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Similar Triangles
A Mathematics 9 Lecture
3
Similar Triangles
What do these pairs of
objects have in common?
4
SAME SHAPES BUT DIFFERENT SIZES
Similar Triangles
What do these pairs of
objects have in common?
They are also called SIMILAR objects
5
Similar Triangles
The Concept of Similarity
Two objects are called similar if they have
the same shape but possibly different
sizes.
Similar Triangles
The Concept of Similarity
You can think of similar objects as one
one being a enlargement or reduction of
the other.
Similar Triangles
The Concept of Similarity
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
Similar Triangles
The Concept of Similarity
Enlargements and Projection
Similar Triangles
QUESTION!
If a polygon is enlarged or reduced,
which part changes and which part
remains the same?
11
Similar Triangles
The Concept of Similarity
Two polygons are SIMILAR if they
have the same shape but not
necessarily of the same size.
Symbol used:
~
F
(is SIMILAR to)
In the figure, ABC is
similar to DEF.
B
Thus ,we write
ABC ~ DEF
A
C
E
D
Similar Triangles
The Concept of Similarity
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)
Similar Triangles
The Concept of Similarity
These two are similar.
Q2
Q1
Corresponding
angles are
congruent
A  E
B  F
C  G
D  H
Corresponding sides are
proportional:
EH EF
FG GH 1




AD AD BC CD 2
Scale factor from
Q1 to Q2 is ½
Similar Triangles
The Concept of Similarity
These two are similar.
Corresponding
angles are
congruent
A  D
B  E
C  F
T1
T2
Corresponding sides are proportional:
DE EF DF


2
AB BC AC
Scale factor from
T1 to T2 is 2
Similar Triangles
The Concept of Similarity
Which pairs are similar? If they are similar,
what is the scale factor?
Similar Triangles
Similar Triangles
Two triangles are SIMILAR if they
have the same shape but not
necessarily of the same size.
Symbol used:
~
F
(is SIMILAR to)
In the figure, ABC is
similar to DEF.
B
Thus ,we write
ABC ~ DEF
A
C
E
D
Similar Triangles
Similar Triangles
http://wps.pearsoned.com.au/wps/media/objects/7029/7198491/opening/c10.gif
Similar Triangles
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
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
ABC
DEF
20
Similar Triangles
Similar Triangles
ABC DEF
A B C
D E F
Congruent Angles
A  D
B  E
C  F
21
Similar Triangles
Similar Triangles

Let’s stress the order of
the letters again. When we
write ABC DEF note
that the first letters are A
and D, and A  D. The
second letters are B and E,
and B  E. The third
letters are C and F, and
C  F.
22
Similar Triangles
Similar Triangle Notation

We can also write the
similarity statement as
BAC EDF
ACB DFE
or CAB FDE
Why?
23
Similar Triangles
Similar Triangle Notation

We CANNOT write the
similarity statement
as
BCA DFE
BAC EFD
Why?
Similar Triangles
Kaibigan, sa
similar triangles,
the
correspondence
of the vertices
matters!!!
Similar Triangles
Proportions from Similar Triangles
ABC
DEF
A B C
D E F
Corresponding
Sides
AB  DE
BC  EF
AC  DF
26
Similar Triangles
Proportions from Similar Triangles
ABC
Corresponding
Sides
AB  DE
BC  EF
AC  DF
DEF
Ratios of
Corresponding
Sides
AB
DE
BC
EF
AC
DF
27
Similar Triangles
Proportions from Similar Triangles

Suppose ABC DEF.
Then the sides of the
triangles are proportional,
which means:
AB AC BC


DE DF EF
Notice that each ratio
consists of corresponding
segments.
28
Similar Triangles
The Similarity Statements
Based on the definition of
similar triangles, we now
have the following
SIMILARITY STATEMENTS:
Congruent
Angles
A  D.
B  E.
C  F.
Proportional
Sides
AB BC AC


DE EF DF
29
Similar Triangles
The Similarity Statements
I
Give the
congruence and
proportionality
statements and
the similarity
statement for the
two triangles
shown.
P
40
110
30
E
K
30
110
O
40
N
30
Similar Triangles
The Similarity Statements
Give the congruence and
I
proportionality statements P
and the similarity statement 110 40
for the two triangles shown.
Congruent Angles
P  O
I  N
K  E
30
E
K
30
Corresponding
PI Sides
 ON
IK  NE
PK  OE
110
O
40
N
31
Similar Triangles
The Similarity Statements
Give the congruence and
I
proportionality statements P
and the similarity statement 110 40
for the two triangles shown.
Congruent
Angles
Proportional
Sides
Similarity
Statement
P  I
I  N
K  E
30
E
PI
IK PK


ON NE OE
PIK
ONE
K
30
110
O
40
N
32
Similar Triangles
The Similarity Statements
Given the triangle similarity
LMN ~ FGH
determine if the given
statement is TRUE or FALSE.
true
LN MN

FG GH
false
FHG  NLM false
MN LN

GH FH
true
GF
HG

ML NM
true
M  G
N  M
false
Similar Triangles
The Similarity Statements
In the figure, SA ON .
Enumerate all the
statements that will
show that SAL NOL.
Note: there is a COMMON
vertex L, so you CANNOT use
single letters for angles!
S
A
L
N
O
34
Similar Triangles
The Similarity Statements
In the figure, SA ON .
Enumerate all the statements
that will show that SAL NOL.
Note: there is a COMMON vertex L, so you
CANNOT use single letters for angles!
Congruent
Angles
Proportional
Sides
S
SAL  LON
ASL  LNO
OLN  SLA
SA AL SL


ON OL NL
A
L
N
O
35
Similar Triangles
The Similarity Statements
K
In the figure, KO AB.
Enumerate all the
statements that will
show that
KOL
A
ABL.
Hint: SEPARATE the two right
triangles and determine the
corresponding vertices.
O
B
L
Similar Triangles
The Similarity Statements
Congruent
Angles
K
K
Proportional
Sides
A
O
O
B
LL
KOL  ABL
LKO  LAB
KLO  ALB
KO KL OL


AB AL BL
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
AB BC AC


DE EF DF
can be separated into the THREE
proportions
AB BC

DE EF
BC AC

EF DF
AB AC

DE DF
38
Similar Triangles
Solving for the Sides
Note The ratios can also be
formed using any of the
following:
a b c
 
d e f
c
d e f
 
a b c
a d
b e
a d
 or  or 
b e
c f
c f
a
b
f
d
e
39
Similar Triangles
Solving for the Sides
F
Given that ABC
8
DEF ,
D
If the sides of the
triangles are as marked
in the figure, find the
missing sides.
6
7
C
E
12
A
B
40
Similar Triangles
Solving for the Sides
Set up the proportions
of the corresponding
sides using the given
sides
For CB:
CB  9
8
D
DF FE

AC CB
8
6

12 CB
8CB  72
F
12
A
6
7
C
E
9
B
41
Similar Triangles
Solving for the Sides
Set up the proportions
of the corresponding
sides using the given
sides
8
D
For AB:
DF DE

AC AB
8
7

12 AB
F
6
7
C
E
8 AB  84
12
21
AB 
or 10.5
2
A
9
10.5
B
42
Similar Triangles
Solving for the Sides
In the figure shown,
solve for x and y.
S
Solution
For x: 16  8
x 10
8 x  160
x  20
y
y
8
For y:

15 10
10 y  120
y  12
16
A
8
L
10
O
15
x
N
Check your understanding
The triangles are similar. Solve for x and z.
3 4

x 12
5 4

z 12
x9
z  15
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
A
D
BC DE
E
DE cuts ABC into
two similar triangles:
ADE ~ ABC
B
C
Similar Triangles
The Proportionality Principles
The Basic Proportionality Theorem
A
A
A
D
E
B
D
E
C
BC DE
B
C
Similar Triangles
The Proportionality Principles
The Basic Proportionality Theorem
A
A
D
E
B
C
D
A
E
B
BC DE
Proportions:
AD AE DE


AB AC BC
C
Similar Triangles
The Proportionality Principles
The Basic Proportionality Theorem
A
D
B
E
C
BC DE
Note The two sides cut
by the line segment are
also cut proportionally;
thus we have
AD AE

DB EC
Similar Triangles
The Proportionality Principles
The Basic Proportionality Theorem
Find the value of x.
Solution
x 28

12 14
x
2
12
x  24
Similar Triangles
The Proportionality Principles
The Basic Proportionality Theorem
In the figure,
KO AB.
Find OL and OB.
Solution
For OL: 12  9
OL 6
9OL  72
OL  8
K
A
12
For OB:
9
OB  OL  BL
 86
OB  2
O
B
6
L
Similar Triangles
The Proportionality Principles
The Basic Proportionality Theorem
Find BU and SB if BC ST .
Similar Triangles
The Proportionality Principles
The Basic Proportionality Theorem
Find BU and SB if
BC ST .
Solution
For BU:
6 BU

24 12
24 BU  72
BU  3
For SB:
SB  SU  BU
 12  3
SB  9
Check your
understanding
If VW QR , find PQ, PV, and PW.
For PQ:
22 12

PQ 6
22
2
PQ
2 PQ  22
PQ  11
For PV:
PV  11  9
PV  2
For PW:
22 PW

11
2
11PW  44
PW  4
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.
CD is the angle
bisector of C.
CB BD

CA DA
Similar Triangles
The Proportionality Principles
Angle Bisectors
Find the value of x.
Solution
15 10

18 x
5 10

6 x
5 x  60
x  12
Similar Triangles
The Proportionality Principles
Angle Bisectors
Find the value of x.
Solution
x
x 21

30 15
x 7

30 5
5 x  210
x  42
Similar Triangles
The Proportionality Principles
Three or more parallel lines divide any
two transversals proportionally.
AB EF CD
AC and BD
are transversals.
a b

c d
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
Similar Triangles
The Proportionality Principles
Parallel Lines and Transversals
Find the value of x.
Solution
x
x
8

28 16
x 1

28 2
2 x  28
x  14
Similar Triangles
The Proportionality Principles
Parallel Lines and Transversals
Find the value of x.
Solution
6 9

4 x
6 x  36
x6
Similar Triangles
The Proportionality Principles
Parallel Lines and Transversals
Find the value of x.
Solution
10
x
x 3

10 5
5 x  30
3
2
x6
Check your
Solve for the indicated variable.
understanding
1. for a
15 25

 a  10
6
a
2. for x and y
20 x
  x5
28 7
y  7  5  12
a
y
x
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
C
A
D
B
Similar Triangles
The Proportionality Principles
Similar Right Triangles
B
C
C
D
C
A
A
D
B
D
B
∆CBD ~ ∆ACD
∆ACD ~ ∆ABC
∆CBD ~ ∆ABC
A
C
Similar Triangles
The Proportionality Principles
a
Similar Right Triangles
C
C
b
a
h
y
D
c
x
h
B
y
c
h y

x h
∆ACD ~ ∆ABC
a x

c a
C
h
D
B
a
Proportions
∆CBD ~ ∆ACD
x
D
b
A
A
B
∆CBD ~ ∆ABC
b y

c b
A
b
C
Similar Triangles
The Proportionality Principles
Similar Right Triangles
b
A
D
c
h  xy
a  xc
a  xc
b 2  yc
b
a
h
y
h  xy
2
C
2
x
B
This result is also called the GEOMETRIC
MEAN THEOREM for similar right triangles
yc
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
GM  16 4  64  8
Similar Triangles
The Proportionality Principles
Similar Right Triangles
Find the value of x.
Solution
x
 18
x
6
 6  3
3
3 2
Similar Triangles
The Proportionality Principles
Similar Right Triangles
Find JM, JK and JL.
Solution
JM  8 2  16  4
JK  8 10  80  4 5
8
2
JL  2 10  20  2 5
Similar Triangles
The Proportionality Principles
Similar Right Triangles
Find the value of x.
Solution
x  9 25
3 5
 15
Similar Triangles
The Proportionality Principles
Similar Right Triangles
Find x.
Solution
12  16 x
144  16 x
x9
Similar Triangles
The Proportionality Principles
Similar Right Triangles
Find x, y and z.
Solution
6  9x
36  9 x
x4
y  4 13
y  2 13
z  9 13
z  3 13
Thank you!