Download 4.3 Proving Triangles Congruent: SSS and SAS

4.3 PROVING TRIANGLES
CONGRUENT:
SSS
SAS
ASA
AAS
ARE THE FOLLOWING STATEMENTS TRUE OR
FALSE?:
A) FIGURES MUST HAVE EXACTLY THE SAME
SIZE TO SHOW CONGRUENCE?
TRUE!
B) FIGURES MUST HAVE EXACTLY THE SAME
SHAPE TO SHOW CONGRUENCE?
TRUE!
FOR TWO TRIANGLES TO BE CONGRUENT,
3
How many pairs of sides need to be congruent? ____
3
How many pairs of angles need to be ? _____
After today, we will no longer need all
of this information to prove congruency!!
SIDE-SIDE-SIDE (SSS ) POSTULATE
If three sides of one triangle are congruent
to three sides of second triangle, then the
two triangles are congruent.
If Side
M
Q
Side
Side
P
Then,
MNP  QRS
S
N
R
SIDE-ANGLE-SIDE (
) POSTULATE
SAS
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
two triangles are congruent.
If Side
Q
X
Angle
S
Y
Side
Then,
P
PQS  WXY
W
INCLUDED ANGLE
What does it mean for an angle to be “included”?
EX. 1 USING THE DIAGRAM, NAME THE INCLUDED ANGLE
BETWEEN THE PAIR OF SIDES GIVEN
a)
b)
c)
ANGLE-SIDE-ANGLE ( ASA) POSTULATE
If two angles and the included side of one
triangle are congruent to the two angels and
included side of a second triangle, then the
two triangles are congruent.
If Angle
M
Q
Side
Angle
P
Then,
MNP  QRS
S
N
R
ANGLE-ANGLE-SIDE ( AAS ) POSTULATE
If two angles and a non-included side of one
triangle are congruent to two angles and the
corresponding non-included side of a second
triangle, then the two triangles are congruent.
If Angle
Q
X
Angle
S
Y
Side
Then,
P
PQS  WXY
W
CHOOSING WHICH CONGRUENCE POSTULATE
 Decide whether enough information is given to prove
conclusion.
 If there is enough information, state the congruence postulate
you would use.
 If not look at the picture to gather more information.
(reflexive property, vertical angles, rt angle congruence, etc.)
EX. 2 DECIDE IF ENOUGH INFORMATION IS GIVEN TO PROVE THE
TRIANGLES CONGRUENT. IF THERE IS ENOUGH INFORMATION,
NAME THE POSTULATE YOU WOULD USE.
a)
b)
∆𝑅𝑆𝑇 𝑎𝑛𝑑 ∆TQR
Yes
ASA
∆𝐽𝐾𝐿 𝑎𝑛𝑑 ∆𝑁𝑀𝐿
YES
AAS
EX. 3 DECIDE IF ENOUGH INFORMATION IS GIVEN TO PROVE THE
TRIANGLES CONGRUENT. IF THERE IS ENOUGH INFORMATION,
NAME THE POSTULATE YOU WOULD USE.
a)
b)
∆𝑈𝑉𝑇 𝑎𝑛𝑑 ∆𝑊𝑉𝑇
NO
Wrong Side!!
∆𝐿𝑀𝑁 𝑎𝑛𝑑 ∆𝑇𝑁𝑀
YES
SAS
EX. 4 DECIDE IF ENOUGH INFORMATION IS GIVEN TO PROVE THE
TRIANGLES CONGRUENT. IF THERE IS ENOUGH INFORMATION,
NAME THE POSTULATE YOU WOULD USE.
a)
b)
∆𝑍𝑌𝑊 𝑎𝑛𝑑 ∆XYW
YES
SSS
∆𝐴𝐵𝐶 𝑎𝑛𝑑 ∆DEC
YES
SAS
EX. 6 DECIDE IF ENOUGH INFORMATION IS GIVEN TO PROVE THE
TRIANGLES CONGRUENT. IF THERE IS ENOUGH INFORMATION,
NAME THE POSTULATE YOU WOULD USE.
a)
b)
∆𝑅𝑆𝑇 𝑎𝑛𝑑 ∆WVU
NO
WRONG ANGLE!
∆𝐽𝐻𝐺 𝑎𝑛𝑑 ∆LKH
YES
SSS
EX. 7 STATE THE THIRD CONGRUENCE THAT MUST BE GIVEN TO
PROVE THAT ∆𝑨𝑩𝑪 ≅ ∆𝑫𝑬𝑭 USING THE GIVEN POSTULATE.
a)
b)
ASA
𝐴𝐵 ≅ 𝐷𝐸
AAS
< 𝐶𝐴𝐵 ≅< 𝐹𝐷𝐸
EX. 7 STATE THE THIRD CONGRUENCE THAT MUST BE GIVEN TO
PROVE THAT ∆𝑨𝑩𝑪 ≅ ∆𝑫𝑬𝑭 USING THE GIVEN POSTULATE.
a)
b)
ASA
SSS
𝐷𝐸 ≅ 𝐽𝐻
𝐵𝐴 ≅ 𝐷𝐶