Download Side-Angle-Side Postulate

Exploring Congruent Triangles
Congruent triangles:
Triangles that are the same size and shape
– Each triangle has six parts, three angles and three sides
– If the corresponding six parts of one triangle are congruent
to the six parts of another triangle, then the triangles are
congruent
• This is abbreviated by CPCTC (corresponding parts
triangles are congruent)
of congruent
– Orientation of the triangles is not important. This means
that the triangles can be flipped, slid and turned around,
and if the corresponding parts are congruent, the triangles
are congruent
D
B
E
C
A
If
ABC @
F
DE F, then the following parts are congruent:
Segments
· AB @ DE
Segments: · BC @ EF
1.
· AC @ DF
2.
3.
Angles:
1.
2.
3.
Angles
· РA @ РD
· РB @ РE
· РC @ РF
Note: The order matters. If ABC @ DEF, it is not the same as
saying ABC @ FED.
EXAMPLE 1:
If ABC @ RQC, name the corresponding congruent sides
and angles.
B
Congruent Sides
Q
C
–
–
–
Congruent Angles
–
–
–
A
R
Example 2
Write the correct congruency
statement.
Compare the sides and the angles.
3.
4.
5
• Do worksheet Parts of congruent triangles and
exploring congruent triangles.
Congruence of triangles is:
Reflexive:
• ABC @ ABC
A
B
B'
C
C'
A'
Congruence of triangles is:
Symmetric
• If ABC @ DEF, then DEF @ ABC
A
C
E
B
D
F
Congruence of triangles is:
Transitive
• If ABC @ DEF and DEF @ LMN, then ABC @ LMN.
A
C
E
B
D
H
F
G
I
Proving Triangle Congruency
There are 4 ways to prove that two triangles are
congruent to each other. Remember, once
you know that two triangles are congruent,
then Corresponding Parts of Congruent
Triangles are Congruent.
Side-Side-Side Postulate (SSS):
If all 3 sides of one triangle are congruent to all 3 sides of
another triangle, then the two triangles are congruent.
Side-Angle-Side Postulate (SAS):
If two sides & the included angle (the angle between the two
sides) of one triangle are congruent to two sides & the
included angle of another triangle, then the two triangles are
congruent.
Ex:
• Do worksheet ways to prove triangle
congruence SAS SSS
A
C
E
B
D
F
Angle-Side-Angle Postulate (ASA):
If 2 angles & the side between them in one
triangle are congruent to 2 angles & the side
between them in another triangle, then the 2
triangles are congruent.
Angle-Angle-Side Postulate (AAS):
If 2 angles & a side not between them in one
triangle are congruent to 2 angles & the
corresponding side not between them in
another triangle, then the 2 triangles are
congruent.
Determine which Postulate or theorem can be used to prove the
2 triangles are congruent. If it’s not possible, write Not Possible.
Remember, you can choose from SSS, SAS, ASA, or AAS.
Hypotenuse-Leg (HL)
If the hypotenuse and a leg of one right triangle
are congruent to the hypotenuse and
corresponding leg of another right triangle,
then the triangle are congruent.
Proofs:
F
I
G
H
E
Given : FE @ HI & G is the midpoint of FHand EI
Prove: FEG @ HIG if EI @ FH
Statements
1) FE @ HI Is the
midpoint of FHand EI
2)
3)
FG @ HGand EG @ IG
FEG @ HIG
Reasons
R
S
Q
Given:
T
RQ TS
RQ @ TS
Prove: QRT @ STR
Statement
1)
RQ TS
RQ @ TS
2) РQRT @ РSTR
3) RT @ TR
4)
QRT @ STR
Reasons
R
E
L
D
W
Given: L is the Midpoint of WE,WR ED
Prove: ΔWRL @ ΔEDL
Statement
1. L is the Midpoint of
WE,WR ED
2. РW @ РE
3.
4.
WL @ EL
РWLR @ РELD
5. ΔWRL @ ΔEDL
Reason
J
K
F
L
M
N
Given: РNKL @ РNJM , KL @
Prove: LN @ MN
Statement
1.
2.
3.
РNKL @ РNJM , KL @ JM
РN @ РN
JMN @ LKN
4. LN @ MN
JM
Reason
L
M
J
K
Given: JK  KM , JM @ KL, ML JK
Prove: ML @ JK
Statement
1. JK  KM , JM @ KL, ML JK
2.
РJKM
3.
KM  ML
4.
РLMK
5.
Is a right angle
Is a right angle
MK MK
6.
JMK @ LMK
7.
ML @ JK
Reason