1. 4.4-4.5 & 5.2: Proving Triangles Congruent
Adapted from:

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
AB =
MO = 5
NO = 7
MN =
By SSS

4. Definition – Included Angle
K is the angle between JK and KL. It is called the included angle of sides JK and KL.
What is the included angle for sides KL and JL? 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) S A S S A S
by SAS

6. Example #2 – SAS – Postulate
Given: N is the midpoint of LW N is the midpoint of SK
Prove:
Statement
Reason
N is the midpoint of LW N is the midpoint of SK
Given 1 1
Definition of Midpoint 2 2 3 3
Vertical Angles are congruent
SAS 4 4

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

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)
by ASA

9. Example #3 – ASA – Postulate
Given: HA || KS
Prove:
Reasons
Statement 1
HA || KS,
Given 1
Alt. Int. Angles are congruent 2 2 3
Vertical Angles are congruent 3 4
ASA Postulate 4

10. Identify the Congruent Triangles.
Identify the congruent triangles (if any). State the postulate by which the triangles are congruent.
by SSS
Note: is not SSS, SAS, or ASA.
by SAS

11. Example
Given:
Prove:
Statement
Reason
1) Given 1)

12. the 2 triangles are CONGRUENT!
AAS (Angle, Angle, Side)
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, . . . F E D A C B
then
the 2 triangles are CONGRUENT!

13. Example
Given:
Prove:
Statement
Reason 1) 1)
Given 2) 2)

14. the 2 triangles are CONGRUENT!
HL (Hypotenuse, Leg)
***** only used with right triangles****
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!

15. Example
Given:
Prove:
Statement
Reason 1) 1)
Given 2) 2)

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

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 A C B D E F

19. 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
No ! SSA doesn’t work

20. 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
YES ! Use the reflexive side CB, and you have SSS

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

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

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

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

25. Homework Assignment