This document discusses different methods for proving that two triangles are congruent, including the side-side-side (SSS), side-angle-side (SAS), angle-side-angle (ASA), and angle-angle-side (AAS) postulates. It provides examples of applying each postulate to determine if two triangles are congruent and practice problems for the reader to identify the postulate used. However, there is no side-side-angle (SSA) or angle-angle-angle (AAA) postulate that can be used to prove triangles are congruent.

1. Proving Triangles
Congruent
Free powerpoints at http://www.worldofteaching.com

2. The Idea of a Congruence
Two geometric figures with
exactly the same size and
shape.
F

3. B
A C E D
How much do you
need to know. . .

4. . . . about two triangles
to prove that they
are congruent?
Corresponding Parts
In Lesson 4.2, you learned that if all
six pairs of corresponding parts (sides

5. and angles) are congruent, then the
triangles are congruent.
• AB ≅ DE
• BC ≅ EF
• AC ≅ DF
D∆ABC ≅ ∆
∀ ∠ A ≅ ∠
DEF
∀ ∠ B ≅ ∠ E

6. ≅ ∠ F
∀ ∠ C

7. Do you need all six ?
NO !

8. Side-Side-Side (SSS)

9. • AB ≅ DE
• BC ≅ EF
• AC ≅ DF
∆ABC ≅ ∆ DEF

10. Side-Angle-Side (SAS)

11. Included Angle
• AB ≅ DE
• ∠ A ≅ ∠ D
• AC ≅ DF
∆ABC ≅ ∆ DEF
included
angle

12. The angle between two sides
∠ G ∠ I ∠ H

13. Included Angle
E
Y S
Angle-
Side-Angle (ASA)
Name the included angle:
YE and ES ∠ E
ES and YS ∠ S
YS and YE ∠ Y

14. Included Side
The side between two angles
GI HI GH

15. Included Side
E
Y S
Name the included side:
∠Y and ∠E YE
∠E and ∠S ES
∠S and ∠Y SY

16. Angle-Angle-Side (AAS)

18. Warning: No SSA Postulate
F
A C D
NOT CONGRUENT
B E
There is no such
thing as an SSA
postulate!

19. Warning: No AAA Postulate
A C F
D
B
E
There is no such
thing as an AAA
postulate!

20. NOT CONGRUENT
The Congruence Postulates
SSS correspondence
ASA correspondence
SAS correspondence
AAS correspondence
SSA correspondence

21. AAA correspondence

22. Name That Postulate
(when possible)

23. Name That Postulate
SAS ASA
SSS
(when possible)
SSA

24. Name That Postulate
ASA
AAA

25. Name That Postulate
SAS
(when possible)
SSA

26. Name That Postulate
Vertical
Reflexive Property SASAngles
SAS

27. Name That Postulate
Vertical
Angles
SAS
Reflexive
Property SSA

28. Name That Postulate
(when possible)

29. Name That Postulate
(when possible)

30. Let’s Practice
Indicate the additional information needed
to enable us to apply the specified
congruence postulate.
For ASA:
∠B ≅ ∠D
For SAS:
AC ≅ FE

31. For AAS: ∠A ≅ ∠F
Indicate the additional information needed
to enable us to apply the specified
congruence postulate.

32. For ASA:
For SAS:
For AAS:

33. Determine if whether each pair of triangles is congruent by
SSS, SAS, ASA, or AAS. If it is not possible to prove
that they are congruent, write not possible.
Ex 4 GK
H J
I

34. ΔGIH ≅ ΔJIK by AAS
Determine if whether each pair of triangles is congruent by
SSS, SAS, ASA, or AAS. If it is not possible to prove
that they are congruent, write not possible.

35. B A
C
E
D
Ex 5

36. ΔABC ≅ ΔEDC by
ASA
Determine if whether each pair of triangles is congruent by
SSS, SAS, ASA, or AAS. If it is not possible to prove
that they are congruent, write not possible.
Ex 6 AE

37. B D
C

38. ΔACB ≅ ΔECD by SAS

39. Determine if whether each pair of triangles is congruent by
SSS, SAS, ASA, or AAS. If it is not possible to prove
that they are congruent, write not possible.
Ex 7 K
M L
J

40. ΔJMK ≅ ΔLKM by SAS or
ASA
Determine if whether each pair of triangles is
congruent by SSS, SAS, ASA, or AAS. If it is
not possible to prove that they are congruent,
write not possible.
Ex 8
JT

41. L
K V U

42. Not possible