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Proving Triangles Congruent NOTES
From yesterday, you learned that you only need 3 pieces of information (combination of angles and sides) to determine if two triangles are congruent. Today, we are going to prove two triangles are congruent using two column proofs.
Steps for triangle congruence proofs:
1. Write the ‘givens.’
2. If your given is not already a ____________________________________, use it to get to one.
(Think…what does that given mean? What is the definition of that word?)
3. Once you have gotten down to congruence statements, check your triangles to see if
you have enough information (3 pieces) to prove that they are __________________.
a. If you do not have enough information , see if you can add
_______________________________ or __________________________________________.
4. State that the triangles are congruent and provide the reason (ASA, SAS, SSS, AAS, HL).
********And always remember… IF YOU WRITE IT, MARK IT!!!! **********
Practice:
1. Given: BC DC and AC EC
Prove: BCA DCE
Statement Reason
Common Statements You Will See/Use in Proofs:
If…. Then… Because…
P is the midpoint of AE
TU bisects STG.
ED XY
XF GH
GTS is isosceles with legs TG and TS.
5. Given: AC BD and AD BC
Prove: ABD BAC
6. Given: GH IH and GF IH
Prove: ABD BAC
7. Given: SR TU and S U
Prove: SRT UTR
Statement Reason
- AC BD 1.^ Given
- AD BC 2. Given
- AB AB 3.^ Reflective Property
- ABD BAC 4.Side-Side-Side (SSS)
Statement Reason
Statement Reason
Name __________________________________ Date _____________
Triangle Congruence Proofs Practice
1. Given: C is the midpoint of BE AND AD.
Prove: ABC DEC
2. Given: BC DA and AC bisects BCD
Prove: ABC CDA
3. Given: X H and XG FH
Prove: XGF HFG
Statement Reason
1. C is midpoint of BE and AD 1.
2. BC EC 2.^ definition of a midpoint
- definition of a midpoint
- ABC DEC 5.
Statement Reason
2. AC bisects BCD 2.^ Given
3. ________ ________
4. AC^ AC 4.
Statement Reason
- Given
- Given
- FGX GFH 3.
Fill in the blank proofs:
Problem 5: Statement Reason
1. I K 1.^ Given
2. IHJ KJH 2. Given
3. HJ HJ 3.
4. HJK JHI 4.
Problem 6: Statement Reason
1. MLN ONL 1. Given
2. OLN _____ 2. Given
- Reflexive Property
4. LNO NLM 4.
Problem 7: Statement Reason
1. PQ QS 1.^ Given
- Given
3. PQT RQS 3.
4. PQT SQR 4.
Problem 8: Statement Reason
1. UV UX 1.^ Given
- VWU XWU 2.^ Given
- Reflexive Property
- V X 4.
5. UWV UWX 5..
Problem 9: Statement Reason
1. Y C 1.
- Given
- (^) 3. Vertical Angles are (^)
4. YZA CBA 4.
R
Create your own!
A. Given: AB CD , BC AD Prove: ABC CDA?
B.Given : AB^ CD and AE^ CE
Pr ove : ABE CDE?
C.
Given : AB BD BCA and BCD are right angles
Pr ove : ABC DBC?
CPCTC
Once you conclude two triangles are congruent, then you can also conclude that corresponding parts of congruent triangles are congruent (CPCTC). CPCTC can be used as a justification after you have proven two triangles are congruent. Look at the example proof below:
Statements Reasons
- SU SK 1. Given
- SR SH 2. Given
- S S 3. Reflexive Property
- (^) SUH SKR 4. SAS
- U K 5. CPCTC
Prove the Isosceles Base Angles Theorem
Given: GB GD, GU bis ec ts DGB Prove:B D
Prove the Converse of the Isosceles Base Angles Theorem: Given: B D, GU bis ec ts DGB
Prove:GB^ GD
Statements Reasons
- GB GD 1. Given
- _________________________ 2. _________________________
- _________________________ 3. Definition of bisects
- GU GU 4. _________________________
- _________________________ 5. _________________________
- B D 6. _________________________
Statements Reasons
- (^) B D 1. Given
- _________________________ 2. _________________________
- _________________________ 3. _________________________
- GU GU 4. _________________________
- _________________________ 5. _________________________
- GB^ GD 6. _________________________
Proof: Given:JHK LHK, JKH LKH
Prove:JK LK
Proof: Given:CW and SD bi sect each other
Prove:CS WD
Proof: Given: AB CB ,D is the midpoint of AC Prove: A C
