This document provides information about proving triangles congruent using various postulates and theorems: 1. It describes the SSS, SAS, ASA, and AAS postulates that can be used to prove any triangles congruent, as well as the HL postulate that applies only to right triangles. 2. Examples are given to demonstrate applying each of the postulates to determine if two triangles are congruent and to find missing corresponding parts. 3. Additional theorems like CPCTC (corresponding parts of congruent triangles are congruent) and properties of vertical angles, isosceles triangles, and midpoints are also explained.

1. 4.4-4.5 & 5.2: Proving
Triangles Congruent
p. 206-221, 245-251
Adapted from:
http://jwelker.lps.org/lessons/ppt/geod_4_4_congruent_triangles.ppt

2. SSS - Postulate
If all the sides of one triangle are congruent to all
of the sides of a second triangle, then the triangles
are congruent. (SSS)

3. Example #1 – SSS – Postulate
Use the SSS Postulate to show the two triangles
are congruent. Find the length of each side.
AC = 5
BC = 7
2
2
AB = 5 + 7 = 74
MO = 5
NO = 7
2
2
MN = 5 + 7 = 74
∆ACB ≅ ∆MON By SSS

4. Definition – Included Angle
J
∠ K is the angle between
JK and KL. It is called the
included angle of sides JK
and KL.
K
L
What is the included angle
for sides KL and JL?
J
∠L
K
L

5. SAS - Postulate
If two sides and the included angle of one triangle
are congruent to two sides and the included angle
of a second triangle, then the triangles are
congruent. (SAS)
L
S
Q
P
A
S
A
J
S
S
K
∆JKL ≅ ∆PQR
by SAS
R

6. Example #2 – SAS – Postulate
K
L
Given: N is the midpoint of LW
N is the midpoint of SK
N
Prove: ∆LNS ≅ ∆WNK
W
S
Statement
Reason
N is the midpoint of LW
N is the midpoint of SK
1
Given
2
LN ≅ NW , SN ≅ NK
2
Definition of Midpoint
3
∠LNS ≅∠WNK
3 Vertical
4
∆LNS ≅ ∆WNK
1
4 SAS
Angles are congruent

7. Definition – Included Side
J
JK is the side between
∠ J and ∠ K. It is called the
included side of angles J
and K.
K
L
J
What is the included side
for angles K and L?
KL
K
L

8. ASA - Postulate
If two angles and the included side of one triangle
are congruent to two angles and the included side
of a second triangle, then the triangles are
congruent. (ASA)
J
X
Y
K
L
∆JKL ≅ ∆ZXY by ASA
Z

9. Example #3 – ASA – Postulate
H
A
W
Given: HA || KS
AW ≅WK
K
Prove: ∆HAW ≅ ∆SKW
S
Reasons
Statement
HA || KS, AW ≅ WK
1
Given
2
∠HAW ≅∠SKW
2
Alt. Int. Angles are congruent
3
∠HWA ≅∠SWK
3 Vertical
4
∆HAW ≅ ∆SKW
1
4
Angles are congruent
ASA Postulate

10. Identify the Congruent Triangles.
Identify the congruent triangles (if any). State the
postulate by which the triangles are congruent.
A
J
R
B
C
H
I
S
K
M
O
L
P
VABC ≅VSTR by SSS
VPNO ≅VVUW by SAS
N V
T
U
W
Note: VJHI is not
SSS, SAS, or ASA.

11. Example
∆AMT is isosceles with vertex ∠MAT
Given:
∠MAT is bisected by AH
Prove: MH ≅ HT
Statement
1)
Reason
∆AMT is isosceles with vertex ∠MAT 1) Given
∠MAT is bisected by AH

12. AAS (Angle, Angle, Side)
• 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

13. Example
Given:
AW || TB
E is the midpoint of WB
Prove: AW ≅ TB
Statement
1) AW || TB
E is the midpoint of WB
2)
Reason
1)
2)
Given

14. HL (Hypotenuse, Leg)
***** only used with right triangles****
• If both hypotenuses and a pair
of legs of two RIGHT triangles
are congruent, . . .
then
the 2 triangles are
CONGRUENT!
A
C
B
D
F
E

15. Given:
AB ≅ CB
Example
∠BDA and ∠BDC are right
∆ABD and ∆CBD are right triangles
Prove: ∠A ≅ ∠C
Statement
1) AB ≅ CB
∠BDA and ∠BDC are right
∆ABD and ∆CBD are right triangles
2)
Reason
1)
2)
Given

16. The Triangle Congruence
Postulates &Theorems
FOR ALL TRIANGLES
SSS
SAS
ASA
AAS
FOR RIGHT TRIANGLES ONLY
HL
Only
this one
is new
LL
HA
LA

17. Summary
• Any Triangle may be proved congruent by:
(SSS)
(SAS)
(ASA)
(AAS)
• Right Triangles may also be proven congruent by HL
( Hypotenuse Leg)
• Parts of triangles may be shown to be congruent by Congruent
Parts of Congruent Triangles are Congruent (CPCTC).

18. Example 1
Given the information in the diagram,
is there any way to determine if
A
CB ≅ DF ?
C
B
D
YES!! ∆CAB ≅ ∆DEF by SAS
so CB ≅ DF by CPCTC
E
F

19. Example 2
A
C
B
• Given the markings on
the diagram, is the pair
of triangles congruent by
one of the congruency
theorems in this lesson?
D
E
F
No ! SSA doesn’t work

20. Example 3
A
C
D
• Given the markings on
the diagram, is the pair
of triangles congruent by
one of the congruency
theorems in this lesson?
B
YES ! Use the
reflexive side CB,
and you have SSS

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

22. Name That Postulate
(when possible)
AAA
SAS
ASA
SSA

23. Name That Postulate
(when possible)
Reflexive
Property
SAS
Vertical
Angles
SAS
Vertical
Angles
SAS
Reflexive
Property
SSA

24. 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
For AAS:
∠A ≅ ∠F

25. Homework Assignment