Triangle Congruence by SSS and SAS

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Transcript Triangle Congruence by SSS and SAS

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: 𝐴𝐢 β‰… 𝐷𝐸
𝐡𝐢 β‰… 𝐹𝐸
E
𝐴𝐡 β‰… 𝐷𝐹
B
C
F
Then: βˆ† 𝐴𝐡𝐢 β‰… βˆ†π·πΉπΈ
Included Angle
Included angleβ€”is an angle formed by two adjacent sides.
A
B
C
< B is the included angle between sides 𝐴𝐡and 𝐡𝐢
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: 𝐴𝐡 β‰… 𝐾𝐿
𝐴𝐢 β‰… 𝐾𝐽
< A β‰…< K
A
B
J
C
K
L
Then: βˆ† 𝐴𝐡𝐢 β‰… βˆ†πΎπΏπ½
Please note both angles must be included between the sides!!!
Exampleβ€”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
βˆ†π·πΈπΉ β‰… βˆ† 𝑇𝑆𝑅
S
2.
A
B
C
βˆ†π΄π΅πΆ β‰… βˆ†ADC
SAS Postulate
SSS Postulate
D
Exampleβ€”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 β‰… <U for SAS
or need 𝑁𝑂 β‰… 𝑉𝑇 for SSS
need <G β‰… <Q for SAS
or need 𝐼𝐻 β‰… 𝑆𝑅 for SSS
Exampleβ€”explain triangle
congruence
Use the SSS or SAS postulate to explain why the triangles are
congruent.
A
B
D
C
It is given: 𝐴𝐡 β‰… 𝐢𝐷 and 𝐴𝐷 β‰… 𝐢𝐡
You know: 𝐴𝐢 β‰… 𝐢𝐴 by reflexive property of congruence
So: βˆ†ADC β‰… βˆ†CBA by SSS Postulate
Exampleβ€”verifying triangle
congruence
Show that the triangles are congruent for the given value of the
variable. βˆ†UVW β‰… βˆ†YXZ, a = 3
U
X
4
V
ZY = a – 1
ZY = 3 – 1
ZY = 2
π‘ˆπ‘Š β‰… π‘Œπ‘
2
W
3
So, βˆ†UVW β‰… βˆ†YXZ
a
3a - 5
Z a–1
XZ = a
XZ =3
π‘‰π‘Š β‰… 𝑋𝑍
Y
XY = 3a - 5
XY = 3(3) - 5
XY = 4
π‘‰π‘ˆ β‰… π‘‹π‘Œ
Proof
Q
Given: 𝑄𝑃 bisects <RQS
𝑄𝑅 β‰… 𝑄𝑆
Prove: βˆ†RQP β‰… βˆ†SQP
P
R
Statements
1.
2.
3.
4.
𝑄𝑃 bisects <RQS
< RQP β‰… < SQP
𝑄𝑅 β‰… 𝑄𝑆
𝑄𝑃 β‰… 𝑄𝑃
5. βˆ†RQP β‰… βˆ†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 β‰… 𝐻𝐹
Prove: βˆ†UVW β‰… βˆ†YXZ
Given:
E
G
F
Statements
1. 𝐸𝐺 || 𝐻𝐹
2. < EGF β‰… <HFG
1.
2.
3. 𝐸𝐺 β‰… 𝐻𝐹
4. 𝐹𝐺 β‰… 𝐺𝐹
3.
4.
5. βˆ†UVW β‰… βˆ†YXZ
5.
H
Reasons
Given
Alternate Interior Angles
Theorem
Given
Reflexive Property of
congruence
SSS Postulate (2, 3, 4)