Unit 5 Proof Practice – Key
(Textbook Chapter 6)
1)
Given: ∠A ≅ ∠B
Prove: ∆ACE ~ ∆BCD
2)
Statements
Reasons
1) ∠A ≅ ∠B
1) Given
2) ∠ACE ≅ ∠BCD
2) Reflexive
3) ∆ACE ~ ∆BCD
3) AA Similarity
Given: ∠S ≅ ∠V
Prove: ∆STX ~ ∆VTU
3)
Statements
Reasons
1) ∠S ≅ ∠V
1) Given
2) ∠STX ≅ ∠VTU
2) Vertical Angles are congruent
3) ∆STX ~ ∆VTU
3) AA Similarity
̅̅̅̅||𝐷𝐶
̅̅̅̅
Given: 𝐴𝐵
Prove: ∆ABE ~ ∆DCE
Statements
Reasons
̅̅̅̅||𝐷𝐶
̅̅̅̅
1) 𝐴𝐵
1) Given
2) ∠BAE ≅ ∠CDE
2) Corresponding Angles Congruence Postulate
∠ABE ≅ ∠DCE
3) ∆ABE ~ ∆DCE
3) AA Similarity
** Can also use vertical angles **
4)
̅̅̅̅̅||𝑂𝑃
̅̅̅̅
Given: 𝑀𝑄
Prove: ∆MNQ ~ ∆PNO
Statements
Reasons
̅̅̅̅
1) ̅̅̅̅̅
𝑀𝑄 ||𝑂𝑃
1) Given
2) ∠M ≅ ∠P
2) Alternate Interior Angles Congruence Theorem
3) ∠MNQ ≅ ∠PNO
3) Vertical Angles are Congruent
4) ∆MNQ ~ ∆PNO
3) AA Similarity
** Can also solve using 2 alternate interior angles **
5)
Given: ∆ABC is a right triangle, ̅̅̅̅
𝐵𝐷 ⊥ ̅̅̅̅
𝐴𝐶
Prove: ∆ABC ~ ∆BDC
Statements
Reasons
1) ∆ABC is a right triangle
1) Given
2) ∠ABC is a right angle
2) Definition of right triangle
̅̅̅̅
̅̅̅̅ ⊥ 𝐴𝐶
3) 𝐵𝐷
3) Given
4) ∠BDC is a right angle
4) Definition of perpendicular
5) ∠ABC ≅ ∠BDC
5) All right angles are congruent
6) ∠C ≅ ∠C
6) Reflexive Property
7) ∆ABC ~ ∆ADB
7) AA Similarity
6)
̅̅̅̅
Given: ̅̅̅
𝐼𝐸 ||𝑉𝑂
𝐼𝐷
𝐸𝐷
Prove: 𝐷𝑉 = 𝐸𝑂
Statements
Reasons
̅̅̅̅
̅̅̅ ||𝑉𝑂
1) 𝐼𝐸
1) Given
2) ∠DIE ≅ ∠DVO, ∠DEI ≅ ∠DOV
2) Corresponding Angles Congruence Postulate
3) ∆DIE ~ ∆DVO
3) AA Similarity
𝐼𝐷
𝐸𝐷
4) 𝐼𝑉 = 𝐸𝑂
7)
4) Corresponding sides in similar triangles are proportional
̅̅̅̅||𝐷𝐸
̅̅̅̅
Given: 𝐴𝐵
𝐷𝐶
𝐷𝐸
Prove: 𝐵𝐶 = 𝐵𝐴
Statements
Reasons
̅̅̅̅||𝐷𝐸
̅̅̅̅
1) 𝐴𝐵
1) Given
2) ∠D ≅ ∠B ∠E ≅ ∠A
2) Alternate Interior Angles Congruence Theorem
3) ∆DCE ~ ∆BCA
3) AA Similarity
𝐷𝐶
𝐷𝐸
4) 𝐵𝐶 = 𝐵𝐴
4) Corresponding sides in similar triangles are proportional
** Can also use vertical angles **
8)
̅̅̅̅||𝐷𝐶
̅̅̅̅
Given: 𝐴𝐵
𝐴𝐸
𝐶𝐸
Prove: 𝐴𝐵 = 𝐶𝐷
Statements
Reasons
̅̅̅̅
1) ̅̅̅̅
𝐴𝐵 ||𝐷𝐶
1) Given
2) ∠BAC ≅ ∠ACD
2) Alternate Interior Angles Congruence Theorem
3) ∠BEA ≅ ∠CED
3) Vertical Angles are Congruent
4) ∆AEB ~ ∆CED
4) AA Similarity
𝐴𝐸
𝐶𝐸
5) 𝐴𝐵 = 𝐶𝐷
5) Corresponding sides in similar triangles are proportional
9)
Given: ∆PQR and ∆UTS
Prove: ∆PQR ~ ∆UTS
Statements
Reasons
1) PQ = 2.1, QR =1.8, RP = 3.6
1) Given
UT = 3.5, TS = 3, RS = 6
𝑃𝑄
2.1
3
=
1.8
=5
=
3.6
3) 𝑈𝑇 =
𝑄𝑅
2) 𝑈𝑇 = 3.5 = 5
𝑄𝑅
𝑇𝑆
𝑅𝑃
𝑅𝑆
𝑃𝑄
3
6
𝑇𝑆
3
3
=5
=
𝑅𝑃
𝑅𝑆
4) ∆PQR ~ ∆UTS
10)
2) Division
3) Substitution
4) SSS Similarity
Given: ∆ABC and ∆DEF are isosceles, ∠C ≅ ∠F
Prove: ∆ABC ~ ∆DEF
Statements
Reasons
1) ∆ABC and ∆DEF are isosceles
1) Given
̅̅̅̅, ̅̅̅̅
2) ̅̅̅̅
𝐶𝐴 ≅ 𝐶𝐵
𝐷𝐹 ≅ ̅̅̅̅
𝐸𝐹
2) Definition of isosceles triangle
3) CA = CB, DF = EF
3) Definition of congruent
𝐶𝐴
𝐵𝐵
4) 𝐷𝐹 = 𝐷𝐹
5)
𝐶𝐴
𝐷𝐹
=
𝐵𝐵
𝐸𝐹
4) Division
5) Substitution
6) ∠C ≅ ∠F
6) Given
7) ∆ABC ~ ∆DEF
7) SAS Similarity
11)
Given: AD = 6, DC = 3, BE = 4, EC = 2
Prove: ∆CDE ~ ∆CAB
Statements
Reasons
1) AD = 6, DC = 3, BE = 4, EC = 2
1) Given
2) AC = AD + DC, BC = BE + EC
2) Segment Addition Postulate
3) AC = 6 + 3, BC = 4 + 2
3) Substitution
4) AC = 9, BC = 6
4) Simplify
𝐶𝐷
3
𝐶𝐷
𝐶𝐸
1 𝐶𝐸
2
1
5) 𝐶𝐴 = 9 = 3, 𝐶𝐵 = 6 = 3
12)
5) Division
6) 𝐶𝐴 = 𝐶𝐵
6) Transitive Property
7) ∠DCE ≅ ∠ACB
7) Reflexive Property
8) ∆CDE ~ ∆CAB
8) SAS Similarity
Given: ∆ABC is isosceles with base ̅̅̅̅
𝐴𝐶
∠1 and ∠2 are right angles
Prove: AE ∙ BD=CE ∙ DC
Statements
Reasons
̅̅̅̅
1) ∆ABC is isosceles with base 𝐴𝐶
1) Given
2) ∠BAC ≅ ∠BCA
2) Base Angles Theorem
3) ∠1 and ∠2 are right angles
3) Given
4) ∠1 ≅ ∠2
4) All right angles are congruent
5) ∆AEC ~ ∆CDB
5) AA Similarity
𝐴𝐸
𝐶𝐸
6) 𝐷𝐶 = 𝐵𝐷
6) Corresponding sides of similar triangles are proportional
7) AE ∙ BD=CE ∙ DC
7) Multiplication
13)
Given: ∆ACE and ∆AFE are isosceles with base ̅̅̅̅
𝐴𝐸
Prove: ∆ABF ~ ∆EDF
Statements
Reasons
1) ∆ACE and ∆AFE are isosceles with base ̅̅̅̅
𝐴𝐸
1) Given
2) ∠CAE ≅ ∠CEA, ∠FAE ≅ ∠FEA
2) Base Angles Theorem
3) ∠CAE = ∠CEA, ∠FAE = ∠FEA
3) Definition of congruent
4) ∠CAE = ∠CAF + ∠FAE
4) Angle Addition Postulate
∠CEA = ∠CEF + ∠FEA
14)
5) ∠CAF + ∠FAE = ∠CEF + ∠FEA
5) Substitution
6) ∠CAF + ∠FAE = ∠CEF + ∠FAE
6) Substitution
7) ∠CAF = ∠CEF
7) Subtraction
8) ∠CAF ≅ ∠CEF
8) Definition of congruent
9) ∠AFB ≅ ∠EFD
9) Vertical angles are congruent
10) ∆ABF ~ ∆EDF
10) AA Similarity
Given: PQ = 6, QR = 6. SR = 4, ∠P ≅ ∠SQR
Prove: TS = 4
Statements
Reasons
1) ∠P ≅ ∠SQR
1) Given
2) ∠R ≅ ∠R
2) Reflexive Property
3) ∆PRT ~ ∆QRS
3) AA Similarity
𝑃𝑅
4) 𝑄𝑅 =
𝑇𝑅
𝑆𝑅
4) Corresponding sides of similar triangles are proportional
5) PQ = 6, QR = 6, SR = 4
5) Given
6) PR = PQ + QR
6) Segment Addition Postulate
7) PR = 6 + 6 = 12
7) Substitution
8)
12
6
=
𝑇𝑅
4
8) Substitution
9) TR = 8
9) Multiplication
10) TR = TS + SR
10) Segment Addition Postulate
11) 8 = TS + 4
11) Substitution
12) TS = 4
12) Subtraction
15)
̅̅̅̅||𝐷𝐸
̅̅̅̅ , DC =5, EC = 10, AC = 12
Given: 𝐴𝐵
Prove: BC = 24
Statements
Reasons
̅̅̅̅
1) ̅̅̅̅
𝐴𝐵 ||𝐷𝐸
1) Given
𝐴𝐶
𝐵𝐶
2) 𝐷𝐶 = 𝐸𝐶
2) Triangle Proportionality Theorem
3) DC =5, EC = 10, AC = 12
3) Given
4)
12
5
=
𝐵𝐶
10
4) Substitution
5) 5 ∙ BC = 120
5) Multiplication
6) BC = 24
6) Division
** Can also prove the triangles are congruent by AA instead of Triangle Proportionality Theorem **
Grading Scale
0 incorrect = 10
1 incorrect = 9
2 – 3 incorrect = 8
4 incorrect = 7
5 – 6 incorrect = 6
7 incorrect = 5