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Sec. 4-2Δ by SSS and SAS
Objective:1) To prove 2 Δs using the SSS and the SAS Postulate
GEOMETRY - CONGRUENT TRIANGLES
GEOMETRY - CONGRUENT TRIANGLES
𝐴𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒h𝑎𝑠6𝑝𝑎𝑟𝑡𝑠
+3 𝑎𝑛𝑔𝑙𝑒𝑠3 𝑠𝑖𝑑𝑒𝑠
R
P
QA
B
C
A PB Q
C R
AB PQBC QRCA RP
𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔𝑎𝑛𝑔𝑙𝑒𝑠𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔𝑠𝑖𝑑𝑒𝑠If ABC PQR then find the corresponding parts𝑒𝑥𝑎𝑚𝑝𝑙𝑒
CPCTC Theorem CPCTC Theorem CCP TC
orrespondingartsongruentrianglesongruent
in
are
ΔABC ΔPQR
AB PQBC QRCA RP
B
CAQ
R
P
A PB QC R
GEOMETRY - CONGRUENT TRIANGLES
• In Sec. 4-1 we learn that if all the sides and all the s are of 2Δs then the Δs are .
• But we don’t need to know all 6 corresponding parts are .
• There are short cuts.
POSTULATE 4-1 (SSS)
POSTULATE Side - Side - Side (SSS) Congruence Postulate
Side MN QR
Side PM SQ
Side NP RS
If
If three sides of one triangle are congruent to three sidesof a second triangle, then the two triangles are congruent.
then MNP QRSS
S
S
GEOMETRY SSS CONGRUENCE POSTULATE
Included – A word used frequently when referring to the s and the sides of a Δ.
• Means – “in the middle of”• What is included between the sides BX and
MX?• X• What side is included between B and M?• BM
B M
X
GEOMETRY
POSTULATE 4-2 (SAS)
POSTULATE Side-Angle-Side (SAS) Congruence Postulate
Side PQ WX
Side QS XY
then PQS WXYAngle Q X
If
If two sides and the included angle of one triangle arecongruent to two sides and the included angle of a second triangle, then the two triangles are congruent.
A
S
S
SAS CONGRUENCE POSTULATE
SAS
S A
S S
S
A
GEOMETRY - CONGRUENT TRIANGLES
AB
C
D
𝑒𝑥𝑎𝑚𝑝𝑙𝑒1
YES, ABC CDA
A
C
GEOMETRY - CONGRUENT TRIANGLES
𝑒𝑥𝑎𝑚𝑝𝑙𝑒2
S S
S
S
S
S
SSSYES,
PQR RSP
P
Q
R
S
GEOMETRY - CONGRUENT TRIANGLES
𝑒𝑥𝑎𝑚𝑝𝑙𝑒3
S
AS
SS
A
SASYES,
PQR SQT
P
Q
R
S
T
GEOMETRY - CONGRUENT TRIANGLES
𝑒𝑥𝑎𝑚𝑝𝑙𝑒3
S
AS SS
A
NO, SASYES,
GEOMETRY - CONGRUENT TRIANGLES
𝑒𝑥𝑎𝑚𝑝𝑙𝑒 4
S
A
S
S
S
A
NO,SASYES,
GEOMETRY ASA CONGRUENCE POSTULATE
S
POSTULATE 4-3 (ASA)
POSTULATE Angle - Side - Angle (ASA) Congruence Postulate
Side PN SR
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.
then MNP QRS
S
Angle N R
Angle P S
If A
S
A
GEOMETRY AAS CONGRUENCE POSTULATE
S
S
POSTULATE 4-4 (AAS)
POSTULATE Angle - Angle - Side (AAS) Congruence Postulate
Side PM SQ
If two angles and the NON included side of one triangle are congruent to two angles and the NON included side of a second triangle, then the two triangles are congruent.
then MNP QRSAngle N R
Angle P S
If A
AS
GEOMETRY - CONGRUENT TRIANGLES
𝑒𝑥𝑎𝑚𝑝𝑙𝑒1
A
S
A
A
A
S
ASAYES,
PQR PST
P
Q
R
S
T
A A
S
A
S
A
GEOMETRY - CONGRUENT TRIANGLES
𝑒𝑥𝑎𝑚𝑝𝑙𝑒2
AASYES,
ABC DCB
C
DB
A
GEOMETRY - CONGRUENT TRIANGLES
𝑒𝑥𝑎𝑚𝑝𝑙𝑒3
A
A
S
A
S
AAASYES,
ABC CDA
A
B
C
D
GEOMETRY - CONGRUENT TRIANGLES
𝑒𝑥𝑎𝑚𝑝𝑙𝑒 4
NO,
AASYES,AA
SA
SA
S S SASYES,
A A
ASAYES,
𝑙𝑒𝑔
𝑙𝑒𝑔
h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
THE THEOREM
THE HYPOTENUSE LEGTHEOREM
GEOMETRY - CONGRUENT RIGHT TRIANGLES
THE THEOREM
h h
𝑙𝑙
GEOMETRY - CONGRUENT RIGHT TRIANGLES
GEOMETRY - CONGRUENT RIGHT TRIANGLES
h
h
𝑙𝑙
𝑒𝑥𝑎𝑚𝑝𝑙𝑒1
𝐻𝐿YES,
h
h
𝑙
𝑙
𝐻𝐿YES,
GEOMETRY - CONGRUENT RIGHT TRIANGLES
𝑒𝑥𝑎𝑚𝑝𝑙𝑒2
GEOMETRY - CONGRUENT RIGHT TRIANGLES
𝑒𝑥𝑎𝑚𝑝𝑙𝑒3
SSS SAS
ASA AAS
HL
CONGRUENCE THEOREMS