Congruent triangles theorem

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-
includedside 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
ACDAB ADBC CDAB DABC CAAC
Are the triangles congruent?
S
S
S
Why?

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

23. Summary:
ASA-Pairs of congruent sides contained
betweentwo congruent angles
SAS-Pairs of congruent angles
contained betweentwo congruent sides
SSS-Three pairs of congruent sides
AAS–Pairs of congruent angles and
the side notcontained betweenthem.

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!!!