Transcript sides and angles

Proving Triangles
Congruent
The Idea of a Congruence
Two geometric figures with
exactly the same size and
shape.
F
B
A
C
E
D
Corresponding Parts
You will learn that if all six pairs of
corresponding parts (sides and angles) are
congruent, then the triangles are congruent.
1. AB  DE
2. BC  EF
3. AC  DF
4.  A   D
5.  B   E
6.  C   F
ABC   DEF
How many of the 6 relationships
do you need. . .
. . .to prove that 2
triangles are congruent?
Do you need ALL six ?
NO!
SSS
SAS
ASA
AAS
BUT ORDER MATTERS!!!!
Side-Side-Side (SSS)
S AB  DE
S BC  EF
S AC  DF
ABC   DEF
Side-Angle-Side (SAS)
S AB  DE
A A   D
S AC  DF
ABC   DEF
included
angle
Included Angle
The angle between two sides
G
I
H
Included Angle
E
Y
S
Name the included angle:
YE and ES
E
ES and YS
S
YS and YE
Y
Angle-Side-Angle (ASA)
A A   D
S AB  DE
ABC   DEF
A B  E
include
d
side
Included Side
The side between two angles
GI
HI
GH
Included Side
E
Y
S
Name the included side:
Y and E
YE
E and S
ES
S and Y
SY
Angle-Angle-Side (AAS)
A A   D
A B  E
ABC   DEF
S BC  EF
Non-included
side
Recap
 SSS
correspondence
 ASA
correspondence
 SAS
correspondence
 AAS
correspondence
 SSA
correspondence
 AAA
correspondence
BUT ORDER MATTERS!!!!
Warning! No SSA Postulate
No ASS Postulate
because the triangles
may or may not be
congruent.
NOT CONGRUENT
Warning! No AAA Postulate
No AAA Postulate because
congruent angles do not
always imply congruent
sides!
NOT CONGRUENT
3
3
8
8
3
8
Name That Postulate
(when possible)
SAS
SSA
ASA
SSS
Name That Postulate
AAA
SAS
(when possible)
ASA
SSA
Name That Postulate
Reflexive
Property
SAS
Vertical
Angles
SAS
(when possible)
Vertical
Angles
SAS
Reflexive
Property
SSA
HW: Name That Postulate
(when possible)
Reflexive
Property
SSS
AS
ASS
AAA
HW: Name That Postulate
(when possible)
Vertical
Angles
ASS
ASA
SAS
Reflexive
Property
Indicate the additional information needed
to enable us to apply the specified
congruence postulate.
For ASA:
B  D
For SAS:
AC  FE
For AAS:
A  F
OYO = On Your Own
Indicate the additional information needed
to enable us to apply the specified
congruence postulate.
For ASA:
N  V
For SAS:
MK  UT
For AAS: M
 U
Hypotenuse-Leg (HL)
QRS and  XYZ are right triangles
H QS  XZ
L RS  YZ
QRS   XYZ
ONLY works for
right triangles!
(when possible)
SAS
Reflexive
Property
AAS
Vertical
Angles
Reflexive
Property
HL
ASS
Another Big Idea: CPCTC
• Stands for “Corresponding Parts of Congruent
Triangles are Congruent”
• Once you have already proved triangles
congruent, you know that all corresponding
sides and angles are congruent.
1. AB  DE
• Recall: If ABC   DEF then
2. BC  EF
3. AC  DF
4.  A   D
5.  B   E
6.  C   F
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