ΜΑΤΗ

2. Similar triangles are triangles that have the same
shape but not necessarily the same size.
A
C
B
D
F
E
ABC  DEF
When we say that triangles are similar we know:
A  D
B  E
C  F
AB
DE
BC
EF
AC
DF
= =

3. 2. SSS Similarity Theorem
 3 pairs of proportional sides
Six of those statements are true as a result of the
similarity of the two triangles. However, if we need to
prove that a pair of triangles are similar how many of
those statements do we need? Because we are working
with triangles and the measure of the angles and sides
are dependent on each other. We do not need all six.
There are three special combinations that we can use
to prove similarity of triangles.
3. SAS Similarity Theorem
 2 pairs of proportional sides and congruent
angles between them
1. AA Similarity Postulate
 2 pairs of congruent angles

4. 8.4 – Similar Triangles
(AA)

5. AA Similarity Postulate - If two angles of one
triangle are congruent to two angles of another
triangle, then the to triangles are similar.
M
N O
Q
P R
70
70
50
50
mN = mR
mO = mP MNO  QRP

7. Example:
Given: ABC and DEF are isosceles right triangles
Prove:
DEF
ABC 
 ~

8. Example:
Given:
Prove:
JS
OH //
SM
HM
JM
OM

J
O
M
H
S

9. 8.5 – Proving that
Triangles are Similar
(SSS and SAS)

10. SSS Similarity Theorem
If corresponding sides of two triangles are
proportional, then the two triangles
are similar.
A
B C
E
F D
25
1
4
5
.
DF
m
AB
m


25
1
6
9
12
.
.
FE
m
BC
m


25
1
4
10
13
.
.
DE
m
AC
m


5
4
12
9.6
ABC  DFE

11. Proof of SSS Similarity Theorem
Given:
Prove:
Draw so that and
Then because ____________.
So by _______.
So
Since PS = ML, then SQ = MN and PQ = LN
Then by ________.
Use CPCTC and AA sim postulate to show
R
S
T
M
L
N
PQ RT
PQ //
P
Q
LM
PS 
STR
SQP 

RTS
PQS 
 ~
S
S 

QP
TR
SQ
ST
PS
RS


NL
TR
MN
ST
LM
RS


LMN
RST 
 ~
LMN
PSQ 


LMN
RST 
 ~

12. SAS Similarity Theorem
If an angle of one triangle is congruent to an angle of
another triangle, and then lengths of the sides
including these angles are proportional, then the
triangle are similar.
G
H I
L
J K
66
0
5
7
5
.
.
LK
m
GH
m


66
0
5
10
7
.
.
KJ
m
HI
m


7
10.5
70
70
mH = mK
GHI  LKJ

14. The SAS Similarity Theorem does not work unless
the congruent angles fall between the proportional
sides. For example, if we have the situation that is
shown in the diagram below, we cannot state that the
triangles are similar. We do not have the information
that we need.
G
H I
L
J K
7
10.5
50
50
Angles I and J do not fall in between sides GH and HI and
sides LK and KJ respectively.

15. Are the following triangles similar? If so,
what similarity statement can be made.
Name the postulate or theorem you used.
F
G
H
K
J
Yes, FGH  KJH because
of the AA~ Postulate

16. Are the following triangles similar? If so,
what similarity statement can be made.
Name the postulate or theorem you used.
M
O R
G
H I
6
10
3
4
No, these are not similar
because

17. Are the following triangles similar? If so,
what similarity statement can be made.
Name the postulate or theorem you used.
A
X Y
B C
20
25
25
30
No, these are not similar
because

18. Are the following triangles similar? If so,
what similarity statement can be made.
Name the postulate or theorem you used.
Yes, APJ  ABC because of
the SSS~ Postulate.
A
P J
B C
3
5
2
3
8
3

19. Explain why these triangles are similar.
Then find the value of x.
3
5
4.5
x
These 2 triangles are similar
because of the AA~ Postulate.
x=7.5

20. Explain why these triangles are similar.
Then find the value of x.
These 2 triangles are similar
because of the AA~ Postulate.
x=2.5
5
70 110
3 3
x

21. Explain why these triangles are similar.
Then find the value of x.
22
14
24
x
These 2 triangles are similar
because of the AA~ Postulate.
x=12

22. Explain why these triangles are similar.
Then find the value of x.
These 2 triangles are similar
because of the AA~ Postulate.
x= 12
x
6
2
9

23. Explain why these triangles are similar.
Then find the value of x.
These 2 triangles are similar
because of the AA~ Postulate.
x=8
15
4
x
5

24. Explain why these triangles are similar.
Then find the value of x.
These 2 triangles are similar
because of the AA~ Postulate.
x= 15
18
7.5 12
x