1. 5.3 Proving Triangles are Congruent – ASA & AAS

2. Objectives:
Show triangles are congruent using ASA and AAS.

3. Key Vocabulary
Included Side

4. Postulates
14 Angle–Side–Angle (ASA) Congruence Postulate

5. Theorems
5.1 Angle-Angle-Side (AAS) Congruence Theorem

6. Definition: Included Side
An included side is the common side of two consecutive angles in a polygon. The following postulate uses the idea of an included side.

7. Example: Included SidedB C X Z Y
The side between 2 angles
INCLUDED SIDE

8. Postulate 14 (ASA): Angle-Side-Angle Congruence 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.

9. Two angles and the INCLUDED side
Angle-Side-Angle (ASA) Congruence Postulate A S A S
Two angles and the INCLUDED side

10. You are given that C  E, B  F, and BC  FE.
Example 1
Determine When To Use ASA Congruence
Based on the diagram, can you use the ASA Congruence Postulate to show that the triangles are congruent? Explain your reasoning. a. b.
You can use the ASA Congruence Postulate to show that ∆ABC  ∆DFE.
SOLUTION a.
You are given that C  E, B  F, and BC  FE.
You are given that R  Y and S  X. b.
You know that RT  YZ, but these sides are not included between the congruent angles, so you cannot use the ASA Congruence Postulate. 10

11. Example 2: Applying ASA Congruence
Determine if you can use ASA to prove the triangles congruent. Explain.
Two congruent angle pairs are given, but the included sides are not given as congruent. Therefore ASA cannot be used to prove the triangles congruent.

12. Your Turn Determine if you can use ASA to prove NKL  LMN. Explain.
By the Alternate Interior Angles Theorem. KLN  MNL. NL  LN by the Reflexive Property. No other congruence relationships can be determined, so ASA cannot be applied.

13. Theorem 5.1 (AAS): Angle-Angle-Side Congruence Theorem
If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the triangles are congruent.

14. AAS A D B F C E OR X H Y Z I J

15. Angle-Angle-Side (AAS) Congruence Theorem
Two Angles and One Side that is NOT included

16. Example 3
Determine What Information is Missing
What additional congruence is needed to show that ∆JKL  ∆NML by the AAS Congruence Theorem?
SOLUTION
You are given KL  ML.
Because KLJ and MLN are vertical angles, KLJ  MLN. The angles that make KL and ML the non-included sides are J and N, so you need to know that J  N. 16

17. Use the AAS Congruence Theorem to conclude that ∆EFG  ∆JHG.
Example 4
Decide Whether Triangles are Congruent
Does the diagram give enough information to show that the triangles are congruent? If so, state the postulate or theorem you would use. b. c. a.
SOLUTION
Given a.
EF  JH
E  J
Given
FGE  HGJ
Vertical Angles Theorem
Use the AAS Congruence Theorem to conclude that ∆EFG  ∆JHG. 17

18. Use the ASA Congruence Postulate to conclude that ∆WUZ  ∆ZXW.
Example 4
Decide Whether Triangles are Congruent b. c. b.
Based on the diagram, you know only that MP  QN and NP  NP. You cannot conclude that the triangles are congruent.
UZW  XWZ
Alternate Interior Angles Theorem c.
Reflexive Prop. of Congruence
WZ  WZ
Alternate Interior Angles Theorem
UWZ  XZW
Use the ASA Congruence Postulate to conclude that ∆WUZ  ∆ZXW. 18

19. Example 5:
Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

20. Example 6:
In addition to the angles and segments that are marked, EGF JGH by the Vertical Angles Theorem. Two pairs of corresponding angles and one pair of corresponding sides are congruent. Thus, you can use the AAS Congruence Theorem to prove that ∆EFG  ∆JHG.

21. Example 7:
Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

22. Example 8:
In addition to the congruent segments that are marked, NP  NP. Two pairs of corresponding sides are congruent. This is not enough information (CBD) to prove the triangles are congruent.

23. BD  BC, AD || EC ∆ABD  ∆EBC Prove Triangles are Congruent
Example 9
Prove Triangles are Congruent
A step in the Cat’s Cradle string game creates the triangles shown. Prove that ∆ABD  ∆EBC. A D B C E
SOLUTION
BD  BC, AD || EC
∆ABD  ∆EBC
Statements
Reasons
Given 1.
BD  BC
Given 2.
AD || EC
D  C 3.
Alternate Interior Angles Theorem
Vertical Angles Theorem 4.
ABD  EBC
ASA Congruence Postulate 5.
∆ABD  ∆EBC

24. Your Turn:
Complete the statement: You can use the ASA Congruence Postulate when the congruent sides are between the corresponding congruent angles.
_____ ? 1.
ANSWER
included
Does the diagram give enough information to show that the triangles are congruent? If so, state the postulate or theorem you would use. 2. 3. 4.
ANSWER
yes; AAS Congruence
Theorem
ANSWER no
ANSWER no

25. Ways To Prove Triangles Are Congruent
Congruence Shortcuts
SSS
SAS
ASA
AAS
Ways To Prove Triangles Are Congruent

26. Congruence Shortcuts

27. AAA and SSA???
Does AAA and SSA provide enough information to determine the exact shape and size of a triangle?

28. AAA and SSA???
Does AAA and SSA provide enough information to determine the exact shape and size of a triangle? NO

29. Not Congruence Shortcuts
NO BAD WORDS
SSA
AAA
Do Not prove Triangle Congruence
NO CAR INSURANCE

30. Triangle Congruence Practice
Your Turn

31. Is it possible to prove the Δs are ?( )) ) )) ) (( (( (
No, there is no AAA !
CBD
Yes, ASA

32. Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write cannot be determined (CBD). G I H J K
ΔGIH  ΔJIK by AAS

33. DEF
NLM
by ____
ASA

34. What other pair of angles needs to be marked so that the two triangles are congruent by AAS?

35. What other pair of angles needs to be marked so that the two triangles are congruent by ASA?

36. Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write cannot be determined (CBD). E A C B D
ΔACB  ΔECD by SAS

37. Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write cannot be determined (CBD). J K L M
ΔJMK  ΔLKM by SAS or ASA

38. Cannot Be Determined (CBD)
Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write cannot be determined (CBD). J T L K V U
Cannot Be Determined (CBD)

42. Cannot Be Determined (CBD) – SSA is not a valid Congruence Shortcut.

43. Yes, ∆TNS ≅ ∆UHS by AAS

44. Review
SSS, SAS, ASA, and AAS use corresponding parts to prove triangles congruent. CPCTC uses congruent triangles to prove corresponding parts congruent.
Remember!

45. Example 10: Using CPCTC
A and B are on the edges of a ravine. What is AB?
One angle pair is congruent, because they are vertical angles. Two pairs of sides are congruent, because their lengths are equal.
Therefore the two triangles are congruent by SAS. By CPCTC, the third side pair is congruent, so AB = 18 mi.

46. Your Turn
A landscape architect sets up the triangles shown in the figure to find the distance JK across a pond. What is JK?
One angle pair is congruent, because they are vertical angles.
Two pairs of sides are congruent, because their lengths are equal. Therefore the two triangles are congruent by SAS. By CPCTC, the third side pair is congruent, so JK = 41 ft.

47. Joke Time Which one came first the egg or the chicken?
I don't care I just want my breakfast served.
What do you call a handsome intelligent sensitive man?
A rumor.
What does a clock do when it's hungry?
Goes back 4 secounds!!!

48. Assignment
Pg #1 – 21 odd, 25 – 45 odd