Download Congruent Triangles

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They have congruent corresponding parts –
their matching sides and angles. When you
name congruent polygons you must list
corresponding vertices in the same order
Example:
A  E
B  H
C  G
D  F
C
G
B
D
H
F
A
E
ABCD  EHGF
AB  EH
BC  HG
CD  GF
AD  EF
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If two angles of one triangle are congruent to
two angles of another triangle, then the the
third angles are congruent.
If A  D and B  E then C  F
E
B
D
A
C
F
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In order to prove triangles congruent with
what we currently know we would have to
prove all angles congruent and all sides
congruent, this would be considered the
definition of congruent triangles ….
But we know some short cuts 

To prove two triangles congruent we can
prove that three sides of one triangle are
congruent to three sides of another triangle.
If:
C
G
IB  JH
BC  HG
CI  GJ
B
H
I
Then:
J
IBC 
By SSS
JHG

If two sides and the included angle of one
triangle are congruent to two sides and the
included angle of another triangle then we
can say those triangles are congruent by SAS
◦ The included angle is the angle between the two
congruent sides.
If:
C
G
IB  JH (Side)
I  J (Included Angle)
CI  GJ (Side)
B
Then:
H
I
J
IBC 
JHG By SAS

If two angles and the included side of one
triangle are congruent to two angles and the
included side of another triangle then the
triangles are congruent.
◦ Included side is the side that connects both congruent
angles.
E
B
F
D
A
C
If:
A  D and B  E, and AB  DE
(Which is the included side)
Then:
ABC 
DEF by ASA

If you have two angle and the non-included
side of one traingle congreunt to two angles
and the corresponding non-inlcued side of
another traingle, then the traingles are
congreunt.
E
B
F
D
A
C
If:
A  D and B  E, and AC  DF
(Which are corresponding
non-included sides)
Then:
ABC 
DEF by AAS
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You can not prove triangles congruent with
AAA or ASS. These two methods do not create
unique triangles, and therefore can not be
used to prove triangles congruent!
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If you know that two triangles are congruent
by CPCTC you can say any of their 6 parts are
congruent.
Example:
B
If
D
F
FED
then by CPCTC you can say :
C
A
ABC 
A   F, B   E, C  D,
AB  FE, BC  ED, AC  FD,
E
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If you have two right triangles, there are
special congruence postulates that can be
used (You do not state the right angle it is
assumed)
LL – two legs are congruent
LA – a leg and another angle besides the right
HA – the hypotenuse and another angle
besides the right
HL – the hypotenuse and a leg
B
Given: AD  BC and E is the MP of
A
Prove:
E
C
D
ADE 
CDE
AC and DB
B
Given: AD  CD and DB bisects ADC
C
A
D
Prove:
ADB 
CDB
B
Given: AD // BC and A  C
A
Prove:
C
D
ABC 
CDB
H
E
Given: FE // GH and FE  GH and
G is MP of FI
Prove:  E   H
F
G
I