Download Using the ASA and AAS Congruence Methods Given

SAS
SSS
SSS
SAS
Sect. 4.4 Proving Triangles are
Congruent: ASA and AAS
Goal 1
Goal 2
Using the ASA and AAS
Congruence Methods
Using Congruence Postulate
and Theorems
Using the ASA and AAS Congruence Methods
Postulate 21 Angle-Side-Angle (ASA) Congruence Postulate
If, in two triangles, two angles and the included
side of one triangle are congruent to two angles
and the included side of the other, then the
triangles are congruent.
Using the ASA and AAS Congruence Methods
A
Example 1
Given: ABC   DCB;  DBC   ACB
Prove: ABC  DCB
Statements
1. ABC   DCB; DBC  ACB
2.
BC  CB
3. ABC  DCB
B
Reasons
1. Given
2. Reflexive
3. ASA
C
D
Using the ASA and AAS Congruence Methods
Theorem 4.5 Angle-Angle-Side (AAS) Congruence Theorem
If, in two triangles, two angles and a non-included
side of one triangle are congruent respectively to
two angles and the corresponding non-included side
of the other, then the triangles are congruent.
Using the ASA and AAS Congruence Methods
Given: B  C; D  F;
B
C
M is the midpoint of DF
Prove: BDM  CFM
D
Statements
1. B  C; D  F;
M
Reasons
1. Given
M is the midpoint of DF
2.
DM  FM
3. BDM  CFM
2. Definition of Midpoint
3. AAS
F
Using the ASA and AAS Congruence Methods
X
Example 3
Given:
WZ
bisects XZY and XWY
Z
W
Prove: WZX  WZY
Statements
1.
Reasons
WZ bisects XZY and XWY 1. Given
2. XZW  YZW;
2. Definition of Angle Bisector
XWZ  YWZ
3.
Y
ZW  ZW
4. WZX  WZY
3. Reflexive
4. ASA
Using Congruence Postulates and Theorems
Methods of Proving Triangles Congruent
SSS
If three sides of one triangle are congruent to three sides of
another triangle, the triangles are congruent.
SAS
If two sides and the included angle of one triangle are congruent
to the corresponding parts of another triangle, the triangles are
congruent.
If two angles and the included side of one triangle are congruent
to the corresponding parts of another triangle, the triangles are
congruent.
If two angles and the non-included side of one triangle are
congruent to the corresponding parts of another triangle, the
triangles are congruent.
ASA
AAS
Using Congruence Postulates and Theorems
AAA works fine to show that triangles are the
same SHAPE (similar), but does NOT work to
show congruent!
You can draw 2 equilateral triangles that are the
same shape but not the same size.
H
D = 60
H = 60
D
G = 60
E = 60
I = 60
F = 60
E
F
G
I
Using Congruence Postulates and Theorems
What about two sides and a not-included angle?
Note that GB and BH are the same length, and that AB and
angle A are the other  parts of Angle – Side – Side.
ABG
m AB = 10.78 cm
m GB = 5.81 cm
mBAG = 28.26
m AG = 6.72 cm
mAGB = 118.55
mABG = 33.19
ABH
m AB = 10.78 cm
m BH = 5.81 cm
mBAH = 28.26
m AH = 12.28 cm
mAHB = 61.45
mABH = 90.29
A
B
G
H
Using Congruence Postulates and Theorems
ASS
It does
NOT
work!!!
Using Congruence Postulates and Theorems
Given: CB  AD ; CB bisects ACD
Prove: ABC  DBC
A
Given: A  E ; B  G ;
AC  EF
Prove: ABC  EGF
E
B
C
G
F
Given: BAC  DAE ; B  D ;
A
AC  AE
Prove: ABC  ADE
B
E
C
D
A
Given: 2  3 ; B  D ;
AC  AE
2
3
Prove: ABE  ADC
B
E
C
D
Given: ○A ; 2  3 ;
B  D
Prove: ABE  ADC
A
2
B
E
3
C
D
Homework
pp. 223 Exs. 8 – 23, 34-38