1. Lesson 4-5: Other Methods of Proving Triangles Congruent (page 140)
Essential Questions
Can you construct a proof using congruent triangles?

2. Corresponding Parts of Congruent Triangles are Congruent
Please do not forget …
CPCTC means …
Corresponding Parts of Congruent Triangles are Congruent
Ways to Prove Triangles Congruent:
SSS Postulate
SAS Postulate
ASA Postulate

3. Theorem 4-3
AAS Theorem
If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. A
Given: ∠B ≅∠E
∠C ≅∠F
Prove: ∆ ABC ≅∆ DEF D C B F E

4. ∠C ≅ ∠F A Given: ∠B ≅ ∠E Prove: ∆ ABC ≅ ∆ DEF D C Proof: B F E
Statements Reasons

5. NOTE: Two-column proofs may be shortened by
writing them in paragraph form which
emphasize the key steps in the proof.
Please note that you will be exposed to each type of proof, but you will be responsible to know how to use two-column proofs.
When given a choice, use a two column proof.

6. right sides hypotenuse
RIGHT TRIANGLE: a triangle with one right angle.
HYPOTENUSE: in a right triangle, the side opposite the ___________ angle.
LEGS (of a right triangle): the other two __________.
right
sides A
hypotenuse
leg
hypotenuse C B
leg

7. Theorem 4-4
HL Theorem
If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent. B E A D C F

8. Ways to Prove ANY Two Triangles Congruent:
SSS Postulate
SAS Postulate
ASA Postulate
AAS Theorem
Ways that will NOT Prove Triangles Congruent:
SSA - except for a special case - what is that?
AAA - only works for car issues … flat tire,
dead battery, out of gas, trip directions, etc.

9. Ways to Prove Two RIGHT Triangles Congruent:
(Look at the Classroom Exercises on page 143, #14.)
HL Theorem
LL Method
SAS Postulate can be used to prove the ∆’s congruent.
∴ the LL Method is NOT needed!

10. Ways to Prove Two RIGHT Triangles Congruent:
(Look at the Classroom Exercises on page 143, #14.)
HL Theorem
LL Method
HA Method
AAS theorem can be used to prove the ∆’s congruent.
∴ the HA Method is NOT needed!

11. Ways to Prove Two RIGHT Triangles Congruent:
(Look at the Classroom Exercises on page 143, #14.)
HL Theorem
LL Method
HA Method
LA Method
ASA Post. or AAS Thm can be used to prove the ∆’s congruent.
∴ the LA Method is NOT needed!

12. Ways to Prove ANY Two Triangles Congruent:
SSS Postulate
SAS Postulate
ASA Postulate
AAS Theorem
Ways to Prove Two RIGHT Triangles Congruent:
HL Theorem
LL Method - not needed
HA Method - not needed
LA Method - not needed

13. Example # 1. State which congruence method(s) can be used to prove the triangles congruent. If no method applies, write none.
none
__________

14. Example # 2. State which congruence method(s) can be used to prove the triangles congruent. If no method applies, write none.
HL Thm
__________

15. Also SAS Post and ASA Post. AAS Thm
Example # 3. State which congruence method(s) can be used to prove the triangles congruent. If no method applies, write none.
Also SAS Post and ASA Post.
AAS Thm
__________

16. Look at the non-overlapping ∆’s! Look at the overlapping ∆’s!
Example # 4. State which congruence method(s) can be used to prove the triangles congruent. If no method applies, write none. W Z X Y
Look at the non-overlapping ∆’s!
Look at the overlapping ∆’s!
AAS Thm
HL Thm

17. Look at overlapping ∆’s!
Example # 5. State which congruence method(s) can be used to prove the triangles congruent. If no method applies, write none. C E D A B
Look at overlapping ∆’s!
∠A ≅ ∠C
AE = DC
AAS Thm
__________

18. Example # 6. State which congruence method(s) can be used to prove the triangles congruent. If no method applies, write none. P T S R Q
none
__________

19. Can you construct a proof using congruent triangles?
Assignment
Written Exercises on pages 143 to 145
RECOMMENDED: 5, 6, 7, 10, 12
REQUIRED: 1, 2, 3, 14, 16, 18
Prepare for Quiz on Lessons 4-4 and 4-5
Can you construct a proof using congruent triangles?