1. SAS
SSS
SAS
SSS

2. Sect. 4.4 Proving Triangles are Congruent: ASA and AAS
Goal Using the ASA and AAS Congruence Methods
Goal Using Congruence Postulate and Theorems

3. Postulate 21 Angle-Side-Angle (ASA) Congruence Postulate
Using the ASA and AAS Congruence Methods
Postulate 21 Angle-Side-Angle (ASA) Congruence Postulate
If, in two triangles, two angles and the included side of one triangle are congruent to two angles and the included side of the other, then the triangles are congruent.
                                                                                                                   

4. Using the ASA and AAS Congruence Methods
Given: ABC   DCB;  DBC   ACB
Prove: ABC  DCB
Statements
Reasons
1. ABC   DCB; DBC  ACB
1. Given
2. Reflexive
3. ABC  DCB
3. ASA 2.

5. Theorem 4.5 Angle-Angle-Side (AAS) Congruence Theorem
Using the ASA and AAS Congruence Methods
Theorem 4.5 Angle-Angle-Side (AAS) Congruence Theorem
If, in two triangles, two angles and a non-included side of one triangle are congruent respectively to two angles and the corresponding non-included side of the other, then the triangles are congruent.

6. Using the ASA and AAS Congruence Methods
Given: B  C; D  F;
M is the midpoint of
Prove: BDM  CFM
Statements
Reasons
1. B  C; D  F;
M is the midpoint of
1. Given 2.
2. Definition of Midpoint
3. AAS
3. BDM  CFM

7. Using the ASA and AAS Congruence Methods
Given: bisects XZY and XWY
Prove: WZX  WZY
Statements
Reasons
bisects XZY and XWY
1. Given
2. XZW  YZW;
XWZ  YWZ
2. Definition of Angle Bisector 3.
3. Reflexive
4. WZX  WZY
4. ASA

8. Methods of Proving Triangles Congruent
Using Congruence Postulates and Theorems
Methods of Proving Triangles Congruent
SSS
If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.
SAS
If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
ASA
If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
AAS
If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

9. Using Congruence Postulates and Theorems
AAA works fine to show that triangles are the same SHAPE (similar), but does NOT work to show congruent!
You can draw 2 equilateral triangles that are the same shape but not the same size.

10. What about two sides and a not-included angle?
Using Congruence Postulates and Theorems
What about two sides and a not-included angle?
Note that GB and BH are the same length, and that AB and angle A are the other  parts of Angle – Side – Side.

11. Using Congruence Postulates and Theorems
ASS
It does NOT work!!!

12. Given: CB  AD ; CB bisects ACD Prove: ABC  DBC
Using Congruence Postulates and Theorems
Given: CB   AD ; CB bisects ACD           Prove: ABC   DBC

13. Given: A  E ; B  G ;
AC  EF
Prove: ABC  EGF
Baby Proof

14. A
Given: BAC  DAE ; B  D ;
AC  AE
Prove: ABC  ADE B E C D

15. Given: 2  3 ; B  D ;
AC  AE
Prove: ABE  ADC

16. Given: ○A ; 2  3 ;
B  D
Prove: ABE  ADC

17. Homework
pp Exs. 8 – 23, 34-38