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Congruence, Triangles & CPCTC
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Congruent triangles have congruent sides and congruent angles.
The parts of congruent triangles that “match” are called corresponding parts.
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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
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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
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Before we start…let’s get a few things straight
C X Z Y INCLUDED ANGLE
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Before we start…let’s get a few things straight
C X Z Y INCLUDED SIDE
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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.
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P R Q S ΔPQS ΔPRS by SAS
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P S U Q R T ΔPQR ΔSTU by SSS
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Not enough Information to Tell
S B A C Not congruent. Not enough Information to Tell
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G I H J K ΔGIH ΔJIK by AAS
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J T L K V U Not possible
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J K U L ΔKJL ΔULM by HL
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T J K L V U Not possible
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Write a proof
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Write a proof
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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.
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The Basic Idea: Given Information Prove Triangles Congruent CPCTC
SSS SAS ASA AAS HL Prove Triangles Congruent CPCTC Show Corresponding Parts Congruent
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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!
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Example A B C L J K Is ABC JKL? YES What’s the reason? SAS
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Example continued ABC JKL What other angles are congruent?
B K and C L What other side is congruent? BC KL
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Example continued Why? CPCTC ABC JKL
What other angles are congruent? B K and C L What other side is congruent? BC KL
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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 .
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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)
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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)
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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
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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
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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)
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