Download 4.3 [Compatibility Mode]

Triangle Congruence using
ASA & AAS
Section 4.3
SAS
SSS
SSS
SAS
Goals / “I can…” for Sect. 4.3
Goal 1
Goal 2
Using the ASA and AAS
Congruence Methods
Using Congruence Postulate
and Theorems
ASA Postulate
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
B
Reasons
1. Given
2.
BC ≅ CB
2. Reflexive
3.
∠ DBC ≅ ∠ACB
3. Given
4. ∆ABC ≅ ∆DCB
4. ASA
C
D
EXAMPLE 1
Use ASA
Decide whether the congruence
statement is true.
Explain your reasoning.
∆ABC ≅ ∆JKL
SOLUTION
(A)
(S)
(A)
∠B ≅ ∠K
BA ≅ KJ
Given
Given
∠A ≅ ∠J
Given
∆ABC ≅ ∆JKL
ASA
AAS Postulate
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
Prove: ∆WZX ≅ ∆WZY
Statements
1.
Z
Reasons
Y
WZ bisects ∠XZY and ∠XWY 1. Given
2. ∠XZW ≅ ∠YZW;
2. Definition of Angle Bisector
∠XWZ ≅ ∠YWZ
3.
W
ZW ≅ ZW
4. ∆WZX ≅ ∆WZY
3. Reflexive
4. ASA
EXAMPLE 3
Use AAS
Decide whether the congruence
statement is true.
Explain your reasoning.
∆RST ≅ ∆TUR
SOLUTION
∠U ≅ ∠S
(A) ∠URT ≅ ∠STR
Given
(S) RT ≅ RT
Reflexive Prop.
(A)
∆RST ≅ ∆TUR
Given
AAS
EXAMPLE 4
Use AAS
Decide whether the congruence
statement is true.
Explain your reasoning.
∆HDE ≅ ∆FGE
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
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
Writing a Proof
P
Given: ∠ S = ∠ Q, RP bisects ∠ SRQ.
Prove: ΔSRP = ΔQRP
Q
S
R
∠ S = ∠ Q, RP bisects ∠ SRQ
∠ SRP = ∠ QRP
RP = RP
∆SRP = ∆QRP
Given
Definition of an angle bisector
Reflexive Property
AAS (Angle-Angle-Side)
Writing a Proof
Given: XQ ll TR, XR bisects QT
Prove: ΔXMQ = ΔRMT
Q
X
M
R
T
XQ ll TR, XR bisects QT
∠Q=∠T
QM = TM
∠ XMQ = ∠ RMT
∆XMQ = ∆RMT
Given
Alternate Interior Angles
Definition of a bisector
Vertical Angles
ASA (Angle-Side-Angle)
Given: ∠BAC ≅ ∠DAE ; ∠B ≅ ∠D ;
A
AC ≅ AE
Prove: ∆ABC ≅ ∆ADE
B
E
C
D
A
Given: ∠2 ≅ ∠3 ; ∠B ≅ ∠D ;
AC ≅ AE
2
Prove: ∆ABE ≅ ∆ADC
B
E
3
C
D