Transcript Similarity Theorems

In geometry, two shapes
are similar when one is a
dilation of the other.
Consider Dr. Evil and Mini Me from
Mike Meyers’ hit movie Austin Powers.
Mini Me is supposed to be an exact
replica of Dr. Evil.
SIMILAR TRIANGLE NOTATION
 The
is
symbol for similarity
.
55
DEF
order of the letters is
important: corresponding
letters should name
congruent angles.
A
45
E
80
55
F
4
ABC
 The
C
80
45
B
D
ABC  DEF
When we say that triangles are similar there are several
conclusions that come from it.
1. All angles are congruent
A  D
B  E
C  F
2. All sides are proportional.
AB
DE
=
BC
EF
=
AC
DF
If we need to PROVE that a pair of triangles are similar
how many of those statements do we need? We do not
need all of them. There are three special combinations
that we can use to prove similarity of triangles.
1. AA Similarity Theorem
 2 pairs of congruent angles
2. SSS Similarity Theorem
 3 pairs of proportional sides
3. SAS Similarity Theorem
 2 pairs of proportional sides and congruent
angles between them
1. AA Similarity Theorem
 2 pairs of congruent angles
Q
M
70
50
N
mN = mR
mO = mP
O
50
70
P
MNO  QRP
R
You may need to find the 3rd angles in order to see if
2 pairs are congruent!
T
X
Y
34
34
59
59
Z
87 59
U
S
mS = 180- (34 + 87)
mS = 180- 121
mS = 59
mT = mX
mS = mZ
TSU  XZY
E
2. SSS Similarity Theorem
 3 pairs of proportional sides
A
9.6
5
B
C
12
mAB

mDF
mBC

mFE
5
 1.25
4
12
 1.25
9.6
F
4
mAC
13

 1.25
mDE 10.4
ABC  DFE
The scale factor is 1.25
D
3. SAS Similarity Theorem
 2 pairs of proportional sides and congruent
angles between them
L
G
70
H
7
I
mGH
5

 0.66
7 .5
mLK
mHI
7

 0.66
mKJ 10.5
70
J
10.5
mH = mK
GHI  LKJ
K
SAS does not work unless angle is in between
proportional sides!
L
G
50
H
7
I
J
50
K
10.5
Angles I and J do not fall in between sides GH and HI and
sides LK and KJ respectively.
Proving Triangles Similar
Steps for proving triangles similar:
1. Mark the Given.
2. Mark …
Shared Angles or Vertical Angles
3. Choose a Method. (AA, SSS , SAS)
Think about what you need for the chosen method and
be sure to include those parts in the proof.
CAN YOU PROVE THESE ARE
SIMILAR? WHY OR WHY NOT???
CAN YOU PROVE THESE ARE
SIMILAR? WHY OR WHY NOT???