1. Proving Triangles are Congruent
EQ: What can you conclude about two triangles when you know that two pairs of corresponding sides and corresponding included angles are congruent?
Today, you will learn to…
* prove that triangles are congruent
* use congruence postulates to solve problems

2. Using 3 segments, can you ONLY create 2 triangles that are congruent?
SSS Experiment
Using 3 segments, can you ONLY create 2 triangles that are congruent?

3. Side-Side-Side Congruence PostulateB
Side-Side-Side Congruence Postulate
If Side AB  XY
Side AC  XZ
Side BC  YZ,
then ΔABC  ΔXYZ
by SSS X Y Z
If 3 pairs of sides are congruent, then the two triangles are congruent.

4. no K L C A J B 1. Does the diagram give enough
info to use SSS Congruence? K L C A no J B

5. LM  MP NM  NM NLM  NPM SSS Congruence 2. Given: LN  NP and
M is the midpoint of LP
Prove: ΔNLM  ΔNPM N L P M
LM  MP
Def of midpoint
NM  NM
Reflexive Property
NLM  NPM
SSS Congruence

6. SAS Experiment Using 2 congruent segments and
1 included angle, can you ONLY create 2 triangles that are congruent?

7. Side-Angle-Side Congruence PostulateB
Side-Angle-Side Congruence Postulate
If Side AB  XY
Angle B  Y
Side BC  YZ,
then ΔABC  ΔXYZ
by SAS X Y Z
If 2 pairs of sides and their included angle are congruent, then the two triangles are congruent.

8. SAS? 5. 4.
NO!
SAS 6. 7.
NO!
SAS

9. A C D A B C D 8. Does the diagram give enough
info to use SAS Congruence? A B C D A C D

10. no W V X Y Z 9. Does the diagram give enough
info to use SAS Congruence? W V no X Y Z

11. no E A B C D 10. Does the diagram give enough
info to use SAS Congruence? A E no B C D

12. VWZ YWX VWZ  YWX 11. X V W Z Y Vertical Angles Th SAS Congruence
Given: W is the midpoint of VY and the midpoint of ZX
Prove: ΔVWZ  ΔYWX X Z W Y V
VW  WY and ZW  WX
Def. of midpoint
VWZ YWX
Vertical Angles Th
VWZ  YWX
SAS Congruence

13. 12. Given: AB  PB , MB  AP Prove: ΔMBA  ΔMBP M P A B MBA  MBP
Def of  lines
ABM & PBM are right s
ABM  PBM
All right s are 
Reflexive Property
MB  MB
MBA  MBP
SAS Congruence

14. What is the best way to get better at proofs?

15. ASA Experiment
Using 2 angles connected by 1 segment, can you ONLY create two triangles that are congruent?

16. Angle-Side-Angle Congruence PostulateB
Angle-Side-Angle Congruence Postulate
If Angle B  Y,
Side BC  YZ,
Angle C  Z
then ΔABC  ΔXYZ
by ASA X Y Z
If 2 pairs of angles and the included sides are congruent, then the two triangles are congruent.

17. Included side? A B C
The included side between
 A and  B is _____ AB

18. Included side? A B C
The included side between
 B and  C is _____ CB

19. Included side? A B C
The included side between
 A and  C is _____ AC

20. ASA? 2. 1.
NO!
ASA 3. 4.
NO!
ASA

21. Δ ABD  Δ by ASA A C D C B A D 5. Does the diagram give enough
info to use ASA Congruence? A B C D
Reflexive Property
Third Angles Theorem
Δ ABD  Δ by ASA A C D

22. yes, Δ ACB  ______ by ASA Δ C A D A D B C
6. Does the diagram give enough
info to use ASA Congruence? A D
Reflexive
Property
Alt. Int.
Angles
Theorem B C
yes, Δ ACB  ______ by ASA
Δ C A D

23. no A D B C 7. Does the diagram give enough info to use ASA Congruence?
Reflexive
Property B C no

24. A C B K L C A J B 8. Does the diagram give enough
info to use ASA Congruence? K L C A J B A C B

25. 9. Determine whether the triangles
are congruent by ASA. L K
Alt. Int. Angles
Theorem J
Vertical Angles
Theorem H G K J L

26. Angle-Angle-Side Congruence TheoremB
Angle-Angle-Side Congruence Theorem
If Angle B  Y
Angle C  Z
Side AB  XY
then ΔABC  ΔXYZ
by AAS X Y Z
If 2 pairs of angles and a pair of nonincluded sides are congruent, then the two triangles are congruent.

27. AAS?
11.
10.
AAS
NO!
12.
13.
AAS
NO!

28. C B A D 14. Does the diagram give enough info to use AAS Congruence?
Reflexive
Property

29. K L C A J B 15. Does the diagram give enough
info to use AAS Congruence? K L C A J B

30. 16. Determine whether the triangles
are congruent by AAS. L K
Alt. Int. Angles
Theorem J
Vertical Angles
Theorem H G K J L

31. SSA Experiment
Using 2 sides and 1 angle that is NOT included, can you ONLY create two triangles that are congruent? NO

32. Using 3 angles, can you ONLY create two triangles that are congruent?
AAA Experiment
Using 3 angles, can you ONLY create two triangles that are congruent? NO
All of the angles are ,
but the s are NOT 

33. Mark the given information on the triangles
Mark the given information on the triangles. What additional congruence would you need to show ABC  XYZ?
17. CB  ZY , AC  XZ
SAS Congruence C A B X Y Z
C  Z

34. Mark the given information on the triangles
Mark the given information on the triangles. What additional congruence would you need to show ABC  XYZ?
18. CB  ZY , AC  XZ
SSS Congruence C A B X Y Z
AB  XY

35. Mark the given information on the triangles
Mark the given information on the triangles. What additional congruence would you need to show ABC  XYZ?
19. CB  ZY , C  Z
SAA Congruence C A B X Y Z
A  X

36. What is the best way to get better at proofs?