you now know about similarity compared to congruency, come up with the wording this theorem.
If the three sides of one triangle are proportional to the three sides of another triangle, then the triangles are similar. A A D D
Ditto. 3 2 6 1 F F E E B B 2 C C 4

68. Example ▲ABC ~ ▲XYZ by SSS ~
The measures of the sides of ▲ABC are 4, 5, 7
The measures of the sides of ▲XYZ are 16, 20 , 28
Are the two triangles similar? Why?
▲ABC ~ ▲XYZ by SSS ~

69. 4 Ways to Prove Triangles Similar
Definition of similarity
AA ~
SAS ~
SSS ~
**PROOFS: Once we have proven that 2 triangles are similar. We can then say what about …
The corresponding angles?
The corresponding sides?

70. White Board Practice
Name the similar triangles and give the postulate or theorem that justifies your answer…

71. A
▲ADE ~ ▲ABC by AA ~
80◦ E D
80◦ C B

72. ▲ABC ~ ▲DEF by SSS ~ F 9 A 18 D 3
4.5 6 E C B 9

73. Z 20 S 15 D R 25 12 T
▲TRS ~ ▲ZRD by SAS ~

74. You want to prove ▲RST ~ ▲ XYZ by SSS ~
State the ratios that you know have to be equal to one another
You want to prove ▲RST ~ ▲ XYZ by SAS ~
If you know LR congruent LX, what else do you need to prove?

75. 7-6: Proportional Lengths
Objectives
Apply the Triangle Proportionality Theorem and its corollary
State and apply the Triangle Angle-bisector Theorem

76. Billy and Bob
Billy and Bob want a foot-long sub from Subway that costs $4
Billy has $1 and Bob has $3
They combine their money and buy the sub
How much of the sub should each person get based on the amount of money they paid?

77. Divided Proportionally$4
Bob $3
12in $2 $1
Billy

78. Divided Proportionally
If points are placed on segments AB and CD so that , then we say that these
segments are divided proportionally. B
This just gets them comfortable with the idea of divided proportionally. Show them several correct proportions that can be written. Also show them an incorrect proportion and why it is incorrect. Most of these are intuitive. D X Y A C

79. Example B D X 4 Y 2 A 2 C 1
*Partners: determine another correct proportion as well as one that wouldn’t work*
AX CY
XB YD =
AX XB
CY YD
AX CY
AB CD = = = =

80. Theorem
If a line parallel to one side of a triangle intersects the other two sides, it divides them proportionally.
Just think of these 2 sides as lines that have been divided proportionally Y
Ditto. B A X Z

81. What can we conclude based on the diagram?
AY BY
XY ZY =
▲AYB ~ ▲XYZ by AA ~
The sides are divided into proportional segments by TH. 7-3
*Find 2 proportions that can be justified by TH. 7-3 Y
Ditto. 4 2 B A 2 1 X Z

82. White Board Practice Are the following proportions possible?
Answer True or False. d b y c x j

83. True or False y j b x = c d j x = T T b d y c = T c b y x = F d b y c

84. Corollary
If three parallel lines intersect two transversals, then they divide the transversals proportionally.
What theorem does this diagram remind you of? R W S X
Ditto.
RS WX
ST XY = T Y

85. Solve for Y 15 y 10 14

86. Theorem
If a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides. Y
WX XY
WZ ZY =
Y has been bisected into congruent angles
Ditto. X Z W

87. White Board Practice
Find X 12 10 24
X = 20 X

88. White Board Practice
Find X 10 5 X 20
X = 15

89. Ch. 7 Test Review Section 7.1 Section 7.2
Putting ratios into simplest form
Find the measure of each angle based on a ratio
i.e. Pg #24 – 29
Section 7.2
Properties of proportions ( purple box pg. 245)
i.e. how can the proportion be changed around and still be equal to the original (i.e. pg. 247 # 1-8)
Find the value of X ( Cross multiply and solve)
i.e. Pg #9 - 20

90. Ch. 7 Test Review Section 7.3 Section 7.4 and 7.5
Understand the definition of similar polygons (~)
Finding the scale factor of similar polys
Compare the lengths of corresponding sides (reduce)
Use the scale factor to find unknown lengths
i.e. Pg #
Section 7.4 and 7.5
**pg. 258 #16**
Proving 2 triangles similar
AA ~ , SAS ~ , SSS ~
i.e. pg # 1 – 6
**REMEMBER ORDER MATTERS WHEN NAMING THE SIMILAR TRIANGLES!!!

91. Ch. 7 Test Review Section 7.6 PROOFS
Understand the 2 theorems and the corollary
i.e. P. 272 # and 20 – 23
PROOFS
Study the following – Understand why a certain statement was given and its reason for it.
i.e. p 255 proof example
P #