1. 5.6 Proving Triangles Congruent using ASA and AAS
Geometry
EQ: What other ways can you prove triangles congruent?

2. Geometry 5.6 Proving Triangles Congruent: ASA, AAS
Goals
Use ASA and AAS postulates to prove two triangles are congruent.
Solve problems using congruence postulates.
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

3. Geometry 5.6 Proving Triangles Congruent: ASA, AAS
Included Sides
AC is included between A and C. C
AB is included between A and B.
CB is included between C and B. B A
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

4. Theorem 5.10: Angle-Side-Angle
If two angles and the included side of one triangle are congruent to corresponding parts of another triangle, then the triangles are congruent. (ASA) A B C X Y Z
ABC  XYZ
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

5. Example Prove ADB  CBD
Proof:
ADB  CBD (Given)
ABD  CDB (Given)
ADB  CBD (ASA) A S
DB  DB (Ref.) A A B
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

6. Theorem 5.11 Angle-Angle-Side
If two angles and a nonincluded side of one triangle are congruent to corresponding parts of another triangle, then the triangles are congruent. A B C D E F
ABC  DEF
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

7. Geometry 5.6 Proving Triangles Congruent: ASA, AAS
Example Is ABC  ADC? A B C D
Yes,by AAS S A A
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

8. Are these triangles congruent?
Yes. Reason:
AAS
OR AAS.
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

9. Are these triangles congruent?
You could even use ASA.
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

10. Overlapping TrianglesC A D A B D C A D A B D
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

11. Geometry 5.6 Proving Triangles Congruent: ASA, AAS
Problem: Is ABD  DCA? C B
85°
85°
15°
15° D A
YES by
AAS
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

12. Geometry 5.6 Proving Triangles Congruent: ASA, AAS
Problem: Is ABD  DCA? C C
85°
15° D B
85° A B
15°
85°
85°
15°
15° D A
YES by
AAS
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

13. Methods to show triangles congruent:
SSS
SAS
ASA
AAS
When trying to determine if triangles are congruent, carefully study the given information, mark the triangles if necessary, then determine what pattern you have.
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

14. Geometry 5.6 Proving Triangles Congruent: ASA, AAS
Problem A S S A B S A C R A A
What other congruence must be given to show that ABC  RST using the AAS postulate?
BC  ST
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

15. Geometry 5.6 Proving Triangles Congruent: ASA, AAS
Problem
Alternate Solution A S B S A A C R A A S
What other congruence must be given to show that ABC  RST using the AAS postulate?
BA  SR
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

16. Geometry 5.6 Proving Triangles Congruent: ASA, AAS
Problem A B S A C S A R A S A
What other congruence must be given to show that ABC  RST using the ASA postulate?
AC  RT
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

17. Geometry 5.6 Proving Triangles Congruent: ASA, AAS
State the third congruence that must be given to prove the given triangles congruent using the indicated postulate or theorem.
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

18. Geometry 5.6 Proving Triangles Congruent: ASA, AAS
1. ASA Theorem. C A B F D E
C  F
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

19. Geometry 5.6 Proving Triangles Congruent: ASA, AAS
2. AAS Theorem. A B C E D F
C  F
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

20. Geometry 5.6 Proving Triangles Congruent: ASA, AAS
3. SSS A C B E F D
CB  FE
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

21. Geometry 5.6 Proving Triangles Congruent: ASA, AAS
4. SAS Theorem. F E A D C B
A  D
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

22. Geometry 5.6 Proving Triangles Congruent: ASA, AAS
Is it possible to prove that the triangles are congruent? If so, state the postulate or theorem you would use. Explain reasoning
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

23. Geometry 5.6 Proving Triangles Congruent: ASA, AASX Z W
Yes
AAS
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

24. Geometry 5.6 Proving Triangles Congruent: ASA, AAS6. C A
Yes
SAS D R
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

25. Geometry 5.6 Proving Triangles Congruent: ASA, AAS7.
The only other thing we know for sure:
SOT  LOF
(Vertical Angles)
But this spells…
SSA
NOT CONGRUENT S T O F L
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

26. Geometry 5.6 Proving Triangles Congruent: ASA, AAS
Given: LA || SN, LR  NR
Prove: LAR  NSR
1. LA || SN
1. Given
2. L  N
2. Alt. int s
3. LR  NR
3. Given
4. LRA  NRS
4. Vert s
5. LAR  NSR
5. ASA
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

27. List four ways to show triangles are congruent.
SSS
ASA
SAS
AAS
List one way that won’t work.
SSA
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

28. Geometry 5.6 Proving Triangles Congruent: ASA, AAS
Important!
To show two triangles congruent, at least one side must be known.
Three angles (AAA) does not show congruence (but it is still useful).
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS

29. Geometry 5.6 Proving Triangles Congruent: ASA, AAS
Summarize
June 10, 2018
Geometry 5.6 Proving Triangles Congruent: ASA, AAS