Proving
Triangles
Similar
Lesson 5-3
(AA~, SSS~, SAS~)
Similar Triangles
Two triangles are similar if
they are the same shape. That
means the vertices can be
paired up so the angles are
congruent. Size does not
matter.
AA Similarity
(Angle-Angle or AA~)
If 2 angles of one triangle are congruent to 2 angles of
another triangle, then the triangles are similar.
E
B
A
C
Given:
Conclusion:
D
A  D
F
and
B  E
ABC ~ DEF
by AA~
SSS Similarity
(Side-Side-Side or SSS~)
If the lengths of the corresponding sides of 2 triangles are
proportional, then the triangles are similar.
E
B
A
Given:
C D
AB BC
AC


DE
EF
DF
Conclusion:
ABC ~ DEF
F
by SSS~
Example: SSS Similarity
(Side-Side-Side)
E
B
5
A
10
8
11
C
D
Given:
AB
BC
AC
8
5
11


DE
EF 22
DF
10 16
16
22
F
Conclusion:
ABC ~ DEF
By SSS ~
SAS Similarity
(Side-Angle-Side or SAS~)
If the lengths of 2 sides of a triangle are proportional to the lengths
of 2 corresponding sides of another triangle and the included angles
are congruent, then the triangles are similar.
E
B
A
C DAB AC
Given: A  D and

DE DF
Conclusion:
ABC ~ DEF
F
by SAS~
Example: SAS Similarity
(Side-Angle-Side)
E
B
5
A
Given:
10
11
C
D
22
F
A  D Conclusion:
AB AC ABC ~ DEF

DE DF
By SAS ~
A
D
B
80
80
E
C
ABC ~ ADE by AA ~ Postulate
Slide from MVHS
C
6
D
3
A
10
E
5
B
CDE~ CAB by SAS ~ Theorem
Slide from MVHS
L
5
3
6
K
6
M
6
N
10
O
KLM~ KON by SSS ~ Theorem
Slide from MVHS
20
A
D
30
16
C
24
B
36
ACB~ DCA by SSS ~ Theorem
Slide from MVHS
L
15
P
25
N
9
A
LNP~ ANL by SAS ~ Theorem
Slide from MVHS
Proving Triangles Similar
Similarity is reflexive, symmetric, and transitive.
Steps for proving triangles similar:
1. Mark the Given.
2. Mark …
Reflexive (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.
Given : DE FG
Problem #1
Pr ove : DEC
FGC
Step 1: Mark the given … and what it implies
Step 2: Mark the vertical angles
Step 3: Choose a method: (AA,SSS,SAS)
Step 4: List the Parts in the order of the method with reasons
Step 5: Is there more?
Statements
Reasons
G
Given
1. DE FG
AA
D
2. D  F
C
E
F
Alternate Interior <s
3. E  G Alternate Interior <s
4. DEC FGC AA Similarity
Given : IJ  3LN
Problem #2 Pr ove :
IJK
JK  3NP
IK  3LP
LNP
Step 1: Mark the given … and what it implies
Step 2: Choose a method: (AA,SSS,SAS)
Step 4: List the Parts in the order of the method with reasons
Statements
Reasons
Step 5: Is there more?
1. IJ = 3LN ; JK = 3NP ; IK = 3LP
Given
SSS
J
K
N
P
2.
IJ
LN
3.
I
L
4.
=3,
IJ
LN
JK
NP
=
=3,
JK
NP
IJK~
IK
LP
=
=3
IK
LP
LNP
Division Property
Substitution
SSS Similarity
Given : G is the midpo int of ED
Problem #3
H is the midpo int of EF
Pr ove :
EGH
EDF
Step 1: Mark the given … and what it implies
Step 2: Mark the reflexive angles
SAS
Step 3: Choose a method: (AA,SSS,SAS)
Step 4: List the Parts in the order of the method with reasons
Next Slide………….
E
Step 5: Is there more?
G
D
H
F
Statements
Reasons
1.
G is the Midpoint of ED
Given
H is the Midpoint of EF
2. EG = DG and EH = HF
Def. of Midpoint
3. ED = EG + GD and EF = EH + HF Segment Addition Post.
4. ED = 2 EG and EF = 2 EH
Substitution
ED
EF
Division Property
5.
6.
EG
=2 and
ED
EH
=2
EF
Substitution
=
EG EH
7. GEHDEF
Reflexive Property
8. EGH~ EDF
SAS Postulate
Similarity is reflexive,
symmetric, and
transitive.
End Slide Show
Choose a Problem.
D
Problem #1
AA
C
E
P
N
G
F
Problem #2
SSS
Problem #3
SAS
J
L
K
E
I
G
D
H
F
The End
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.