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Geometry Lesson 4.4
Proving Triangles are
Congruent: ASA and AAS
Warm Up: Review of SSS / SAS

Write the method you would use to show the two
triangles are congruent based on the information
given in the diagram and any other information
you can determine
(a)
(c)
SSS
SSS
(b)
(d)
SAS
SAS
1. Angle-Side-Angle (ASA)

If two angles and the included side of one
triangle are congruent to two angles and
the included side of a second triangle, then
the two triangles are congruent
AC is the
included side
of A & C
B
A
C
D
E
Angle: A  D
Side: AC  DF
Angle: C  F
 ABC  DEF
Order is
F Important!!
2. Angle-Angle-Side (AAS)

If two angles and a non-included side of
one triangle are congruent to two angles
and a non-included side of a second
triangle, then the triangles are congruent
BC is a nonincluded side
of A & C
B
C
A
D
E
Angle: A  D
Angle: C  F
Side: BC  EF
 ABC  DEF
F
Order is
Important!!
Example 1a/b: Can You Prove It?

Is it possible to prove the s are ? How?
E
H
N
Q
G
F
J
Know: EF  JH & E  J
Can show: EGF  HGJ
(Vertical s)
 EGF  JGH by AAS
M
P
Know: NQ  PM
Can show: NP  NP
(Reflexive)
Not enough info.
What do you know about the figures?
What can you show to help show congruence?
Example 1c: Can you Prove It?

Is it possible to prove the s are ? How?
U
4
W
3
1
X
Z
2
Know: UW || XZ
Can show:
1  3
2  4 (alt. int. s)
WZ  WZ (reflexive)
 UZW  XWZ by ASA
What do you know about the figures?
What can you show to help show congruence?
Practice 1

Is it possible to prove the s are ? How?
C
H
E
I
D
G
F
Know: C  F & G  E
Can show: CDG  FDE
(vertical s)
 AAA not enough
K
J
Know: KHJ  IJH
KJH  IHJ
Can show: HJ  HJ
 KHJ  IJH by ASA
What do you know about the figures?
What can you show to help show congruence?
Example 2: What is Needed?

What information do you need to show the
triangles are  using the specified method?
(a)
(b)
ASA
AAS
C  F
C  F
Practice 2: What is Needed?

What information do you need to show the
triangles are  using the specified method?
N
(a) E
(b)
AAS
F
SAS
D
M
DF  MO
O
E
N
F
D
M
D  M
O
Example 3a: Creating a Proof


C
Given: AD || EC; BD  BC
A
Prove: ABD  EBC
Plan for proof: We have two
 sides and two || lines. Our
options are SSS, SAS, ASA,
or AAS. What do we need?
Statement Reason
B
D
AD || EC & BD  BC Given
ABD  EBC Vertical Angles are 
D  C Alt. Int. Angles are 
ABD  EBC ASA
E
Example 3b: Creating a Proof


Given: B  C, D  F, M is the midpoint of DF
Prove: BDM  CFM B
C
Plan for proof:
We have two  angles
and a midpoint. Our
options are ASA or
AAS. What’s needed? D
Statement
Reason
B  C, D  F
M is midpoint of DF
DM  FM
BDM  CFM
M
Given
Given
Definition of a midpoint
AAS
F
Practice 3a: Creating a Proof


Given: LA || SN, LR  NR
Prove: LAR  NSR
Plan for proof:
SSS?
SAS?
ASA?
AAS?
Statement
LA || SN, LR  NR
LRA  NRS
ALR  SNR
LAR  NSR
Reason
Given
Definition of vertical angles
alt. int. s are 
ASA
Practice 3b: Creating a Proof


Given: AC || BD, AB || CD
Prove: ABC  DCB
Plan for proof:
Statement
Reason
AC || BD, AB || CD
ACB  DBC
Given
Alt. int. s 
ABC  DCB
BC  BC
ABC  DCB
Alt. int. s 
SSS? SAS?
ASA? AAS?
Reflexive Property
ASA
Assignment
4.4 (pg. 223-224)
#8-22 EVEN