Download Proving Triangles Congruent

Proving Triangles
Congruent
Side-Side-Side Postulate
Side-Angle-Side Postulate
Angle-Side-Angle Postulate
Angle-Angle-Side Theorem
• How much do you need to know to prove that triangles
are congruent?
• We know that if we can show all three sides and all
three angles are congruent, we can show that the
triangles are congruent.
Postulate
Side-Side-Side Postulate
If three sides of one triangle are congruent
to three sides of a second triangle, then the
triangles are congruent.
M
P
If Side MN ≅ QR
Side NP ≅ RS
Q
N
Side PM ≅ SQ
Then ∆MPN ≅ ∆QRS by SSS
R
S
Postulate
Side-Angle-Side Postulate
If two sides and the included angle of one triangle
are congruent to two sides and the included angle of
a second triangle, then the triangles are congruent.
X
If Side PQ ≅ WX
Q
Angle ∠ Q ≅ ∠X
Side QS ≅ XY
W
P
then ∆PQS ≅ ∆WXY by SAS
S
Y
Postulate
Angle-Side Angle (ASA) Congruence Postulate
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.
If Angle ∠ A ≅ ∠D
B
Side AB ≅ DE
Angle ∠B ≅ ∠E
then ∆ABC ≅ ∆DEF by ASA
A
C
E
D
F
EXAMPLE 1
Is it possible to prove that the triangles are congruent?
If so, state the postulate or theorem.
a.
C
E
H
b.
I
D
K
G
J
F
NO, there is no AAA
theorem or postulate
of congruence.
YES, HJ ≅ HJ by the
reflexive property, so
the triangles are
congruent by ASA
Angle-Angle-Side (AAS) Congruence Theorem
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 two triangles are congruent.
If Angle ∠A ≅ ∠D,
Angle ∠C ≅ ∠F, and
B
Side AB≅ DE,
Then ∆ABC ≅ ∆DEF by AAS
A
C
E
D
F
Example 2
Determine if there is enough information to
determine if the pairs of triangles are congruent. If
there is enough information, indicate the postulate
or theorem that can be used.
Yes, SAS
a)
b) Determine if there is enough information to
determine if the pairs of triangles are congruent. If
there is enough information, indicate the postulate
or theorem that can be used.
Yes, ASA
c) Determine if there is enough information to
determine if the pairs of triangles are congruent. If
there is enough information, indicate the postulate
or theorem that can be used.
No
Name the included side between:
1.
XZ
∠X and ∠Z _____
T
Z
S
R
X
Y
F
A
C
B
D
E
J
L
M
K
N
O
NO
2. ∠N and ∠O _______
T
Z
S
R
X
Y
F
A
C
B
D
E
J
L
M
K
N
O
3.
∠A and ∠B ________
AB
T
Z
S
R
X
Y
F
A
C
B
D
E
J
L
M
K
N
O
Name the include angle between:
4.
TR and ST ________
∠T
T
Z
S
R
X
Y
F
A
C
B
D
E
J
L
M
K
N
O
∠K
5. KL and KJ
________
T
Z
S
R
X
Y
F
A
C
B
D
E
J
L
M
K
N
O
E
________
6. FE and DE ∠
T
Z
S
R
X
Y
F
A
C
B
D
E
J
L
M
K
N
O
Tests for Congruent Triangles
SSS
__________
postulate
ASA
__________
postulate
SAS
__________
postulate
AAS
__________
theorem
Name the postulate or theorem that you can use to
prove the triangles congruent and write a
congruence statement.
7
U
8
V
S
Y
W
X
X
T
T
W
ASA
∆TUV ≅ ∆WXY
U
V
SSS
∆STU ≅ ∆VXW
9
C
H
G
10
AAS
J
H
D
E
K
P
F
SAS
∆CDE ≅ ∆FGH
∆JHK ≅ ∆MLP
M
L
Mark the third congruence that must be
given to prove ∆ABC ≅ ∆DEF using the
indicated postulate or theorem.
11. ASA Postulate
F
D
B
E
A
C
A
12. AAS Theorem
B
C
D
E
F
D
B
13. SSS Postulate
C
F
A
E
E
14. SAS Postulate
D
A
F
B
C
Determine if enough information is given to prove the
triangles congruent. If there is, state the postulate or the
theorem and write a congruence statement.
15.
A
E
ASA
∆ACB ≅ ∆DCE
C
B
16.
D
Y
Z
SSS
∆XYZ ≅ ∆ZWX
X
W
G
17.
NO
K
19.
H
18.
J
U
K
H
N
AAS
∆JKU ≅ ∆LKU
L
L
SAS
∆JKH ≅ ∆KJLJ
20.
AAS
T
N
∆TSN ≅ ∆USH
K
L
H
S
U
21.
22.
C
M
H
SAS
∆ABC ≅ ∆DCA
Q
U
SAS
∆MQU ≅ ∆CUQ
HOMEWORK: Chapter 13 Review