Download Postulate 4-1

4-2 Triangle Congruence using SSS and SAS
Postulate 4-1
Side-Side-Side (SSS) Postulate
If the three sides of one triangle are congruent to the three sides of another triangle,
then the two triangles are congruent.
A
D
DFG
ABC
C
B
G
F
Postulate 4-2
Side-Angle-Side (SAS) Postulate
If two sides and the included angle of one triangle are congruent to two sides and the
included angle of another triangle, then the two triangles are congruent.
D
A
C
B
1. In
DFG
ABC
G
F
ABP , which sides include
B
?
A
P
2. In
YXP which angle is between PX and XY ?
Y
B
X
3. Which triangles can you prove congruent?
Tell whether you would use SSS or ASA Postulate.
D
4. What other information do you need to prove
DWO  DWG
?
O
5. Can you prove
SED  BUT
from the information given? Explain.
1
W
G
Triangle Congruence: ASA and AAS
Theorem
Angle-Side-Angle (ASA) Postulate
If two angles and the included side of one triangle are congruent to two angles and the included
side of another triangle, then the two triangles are congruent.
D
A
B
____
G
C
F
______ by
_____
Ex 1: Given
M
4
30
P
0
4
N
O
60
Q
0
R
Write a congruence statement for the triangles above.
Ex 2: Which side is included between  R and  F in  FTR?
Ex 3: Which angles in  STU include US ?
Theorem
Angle-Angle-Side (AAS) Theorem
If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding
nonincluded side of another triangle, then the triangles are congruent.
D
A
G
C
B
DFG
F
Ex 4:
Given:
D
A
5
420
B
AB
C
0
38
0
C
F
5 100
420
G
Write a congruence Statement for the triangles above.
Ex 5:
Can you prove ASA or AAS? If yes, state the triangle congruence and the postulate or theorem
you used.
Given: that MNOP is a parallelogram.
P
Y
L
A
2
Section 4-6: Right Triangle Congruence
Use the following theorems for RIGHT TRIANGLES Only! Since we already know that we
have a congruent angle, we only have to prove two other things are congruent.
Fill in the name for the parts of the right triangle below.
Congruence Patterns: LL, HA, LA and HL
Leg – Leg (LL): When both sets of legs are congruent, we have congruent triangles.
Hypotenuse – Angle (HA): When the hypotenuse of both triangles and another angle outside of
the right angle are congruent, we have congruent triangles.
Leg – Angle (LA): To prove LA, we need one leg and one other angle.
OR
Hypotenuse – Leg (HL): When the hypotenuse of both triangles and on the two legs is
congruent, we can prove congruence using HL.
3
Examples:
Using only the right triangle theorems, LL, HA, HL and LA, prove that the following triangles
are congruent.
1.
2.
3.
4. FG  HK , F  G
5. LJK is isosceles with
base LK , GJL  IJK
GL  GI , KI  GI
6. Q is the midpoint of PX.
F
H
K
G
J
I
G
P
K
L
4
Q
Z
X
Sample Proofs for 4-2 (SSS and SAS)
Given:
Prove:
AB DE , AB  DE , BC  FE
B
∆ABF  ∆DEC
D
C
F
A
E
B
Given:
Prove:
ABC is isosceles with base AC
D is the midpoint of AC
ABD  CBD
A
5
D
C
Sample Proofs Using ASA and AAS
1.
Given:
B
BD bisects ADC
BD bisects ABC
Prove: ABD  CBD
A
C
D
2.
Given: D is the midpoint of MT
B
Z
MB DZ
B  Z
Prove: MDB  DTZ
M
6
D
T
Sample right triangle proofs:
1. Given: BX  DY
D
Y
A
BA  AY
DC  XC
AY XC
Prove: ABY  CDX
B
C
X
D
2. Given:
NT  DA
DA bisects NT
ND  TA
Prove: SND  STA
N
S
A
7
T
Examples for CPCTC – Corresponding Parts of Congruent Triangles are Congruent
When you prove two triangles congruent, you use only three pairs of parts. After the two triangles have
been shown to be congruent, the remaining three pairs of parts are congruent by CPCTC. You will need
to use CPCTC if you are to prove something other than congruent triangles.
Example 1
Given: C is the midpoint of AE and BD
A
Prove: AB  ED and AB ED
B
C
D
E
8
Example 2
Given: ∆UJY is isosceles triangle with vertex  J
JR bisects  UJY
J
Prove: R is the midpoint of UY and JR  UY
U
9
R
Y
10
11