Congruence, Triangles & CPCTC

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

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

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

Before we start…let’s get a few things straight C X Z Y INCLUDED ANGLE

Before we start…let’s get a few things straight C X Z Y INCLUDED SIDE

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.

P R Q S ΔPQS  ΔPRS by SAS

P S U Q R T ΔPQR  ΔSTU by SSS

Not enough Information to Tell S B A C Not congruent. Not enough Information to Tell

G I H J K ΔGIH  ΔJIK by AAS

J T L K V U Not possible

J K U L ΔKJL  ΔULM by HL

T J K L V U Not possible

Write a proof

Write a proof

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.

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

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!

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

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

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

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 .

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)

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)

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

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

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)