1. Congruence, Triangles & CPCTC

2. Congruent triangles have congruent sides and congruent angles.
The parts of congruent triangles that “match” are called corresponding parts.

3. Alt Int Angles are congruent given parallel lines
Overlapping sides are congruent in each triangle by the REFLEXIVE property
Alt Int Angles are congruent given parallel lines
Vertical Angles are congruent

4. The Only Ways To Prove That Triangles Are Congruent
SSS
SAS
ASA
AAS HL
The Only Ways To Prove That Triangles Are Congruent
NO BAD WORDS

5. Before we start…let’s get a few things straightC X Z Y
INCLUDED ANGLE

6. Before we start…let’s get a few things straightC X Z Y
INCLUDED SIDE

7. On the following slides, we will determine if the triangles are congruent. If they are, write a congruency statement explaining why they are congruent. Then, state the postulate (rule) that you used to determine the congruency.

8. P R Q S
ΔPQS  ΔPRS by SAS

9. P S U Q R T
ΔPQR  ΔSTU by SSS

10. Not enough Information to TellS B A C
Not congruent.
Not enough Information to Tell

11. G I H J K
ΔGIH  ΔJIK by AAS

12. J T L K V U
Not possible

13. J K U L
ΔKJL  ΔULM by HL

14. T J K L V U
Not possible

15. Write a proof

16. Write a proof

17. CPCTC is an abbreviation for the phrase “Corresponding Parts of Congruent Triangles are Congruent.”
It can be used as a justification in a proof after you have proven two triangles congruent because by definition, corresponding parts of congruent triangles are congruent.

18. The Basic Idea: Given Information Prove Triangles Congruent CPCTC
SSS
SAS
ASA
AAS HL
Prove Triangles Congruent
CPCTC
Show Corresponding Parts Congruent

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

20. Example A B C L J K
Is ABC  JKL?
YES
What’s the reason?
SAS

21. Example continued ABC  JKL What other angles are congruent?
B  K and C  L
What other side is congruent?
BC  KL

22. Example continued Why? CPCTC ABC  JKL
What other angles are congruent?
B  K and C  L
What other side is congruent?
BC  KL

23. Proofs
1) Ask: to show angles or segments congruent, what triangles must be congruent?
2) Prove triangles congruent, (SSS, SAS, ASA, AAS)
4) CPCTC to show angles or segments are congruent .

24. Example
Given: HJ || LK and JK || HL
Prove: H  K H J K L
Plan: Show JHL  LKJ by ASA, then use CPCTC.
HJL  KLJ
(Alt Int s)
LJ  LJ
(Reflexive)
HLJ  KJL
(Alt Int s)
JHL  LKJ
(ASA)
H  K
(CPCTC)

28. Example 2
Since MS || TR, M  T (Alt. Int. s) M R A
SAM  RAT (Vert. s)
MS  TR (Given) S T
Given: MS || TR and MS  TR
SAM  RAT (AAS)
Prove: A is the midpoint of MT.
MA  AT (CPCTC)
Plan: Show the triangles are congruent using AAS, then MA =AT. By definition, A is the midpoint of segment MT.
A is the midpoint of MT (Def. midpoint)

29. Statements
Reasons
Example
MP bis. LMN (Given) P N L M
NMP  LMP (def.  bis.)
LM  NM (Given)
PM  PM (Ref)
PMN  PML (SAS)
LP  NP (CPCTC)
Given: MP bisects LMN and LM  NM
Prove: LP  NP

30. Given: AB  DC, AD  BC Prove: A  C Statements Reasons A B
3. BD  BD
3. Reflexive D C
4. ABD  CDB
4. SSS
5. A  C
5. CPCTC

31. Show B  E (given) 1. AC  DC (given) 2. A  D (vert s)
Statements
Reasons A B C E D
(given)
1. AC  DC
2. A  D
(given)
3. ACB  DCE
(vert s)
4. ACB  DCE
(ASA)
5. B  E
(CPCTC)