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Congruent Triangles
 Can I have a volunteer read today’s
objective?
SWBAT justify that two triangles are
congruent using the congruence postulate
theorems.
Quick Review: 6 minutes
If you get stuck, try to think about these
questions or hints.
What does opposite mean?
The smallest side is opposite the smallest angle.
The biggest side is opposite the biggest angle.
Triangles have the EXACT same measures for all
angles and sides.
POINTS I MUST KNOW !!!
Remember…
Tests try to trip you up! Watch out so
you can beat them! For Geometry, that
means
Never assume an angle is right unless you
see the square
Use what you know, not what it looks like.
This is really important for size &
measurements!
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.
Example
30°
30
°
1. Why aren’t these triangles congruent?
2. What do we call these triangles?
The Triangle Congruence
Postulates &Theorems
FOR ALL TRIANGLES
SSS
SAS
ASA
AAS
FOR RIGHT TRIANGLES ONLY
HL
LL
HA
LA
1. Write the theorem or postulate that matches
with the picture.
2. Write the congruency statement. Be sure to
match up the appropriate parts.
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
ASA
C
B
D
E
F
Example 2
S S A
A
C
B
D
E
F
Example 3
SSS
A
C
B
D
D
Example 4
SAS
F
E
A
C
B
Example 5
B
A
AB CD S
BC DA S
AC CA S
C
D
Example 6
A
SSS
D
B
C
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
R
H
S
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