The document is a lecture on similar triangles. It defines similar triangles as having the same shape but different sizes, and discusses how similar triangles have corresponding angles that are congruent and corresponding sides that are proportional. It provides examples of similar triangles and statements showing their similarity. It also covers using proportions of corresponding sides to solve for missing sides in similar triangles and several proportionality principles related to similar triangles, including the basic proportionality theorem involving parallel lines cutting across a triangle.

1. The session shall begin
shortly…

3. Similar Triangles
A Mathematics 9 Lecture
3

4. 4
Similar Triangles
What do these pairs of
objects have in common?
SAME SHAPES BUT DIFFERENT SIZES

5. 5
Similar Triangles
What do these pairs of
objects have in common?
They are also called SIMILAR objects

6. The Concept of Similarity
Similar Triangles
Two objects are called similar if they have
the same shape but possibly different
sizes.

7. The Concept of Similarity
Similar Triangles
You can think of similar objects as one
one being a enlargement or reduction of
the other.

8. The Concept of Similarity
Similar Triangles
You can think of similar objects as one
being an enlargement or reduction of the
other (zoom in, zoom out).
The degree of enlargement or reduction is
called the SCALE FACTOR

10. The Concept of Similarity
Similar Triangles
Enlargements and Projection

11. 11
Similar Triangles
QUESTION!
If a polygon is enlarged or reduced,
which part changes and which part
remains the same?

12. The Concept of Similarity
Similar Triangles
Two polygons are SIMILAR if they
have the same shape but not
necessarily of the same size.
Symbol used: ~ (is SIMILAR to)
A C
B
DE
F
In the figure, ABC is
similar to DEF.
Thus ,we write
ABC ~ DEF

13. The Concept of Similarity
Similar Triangles
Two polygons are SIMILAR if they
have the same shape but not
necessarily of the same size.
If they are similar, then
1. The corresponding angles remain
the same (or are CONGRUENT)
2. The corresponding sides are related
by the same scale factor (or, are
PROPORTIONAL)

14. The Concept of Similarity
Similar Triangles
Q1
Q2
These two are similar.
Corresponding
angles are
congruent
A  E
B  F
C  G
D  H
Corresponding sides are
proportional:
1
2
EH EF FG GH
AD AD BC CD
    Scale factor from
Q1 to Q2 is ½

15. The Concept of Similarity
Similar Triangles
T1
T2
These two are similar.
Corresponding
angles are
congruent
A  D
B  E
C  F
Corresponding sides are proportional:
2
DE EF DF
AB BC AC
   Scale factor from
T1 to T2 is 2

16. Similar Triangles
The Concept of Similarity
Which pairs are similar? If they are similar,
what is the scale factor?

17. Similar Triangles
Similar Triangles
Two triangles are SIMILAR if they
have the same shape but not
necessarily of the same size.
Symbol used: ~ (is SIMILAR to)
A C
B
DE
F
In the figure, ABC is
similar to DEF.
Thus ,we write
ABC ~ DEF

18. Similar Triangles
Similar Triangles
http://wps.pearsoned.com.au/wps/media/objects/7029/7198491/opening/c10.gif

19. Similar Triangles
Two triangles are SIMILAR if all of
the following are satisfied:
1. The corresponding
angles are
CONGRUENT.
2. The corresponding
sides are
PROPORTIONAL.
Similar Triangles

20. Similar Triangles
 The two triangles shown
are similar because they
have the same three angle
measures.
 The order of the letters is
important: corresponding
letters should name
congruent angles.
 For the figure, we write
20
ABC DEF 
Similar Triangles

21. Similar Triangles
21
ABC DEF 
Similar Triangles
A B C D E F
Congruent Angles
A D  
B E  
C F  

22.  Let’s stress the order of
the letters again. When we
write note
that the first letters are A
and D, and The
second letters are B and E,
and The third
letters are C and F, and
22
ABC DEF 
.A D  
.B E  
.C F  
Similar Triangles
Similar Triangles

23.  We can also write the
similarity statement as
23
ACB DFE 
 BAC EDF
or CAB FDE 
Similar Triangle Notation
Similar Triangles
Why?

24.  BCA DFE
Similar Triangle Notation
Similar Triangles
 We CANNOT write the
similarity statement
as
 BAC EFD
Why?

25. Kaibigan, sa
similar triangles,
the
correspondence
of the vertices
matters!!!
Similar Triangles

26. 26
ABC DEF 
Similar Triangles
A B C D E F
Corresponding
Sides
AB DE
BC EF
AC DF
Proportions from Similar Triangles

27. 27
ABC DEF 
Similar Triangles
Corresponding
Sides
AB DE
BC EF
AC DF
Proportions from Similar Triangles
Ratios of
Corresponding
Sides
AB
DE
BC
EF
AC
DF

28.  Suppose
Then the sides of the
triangles are proportional,
which means:
28
.ABC DEF 
AB AC BC
DE DF EF
 
Notice that each ratio
consists of corresponding
segments.
Similar Triangles
Proportions from Similar Triangles

29. The Similarity Statements
Based on the definition of
similar triangles, we now
have the following
SIMILARITY STATEMENTS:
29
Congruent
Angles
.A D  
.B E  
.C F  
Proportional
Sides
Similar Triangles
 
AB BC AC
DE EF DF

30. 30O N
E
P
K
I
110
110
30
30
40
40
Similar Triangles
Give the
congruence and
proportionality
statements and
the similarity
statement for the
two triangles
shown.
The Similarity Statements

31. The Similarity Statements
31O N
E
P
K
I
110
110
30
30
40
40
Similar Triangles
Give the congruence and
proportionality statements
and the similarity statement
for the two triangles shown.
Congruent Angles
P O  
I N  
K E  
Corresponding
SidesPI ON
IK NE
PK OE

32. 32O N
E
P
K
I
110
110
30
30
40
40
Similar Triangles
Give the congruence and
proportionality statements
and the similarity statement
for the two triangles shown.
Congruent
Angles
  P I
  I N
  K E
Proportional
Sides  
PI IK PK
ON NE OE
Similarity
Statement  PIK ONE
The Similarity Statements

33. Similar Triangles
Given the triangle similarity
LMN ~ FGH
determine if the given
statement is TRUE or FALSE.
M G   true
FHG NLM   false
N M   false
LN MN
FG GH
 false
MN LN
GH FH
 true
GF HG
ML NM
 true
The Similarity Statements

34. In the figure,
Enumerate all the
statements that will
show that
34
.SA ON S A
L
O N
. SAL NOL
Similar Triangles
The Similarity Statements
Note: there is a COMMON
vertex L, so you CANNOT use
single letters for angles!

35. In the figure,
Enumerate all the statements
that will show that
35
.SA ON
S A
L
O N
. SAL NOL
Similar Triangles
The Similarity Statements
Congruent
Angles
  SAL LON
  ASL LNO
  OLN SLA
Proportional
Sides  
SA AL SL
ON OL NL
Note: there is a COMMON vertex L, so you
CANNOT use single letters for angles!

36. Similar Triangles
In the figure,
Enumerate all the
statements that will
show that
.KO AB
. KOL ABL
O B L
K
A
Hint: SEPARATE the two right
triangles and determine the
corresponding vertices.
Similar Triangles
The Similarity Statements

37. O B L
K
A
Similar TrianglesSimilar Triangles
The Similarity Statements
O L
K
Congruent
Angles
  KOL ABL
  LKO LAB
  KLO ALB
Proportional
Sides  
KO KL OL
AB AL BL

38. Similar Triangles
Solving for the Sides
The proportionality of the sides
of similar triangles can be used
to solve for missing sides of
either triangle. For the two
triangles shown, the statement
38
 
AB BC AC
DE EF DF
can be separated into the THREE
proportions

AB AC
DE DF

BC AC
EF DF

AB BC
DE EF

39. Similar Triangles
Solving for the Sides
Note The ratios can also be
formed using any of the
following:
39
a
b
c
d
e
f
 
a b c
d e f
 
d e f
a b c
  
a d b e a d
or or
b e c f c f

40. Given that
If the sides of the
triangles are as marked
in the figure, find the
missing sides.
40
A B
C
D E
F
,ABC DEF 
68
7
12
Similar Triangles
Solving for the Sides

41. 41
A B
C
D E
F
68
7
12 9

DF FE
AC CB
Similar Triangles
Solving for the Sides
Set up the proportions
of the corresponding
sides using the given
sides
For CB:
8 6
12

CB
8 72CB
9CB

42. 42
A B
C
D E
F
68
7
12 9
10.5
Similar Triangles
Solving for the Sides
Set up the proportions
of the corresponding
sides using the given
sides

DF DE
AC AB
For AB:
8 7
12

AB
8 84AB
21
10.5
2
AB or

43. S A
L
O N
8
10
16
x
y
Similar Triangles
Solving for the Sides
In the figure shown,
solve for x and y.
Solution
15
16 8
10

x
For x:
8 160x
20x
8
15 10

y
For y:
10 120y
12y

44. Check your understanding
The triangles are similar. Solve for x and z.
3 4
12

x
9x
5 4
12

z
15z

45. Similar Triangles
The Proportionality Principles
A line parallel to a side of a triangle
cuts off a triangle similar to the
given triangle.
This is also called the BASIC PROPORTIONALITY
THEOREM
BC DE
cuts ABC into
two similar triangles:
DE
~ ADE ABC
A
B C
D E

46. A
B C
D E
Similar Triangles
The Proportionality Principles
The Basic Proportionality Theorem
A
D E
B C
A
BC DE

47. A
B C
D E
Similar Triangles
The Proportionality Principles
The Basic Proportionality Theorem
A
D E
B C
A
BC DE
 
AD AE DE
AB AC BC
Proportions:

48. A
B C
D E
Similar Triangles
The Proportionality Principles
The Basic Proportionality Theorem
BC DE 
AD AE
DB EC
Note The two sides cut
by the line segment are
also cut proportionally;
thus we have

49. Similar Triangles
The Proportionality Principles
The Basic Proportionality Theorem
Find the value of x.
Solution
28
12 14

x
2
12

x
24x

50. Similar Triangles
The Proportionality Principles
The Basic Proportionality Theorem
O B L
K
A
12
6
9
In the figure,
Find OL and OB.
.KO AB
Solution
12 9
6

OL
For OL:
9 72OL
8OL
For OB:
 OB OL BL
8 6 
2OB

51. Similar Triangles
The Proportionality Principles
The Basic Proportionality Theorem
Find BU and SB if .BC ST

52. Similar Triangles
The Proportionality Principles
The Basic Proportionality Theorem
Find BU and SB if
.BC ST
Solution
6
24 12

BUFor BU:
24 72BU
3BU
 SB SU BU
For SB:
12 3 
9SB

53. Check your
understandingIf , find PQ, PV, and PW.VW QR
22 12
6

PQ
For PQ:
22
2
PQ
2 22PQ
11PQ
For PV:
11 9 PV
2PV
22
11 2

PW
For PW:
11 44PW
4PW

54. Similar Triangles
The Proportionality Principles
A bisector of an angle of a triangle
divides the opposite side into segments
which are proportional to the adjacent
sides.
is the angle
bisector of C.
CD

CB BD
CA DA

55. Similar Triangles
The Proportionality Principles
Angle Bisectors
Find the value of x.
Solution 15 10
18

x
5 10
6

x
5 60x
12x

56. Similar Triangles
The Proportionality Principles
Angle Bisectors
Find the value of x.
Solution 21
30 15

x
7
30 5

x
5 210x
42x
x

57. Similar Triangles
The Proportionality Principles
Three or more parallel lines divide any
two transversals proportionally.
AB EF CD
and
are transversals.
AC BD

a b
c d

58. Similar Triangles
The Proportionality Principles
Three or more parallel lines divide any
two transversals proportionally.
Note The cut segment and the length of the
segment themselves are also proportional; thus
we have

 
a c
a b c d

 
b d
a b c d

59. Similar Triangles
The Proportionality Principles
Parallel Lines and Transversals
Find the value of x.
Solution 8
28 16

x
1
28 2

x
2 28x
14x
x

60. Similar Triangles
The Proportionality Principles
Parallel Lines and Transversals
Find the value of x.
Solution 6 9
4

x
6 36x
6x

61. Similar Triangles
The Proportionality Principles
Parallel Lines and Transversals
Find the value of x.
Solution
3
10 5

x
x
10
3
2
5 30x
6x

62. Check your
understandingSolve for the indicated variable.
2. for x and y y
20
5
28 7
7 5 12
  
  
x
x
y
1. for a
a15 25
10
6
  a
a
x

63. Similar Triangles
The Proportionality Principles
The altitude to the hypotenuse of a
right triangle divides the triangle into
two triangles that are similar to the
original and each other
A B
C
D

64. Similar Triangles
The Proportionality Principles
A B
C
D
C D
B
A D
C
A C
B
∆CBD ~ ∆ACD
∆ACD ~ ∆ABC
∆CBD ~ ∆ABC
Similar Right Triangles

65. Similar Triangles
The Proportionality Principles
A B
C
D
C D
B
A D
C
A C
B
Similar Right Triangles
h
ab
y x
c
x
h
h
y
a
b
a
b
c

h y
x h

a x
c a

b y
c b
Proportions
∆CBD ~ ∆ACD ∆ACD ~ ∆ABC ∆CBD ~ ∆ABC

66. Similar Triangles
The Proportionality Principles
A B
C
D
h
ab
y x
c
2
h xy h xy
2
a xc a xc
2
b yc b yc
Similar Right Triangles
This result is also called the GEOMETRIC
MEAN THEOREM for similar right triangles

67. Similar Triangles
The Proportionality Principles
Similar Right Triangles
The GEOMETRIC MEAN of two
positive numbers a and b is
GM ab
The geometric mean of 16
and 4 is
16 4GM 64 8

68. Similar Triangles
The Proportionality Principles
Similar Right Triangles
Find the value of x.
Solution
36
x
  6 3x
18
3 2

69. Similar Triangles
The Proportionality Principles
Similar Right Triangles
Find JM, JK and JL.
Solution
8 2
8 2 16 4  JM
8 10 80 4 5  JK
2 10 20 2 5  JL

70. Similar Triangles
The Proportionality Principles
Similar Right Triangles
Find the value of x.
Solution
9 25x
3 5
15

71. Similar Triangles
The Proportionality Principles
Similar Right Triangles
Find x.
Solution
12 16 x
144 16 x
9x

72. Similar Triangles
The Proportionality Principles
Similar Right Triangles
Find x, y and z.
Solution
6 9 x
36 9 x
4x
4 13y
2 13y
9 13z
3 13z

74. Thankyou!