Download 9-4 the aaa similarity postulate

SIMILARITY
Anggota:
Desinta Yosopranata
Devi Iriyani
Elok Nur Afiyati
M. Syafaur Rahmat
Solekhah
(4101414008)
(4101414026)
(4101414020)
(4111411009)
(4101410036)
4 cm
8 cm
5 cm
10 cm
Both of these photographs are of the same subject, but one is larger
than the other. Both pictures have the 'same shape'. If we compare the
ratio of width to length of each picture, we see that the ratios are
equal.
This equation is called a proportion
4 8
because it is made up of two equal ratios
5

10
4
5
and
8
10
A
proportion is an equality
between two ratios. The
a
c
ratios b and d are
proportional if
a
c
b  0, d  0
b = d .
 Example
1
In a proportion cross products are equal.
4
2

or 4x4  2x8
8
4
a c
if b  d then a x d = b x c
Theorem 9-1
 Example
2
In a proportion, I can be added to both sides.
9 12
9 3 12 4
9  3 12  4

 

If 3 4 then 3 3 4 4 or 3  4
ab cd
a c

if b d then b  d
Theorem 9-2
 Example
3
In a proportion, I can be subtracted from
both sides.
9 12
9 3 12 4
9  3 12  4

  
If 3 4 , then 3 3 4 4 or 3  4
a b cd
a c

If
,
then

b
d
Theorem 9-3
b d
 Example
If
a c

b d
4
,then
9 12

3 4
9 3

12 4
Theorem 9-4
If
a c

b d
,then
a
b

c
d
 Example
5
9 12
If 9 x 4 = 3 x 12, then 3  4
If a x d = b x c, then
Theorem 9-5
a
c

b
d
MN BC
AM
AN
 1,
1
MB
NC
AM AN

MB NC
XY DE
FX 1 FY 1
 ,

XD 2 YE 2
FX FY

XD YE
Theorem 9-6
Side-Splitting Theorem.
If a line is parallel to one side of triangle and
intersect the other two sides, then it divides
the two sides proportionally.
PROOF
Given:
XY BC
XY berpotongan dengan garis yang
memuat CA , XY berpotongan
dengan garis yang memuat AB
Prove :
AX AY

XB YC
Answers :
A
Y
X
C
B
Statements
1.
XY BC
Reasons
1. Given
2. XY Intersecting lines
containing CA
2. Given
3. XY Intersecting lines
containing AB
3. Given
4.
X  B ,X  B
4.
5.
A  A
5. Berimpit
6.
ABC  XYA
6. AAA similarity postulate
7.
AX AY

XB YC
7. Restatement 6
XY BC
 Example
1
In this figure MN
By theorem 9-6,
hence

or
. Find NE!
CM
CN

MD
NE ,
DE
3
4

5
x
3x  20
x
20
3
 Example
2
Find FJ in the figure shown if GF KJ.
GK
FJ

By theorem 9-6, KH JH ,
2
x

3 6 x
3x  12  2x
or

5x  12
x
12
5
Theorem 9-7
If a line intersect two sides of a triangle and
divides them proportionally, then the line is
parallel to the third side.
BUKTI
Given :XY intersecting lines containing AC
XY intersecting lines containing AB
AX
AY

XB YC
Prove : XY BC
A
Y
X
C
B
Statements
Reasons
1. XY Intersecting lines
containing AC
1. Given
2.
Intersecting lines
XY
containing AB
2. Given
AX
AY

3. XB YC
3. Given
4.
A  A
4. Berimpit
5. Y  C , X  B
6.
ABC
7.
XY BC
XYA
5.
AX
AY

XB YC
6. AAA similarity postulate
7. Teorema kesejajaran garis
yaitu Y  C (sehadap)
 ABCDE


A



is similar to A’ B’ C’ E’.
E 
 E'
A'  
 B'
 B






 D'

C'






D



C
A  A' , B  B' , C  C '
D  D' , E  E '
AB
BC
CD
DE
AE




2
A' B' B' C ' C ' D' D' E ' A' E '
 WXYZ
is similar to W’X’Y’Z’
Z
Z'
W
Y
X
W'
W  W ' , X  X ' ,
Y  Y ' , Z  Z '
WX
XY
YZ
WZ
1




W ' X ' X 'Y ' Y ' Z ' W ' Z ' 2
X'
Y'
Notice that all corresponding angles are
congruent and that the ratios of
corresponding sides are equal.
The symbol “
“ means “is similar to.”
ABCD
A’B’C’D’ means that ABCD is similar
to A’B’C’D’.
C
B
D
D'
A'
C'
B'
A
 Two
polygons are similar if there is a
correspondence between vertices such that
corresponding angles are congruent and
corresponding sides are proportional.
 Example1
If we are given that ABCD
A’B’C’D’, then
we can conclude:
1. A  A' , B  B' , C  C ' , D  D'
AB
BC
CD
AD



2. A' B' B' C ' C ' D' A' D'
D'
B
A
C
C'
D
B'
A'
 Example
2
If we are given that
1. A  A' , B  B' , C  C ' , D  D' and
2. AB  BC  CD  AD
A' B'
B' C '
C ' D'
A' D'
than we can conclude that ABCD
A’B’C’D’.
If you have a scaled map, then a figure on
the map is similar to the figure it represents.
Given a scaled map of locations A,B, and C,
then
.
If A’C’=36 mm, A’B’=24mm, and AB=32 m,
find AC.
AB
AC
32 AC
Solution:

or

A' B'
Therefore AC =
C
A
32m
B
A' C '
24
36
32 36
 48
24
 In
ΔABC and ΔDEF, <A
<C  <F.
C

<D, <B  <E, and
F
D
B
A
Observe that
AB BC CA


DE EF FD
E
 It
appears that whenever all three angles of
one triangle are congruent to all three angles
of another triangle, then the ratios of
corresponding sides are also equal. We
accept this as a postulate
AAA
Similary Postulate
If three angles of one triangle are congruent
to three angles of another triangles, then the
triangles are similar.
Theorem 9-8
AA Similarity Theorem. If two angles of one triangle
are congruent to two angles of another triangle,
then the triangles are similar
Proof
Given : ΔABC and ΔDEF with ˂A  <D, <B  <E
Prove : ΔABC is similar to ΔDEF.
C
F
D
A
B
E
Statements
Reasons
1.
A  D
1. GIVEN
2.
B  E
2. GIVEN
C  F
3. Theorem 6-4 (The sum of the
measure of the angles of a
triangle is 180)
3.
4.
4. . AAA Similarity
Theorem 9-9
Two right triangles are similar if an acute angle of
one triangle is congruent to an acute angle of the
other triangle
Proof
Given: B  E, C  F , A  D
Prove : ΔABC
ΔDEF.
A
D
B
C
F
E
Statements
Reason
B  E
1. Given
2. C  F
2. Given
1.
3.
3. Theorem 6-4 (The sum of the
measure of the angles of a
triangle is 180 )
A  D
4. ABC
DEF
4. AAA Similarity