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Geometry
Triangle Congruence Theorems
Congruent Triangles
Congruent triangles have three
congruent sides and and three
congruent angles.
However, triangles can be proved
congruent without showing 3 pairs of
congruent sides and angles.
The Triangle Congruence
Postulates &Theorems
FOR ALL TRIANGLES
SSS
SAS
ASA
AAS
FOR RIGHT TRIANGLES ONLY
HL
LL
HA
LA
Theorem
If two angles in one triangle are
congruent to two angles in
another triangle, the third angles
must also be congruent.
Think about it… they have to add
up to 180°.
A closer look...
If two triangles have two
pairs of angles congruent,
then their third pair of
angles is congruent.
But do the two triangles
have to be congruent?
85°
85°
30°
30°
Example
30°
30°
Why aren’t these triangles congruent?
What do we call these triangles?
So, how do we prove
that two triangles really
are congruent?
ASA (Angle, Side, Angle)
A
C
D
If two angles and the
included side of one
triangle are congruent
to two angles and the
B
included side of another
triangle, . . .
F
E
then
the 2 triangles are
CONGRUENT!
AAS (Angle, Angle, Side)
Special case of ASA
If two angles and a nonincluded side of one triangle
are congruent to two angles
and the corresponding non- C
included side of another
triangle, . . .
then
the 2 triangles are
CONGRUENT!
A
D
B
F
E
SAS (Side, Angle, Side)
A
C
D
If in two triangles, two
sides and the included
angle of one are
congruent to two sides
B and the included angle
of the other, . . .
F
E
then
the 2 triangles are
CONGRUENT!
SSS (Side, Side, Side)
A
C
B
D
F
E
In two triangles, if 3
sides of one are
congruent to three sides
of the other, . . .
then
the 2 triangles are
CONGRUENT!
HL (Hypotenuse, Leg)
If both hypotenuses and a
pair of legs of two RIGHT
triangles are congruent, . . .
A
C
B
D
then
the 2 triangles are
CONGRUENT!
F
E
HA (Hypotenuse, Angle)
If both hypotenuses and a
pair of acute angles of two
RIGHT triangles are
congruent, . . .
then
the 2 triangles are
CONGRUENT!
A
C
B
D
F
E
LA (Leg, Angle)
If both hypotenuses and a
pair of acute angles of two
RIGHT triangles are
C
congruent, . . .
A
B
D
then
the 2 triangles are
CONGRUENT!
F
E
LL (Leg, Leg)
A
If both pair of legs of two
RIGHT triangles are
congruent, . . .
then
the 2 triangles are
CONGRUENT!
C
B
D
F
E
Example 1
A
C
B
D
E
F
Given the markings on
the diagram, is the
pair of triangles
congruent by one of
the congruency
theorems in this
lesson?
Example 2
A
C
B
D
E
F
Given the markings on
the diagram, is the pair
of triangles congruent
by one of the
congruency theorems
in this lesson?
Example 3
A
C
B
D
Given the markings on
the diagram, is the pair
of triangles congruent
by one of the
congruency theorems
in this lesson?
D
Example 4
Why are the two
triangles congruent?
SAS
F
E
A
What are the
corresponding
vertices?
A   D
C   E
C
B
B   F
Example 5
A
Why are the two
triangles
congruent?
SSS
D
B
What are the
corresponding
A   C
vertices?
ADB   CDB
ABD   CBD
C
Example 6
Given: AB CD
BC AD
B
A
Are the triangles congruent?
Why?
AB CD S
BC DA S
AC CA S
C
D
Example 7
Q
P
Given: PS  QR
T
mQSR = mPRS = 90°
R
S

Are the Triangles Congruent?
Why?
QSR  PRS = 90°
QR  PS
SR  RS
H
L
Summary:
ASA - Pairs of congruent sides contained
between two congruent angles
AAS – Pairs of congruent angles and
the side not contained between them.
SAS - Pairs of congruent angles
contained between two congruent sides
SSS - Three pairs of congruent sides
Summary --for Right Triangles Only:
HL – Pair of sides including the
Hypotenuse and one Leg
HA – Pair of hypotenuses and one acute
angle
LL – Both pair of legs
LA – One pair of legs and one pair of
acute angles
THE END!!!