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

2. Two geometric figures with
exactly the same size and
shape.
The Idea of a CongruenceThe Idea of a Congruence
A C
B
DE
F

3. How much do youHow much do you
need to know. . .need to know. . .
. . . about two triangles
to prove that they
are congruent?

4. In Lesson 4.2, you learned that if all
six pairs of corresponding parts (sides
and angles) are congruent, then the
triangles are congruent.
Corresponding PartsCorresponding Parts
∆ABC ≅ ∆
DEF
B
A
C
E
D
F
1. AB ≅ DE
2. BC ≅ EF
3. AC ≅ DF
4. ∠ A ≅ ∠ D
5. ∠ B ≅ ∠ E
6. ∠ C ≅ ∠ F

5. Do you needDo you need all six ?all six ?
NO !

6. Side-Side-Side (SSS)Side-Side-Side (SSS)
1. AB ≅ DE
2. BC ≅ EF
3. AC ≅ DF
∆ABC ≅ ∆ DEF
B
A
C
E
D
F

7. Side-Angle-Side (SAS)Side-Angle-Side (SAS)
1. AB ≅ DE
2. ∠A ≅ ∠ D
3. AC ≅ DF
∆ABC ≅ ∆ DEF
B
A
C
E
D
F
included
angle

8. The angle between two sides
Included AngleIncluded Angle
∠ G ∠ I ∠ H

9. Name the included angle:
YE and ES
ES and YS
YS and YE
Included AngleIncluded Angle
SY
E
∠ E
∠ S
∠ Y

10. Angle-Side-Angle-Side-AngleAngle (ASA)(ASA)
1. ∠A ≅ ∠ D
2. AB ≅ DE
3. ∠ B ≅ ∠ E
∆ABC ≅ ∆ DEF
B
A
C
E
D
F
include
d
side

11. The side between two angles
Included SideIncluded Side
GI HI GH

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

13. Angle-Angle-Side (AAS)Angle-Angle-Side (AAS)
1. ∠A ≅ ∠ D
2. ∠ B ≅ ∠ E
3. BC ≅ EF
∆ABC ≅ ∆ DEF
B
A
C
E
D
F
Non-included
side

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

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

16. The Congruence PostulatesThe Congruence Postulates
SSS correspondence
ASA correspondence
SAS correspondence
AAS correspondence
SSA correspondence
AAA correspondence

17. Name That PostulateName That Postulate
SASSAS
ASAASA
SSSSSSSSASSA
(when possible)

18. Name That PostulateName That Postulate(when possible)
ASAASA
SASSAS
AAAAAA
SSASSA

19. Name That PostulateName That Postulate(when possible)
SASSAS
SASSAS
SASSAS
Reflexive
Property
Vertical
Angles
Vertical
Angles
Reflexive
Property SSASSA

20. Name That PostulateName That Postulate(when possible)

21. (when possible)
Name That PostulateName That Postulate

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

23. Indicate the additional information needed
to enable us to apply the specified
congruence postulate.
For ASA:
For SAS:
For AAS:

24. 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.
ΔGIH ≅ ΔJIK by AAS
G
I
H J
K
Ex 4

25. ΔABC ≅ ΔEDC by ASA
B A
C
ED
Ex 5
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.

26. ΔACB ≅ ΔECD by SAS
B
A
C
E
D
Ex 6
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.

27. ΔJMK ≅ ΔLKM by SAS or ASA
J K
LM
Ex 7
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.

28. Not possible
K
J
L
T
U
Ex 8
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.
V

29. Hypotenuse-Leg Congruence Theorem: HL
• Hypotenuse-Leg Congruence Theorem (HL)
– If the hypotenuse and a leg of a right triangle
are congruent to the hypotenuse and the leg of a
second right triangle, then the two triangles are
congruent.
– If ABC and DEF are right triangles, and
AC = DF, and
BC = EF, then
ABC = DEF
~
~
~
A
B C
D
E F

30. Determine When to Use HL
• Is it possible to show that the two triangles
are congruent using the HL Congruence
Theorem? Explain your reasoning.
• The two triangles are right triangles. You know
that JH = JH (Hypotenuse). You know that JG =
HK (Leg). So, you can use the HL Congruence
theorem to prove that
JGH = HKJ.
G H
J K
~
~ ~

31. Triangle Congruence
Postulates and Theorems
1. SSS (Side-Side-Side)
2. SAS (Side-Angle-Side)
3. ASA (Angle-Side-Angle)
4. AAS (Angle-Angle-Side)
5. HL (Hypotenuse-Leg)