1. Geometry
Triangle Congruence Theorems

2. 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.
Congruent Triangles

3. The Triangle Congruence
Postulates &Theorems
LAHALLHL
FOR RIGHT TRIANGLES ONLY
AASASASASSSS
FOR ALL TRIANGLES

4. 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°.

5. 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° 30°
85° 30°

6. Example
30°
30°
Why aren’t these triangles congruent?
What do we call these triangles?

7. So, how do we prove
that two triangles really
are congruent?

8. ASA (Angle, Side, Angle)
If two angles and the
included side of one
triangle are congruent
to two angles and the
included side of another
triangle, . . .
then
the 2 triangles are
CONGRUENT!
F
E
D
A
C
B

9. AAS (Angle, Angle, Side)
Special case of ASA
If two angles and a non-
included side of one triangle
are congruent to two angles
and the corresponding non-
included side of another
triangle, . . .
then
the 2 triangles are
CONGRUENT!
F
E
D
A
C
B

10. SAS (Side, Angle, Side)
If in two triangles, two
sides and the included
angle of one are
congruent to two sides
and the included angle
of the other, . . .
then
the 2 triangles are
CONGRUENT!
F
E
D
A
C
B

11. SSS (Side, Side, Side)
In two triangles, if 3
sides of one are
congruent to three sides
of the other, . . .
F
E
D
A
C
B
then
the 2 triangles are
CONGRUENT!

12. HL (Hypotenuse, Leg)
If both hypotenuses and a
pair of legs of two RIGHT
triangles are congruent, . . .
A
C
B
F
E
D
then
the 2 triangles are
CONGRUENT!

13. HA (Hypotenuse, Angle)
If both hypotenuses and a
pair of acute angles of two
RIGHT triangles are
congruent, . . .
then
the 2 triangles are
CONGRUENT!
F
E
D
A
C
B

14. LA (Leg, 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
F
E
D

15. LL (Leg, Leg)
If both pair of legs of two
RIGHT triangles are
congruent, . . .
then
the 2 triangles are
CONGRUENT!
A
C
B
F
E
D

16. Example 1
Given the markings on
the diagram, is the
pair of triangles
congruent by one of
the congruency
theorems in this
lesson?
F
E
D
A
C
B

17. Example 2
Given the markings on
the diagram, is the pair
of triangles congruent
by one of the
congruency theorems
in this lesson?
A
C
B
F
E
D

18. Example 3
Given the markings on
the diagram, is the pair
of triangles congruent
by one of the
congruency theorems
in this lesson?
D
A
C
B

19. Example 4
Why are the two
triangles congruent?
What are the
corresponding
vertices?
A
B
C
D
E
F SAS
A   D
C   E
B   F

20. Example 5
Why are the two
triangles
congruent?
What are the
corresponding
vertices?
A
B
C
D
SSS
A   C
ADB   CDB
ABD   CBD

21. Example 6
Given:
B C
D
A
CDAB
ADBC
CDAB
DABC
CAAC
Are the triangles congruent?
S
S
S
Why?

22. Example 7
Given: QRPS 
R
H
SRSSR
 Are the Triangles Congruent?
QSR  PRS = 90
Q
RS
P
T
mQSR = mPRS = 90
PSQR 
Why?

23. Summary:
ASA - Pairs of congruent sides contained
between two congruent angles
SAS - Pairs of congruent angles
contained between two congruent sides
SSS - Three pairs of congruent sides
AAS – Pairs of congruent angles and
the side not contained between them.

24. 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

25. THE END!!!