Triangle Congruence:
by SSS and SAS
Geometry (Holt 4-5)
K. Santos
Side-Side-Side (SSS)
Congruence Postulate (4-5-1)
If the three sides of one triangle are congruent to the three
sides of another triangle , then the two triangles are
congruent.
A
D
Given: 𝐴𝐶 ≅ 𝐷𝐸
𝐵𝐶 ≅ 𝐹𝐸
E
𝐴𝐵 ≅ 𝐷𝐹
B
C
F
Then: ∆ 𝐴𝐵𝐶 ≅ ∆𝐷𝐹𝐸
Included Angle
Included angle—is an angle formed by two adjacent sides.
A
B
C
< B is the included angle between sides 𝐴𝐵and 𝐵𝐶
Side-Angle-Side (SAS)
Congruence Postulate (4-5-2)
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.
Given: 𝐴𝐵 ≅ 𝐾𝐿
𝐴𝐶 ≅ 𝐾𝐽
< A ≅< K
A
B
J
C
K
L
Then: ∆ 𝐴𝐵𝐶 ≅ ∆𝐾𝐿𝐽
Please note both angles must be included between the sides!!!
Example—Writing a
congruence statement
Write a congruence statement for the congruent triangles and
name the postulate you used to know the triangles were
congruent.
1. D
F
R
E
T
∆𝐷𝐸𝐹 ≅ ∆ 𝑇𝑆𝑅
S
2.
A
B
C
∆𝐴𝐵𝐶 ≅ ∆ADC
SAS Postulate
SSS Postulate
D
Example—what other
information is needed
What other information do you need to prove the two triangles
congruent by SSS or SAS?
1. M
T
2. G
H
U
N
O
V
Q
R
I
S
Need <M ≅ <U for SAS
or need 𝑁𝑂 ≅ 𝑉𝑇 for SSS
need <G ≅ <Q for SAS
or need 𝐼𝐻 ≅ 𝑆𝑅 for SSS
Example—explain triangle
congruence
Use the SSS or SAS postulate to explain why the triangles are
congruent.
A
B
D
C
It is given: 𝐴𝐵 ≅ 𝐶𝐷 and 𝐴𝐷 ≅ 𝐶𝐵
You know: 𝐴𝐶 ≅ 𝐶𝐴 by reflexive property of congruence
So: ∆ADC ≅ ∆CBA by SSS Postulate
Example—verifying triangle
congruence
Show that the triangles are congruent for the given value of the
variable. ∆UVW ≅ ∆YXZ, a = 3
U
X
4
V
ZY = a – 1
ZY = 3 – 1
ZY = 2
𝑈𝑊 ≅ 𝑌𝑍
2
W
3
So, ∆UVW ≅ ∆YXZ
a
3a - 5
Z a–1
XZ = a
XZ =3
𝑉𝑊 ≅ 𝑋𝑍
Y
XY = 3a - 5
XY = 3(3) - 5
XY = 4
𝑉𝑈 ≅ 𝑋𝑌
Proof
Q
Given: 𝑄𝑃 bisects <RQS
𝑄𝑅 ≅ 𝑄𝑆
Prove: ∆RQP ≅ ∆SQP
P
R
Statements
1.
2.
3.
4.
𝑄𝑃 bisects <RQS
< RQP ≅ < SQP
𝑄𝑅 ≅ 𝑄𝑆
𝑄𝑃 ≅ 𝑄𝑃
5. ∆RQP ≅ ∆SQP
1.
2.
3.
4.
S
Reasons
Given
Definition of angle bisector
Given
Reflexive property of
congruence
5. SAS Postulate (3, 2, 4)
Proof
𝐸𝐺 || 𝐻𝐹
𝐸𝐺 ≅ 𝐻𝐹
Prove: ∆UVW ≅ ∆YXZ
Given:
E
G
F
Statements
1. 𝐸𝐺 || 𝐻𝐹
2. < EGF ≅ <HFG
1.
2.
3. 𝐸𝐺 ≅ 𝐻𝐹
4. 𝐹𝐺 ≅ 𝐺𝐹
3.
4.
5. ∆UVW ≅ ∆YXZ
5.
H
Reasons
Given
Alternate Interior Angles
Theorem
Given
Reflexive Property of
congruence
SSS Postulate (2, 3, 4)