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Name ___________________________
UNIT 2 REVIEW: Proofs
DIRECTIONS: Justify each statement with a definition, property, postulate, or theorem.
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1) ∠5 ≅ ∠5
1)
2) If BF ≅ BE and BE ≅ BD , then BF ≅ BD .
2)
3) If B is the midpoint of AC , then AB ≅ BC
3)
€ 4) If ∠6 ≅€∠5 , then BD bisects
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∠EBC .
4)
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5)
€
5) ∠ABD and ∠5 are supplementary.
€m∠EBC = 90
€ 6) If m∠7 +
€ , then ∠7 and ∠EBC are
complementary.
€
€
€ 7) If AB=BC, then BC=AB.
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€
€
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6)
7)
8) If m∠5 = m∠6 , then 2 • m∠5 = 2 • m∠6
8)
9) AB + BC = AC
9)
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10) m∠8 + m∠7 = m∠ABE
10)
1
x −5
2
Prove: x = 30
11) Given: 10 =
Statements:
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Reasons:
€
€
1
x −5
2
1)
10 =
2)
#1
&
20 = 2% x − 5(
$2
'
3)
20 = x −10
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4)
30 = x
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5)
x = 30
1) ______________________________________
2) ______________________________________
3) ______________________________________
4) ______________________________________
5) ______________________________________
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€
12) Given: m∠CDE = x , m∠EDF = 3x + 20
Prove: x = 40
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Statements:
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Reasons:
€
1)
m∠CDE = x , m∠EDF = 3x + 20
1) ______________________________________
2)
∠CDE and ∠EDF are supplementary
2) ______________________________________
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3) m ∠CDE
€ + m ∠EDF = 180
3) ______________________________________
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4)
x€
+ (3x + 20) = 180
4) ______________________________________
5)
4 x€+ 20 = 180
5) ______________________________________
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€
6)
4 x = 160
6) ______________________________________
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7)
x = 40
7) ______________________________________
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13) Given: C is the midpoint of AD
AC = 4 x , and CD = 2x +12
Prove: x = 6
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€
€
Statements:
€
1)
€
2)
C is the midpoint of AD ,
AC = 4 x , and CD = 2x +12
AC ≅ CD
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€
3) AC = CD
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€
4)
4 x = 2x +12
5)
2x = 12
6)
x =6
Reasons:
1) ______________________________________
2) ______________________________________
3) ______________________________________
4) ______________________________________
5) ______________________________________
6) ______________________________________
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14) Given: m∠AOC = m∠BOD
Prove: m∠AOB = m∠COD
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Statements
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1. m∠AOC = m∠BOD
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m∠AOB + m∠BOC = m∠AOC
2.
m∠BOC + m∠COD = m∠BOD
Reasons
1. ______________________________
2. ______________________________
3.
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Substitution Property of Equality
3.
4. ______________________________
4. m∠BOC = m∠BOC
5. ______________________________
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€
5. m∠AOB = m∠COD
15) Given: ∠1 and ∠2 are complementary
∠1 and ∠3 are complementary
€ Prove:
€ ∠2 ≅ ∠4
€
€
Statements:
€
Reasons:
16) Given: MI=LD
Prove: ML=ID
Statements:
Reasons:
17) Given: ∠2 ≅ ∠6
Prove: ∠4 ≅ ∠7
Statements:
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€
Reasons:
18) Complete the proof the Congruent Supplements Theorem using definitions, properties,
and postulates. (You can’t use the theorem in the proof of the theorem!)
Given: ∠3 and ∠1 are supplementary
∠3 and ∠2 are supplementary
Prove: ∠1 ≅ ∠2
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€
€ Statements:
€
€
€
€
€
€
1.
∠3 and ∠1 are supplementary
∠3 and ∠2 are supplementary
Reasons:
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
m∠3 + m∠1 = 180
€ m∠3 + m∠2 = 180
€
UNIT 2 REVIEW ANSWERS
1) Reflexive Property of Equality
2) Transitive Property of Congruence
3) Definition of Midpoint
4) Definition of Angle Bisector
5) Linear Pair Postulate
6) Definition of Complementary Angles
7) Symmetric Property of Equality
8) Multiplication Property of Equality
9) Segment Addition Postulate
10) Angle Addition Postulate
11)
Statements:
Reasons:
1
x −5
2
#1
&
2) 20 = 2% x − 5(
$2
'
1) Given
1)
€
€
€
12)€
€
€
€
€
€
€
13)
€
€
3)
10 =
20 = x −10
4)
30 = x
5)
x = 30
Statements:
€
5) Symmetric Property of Equality
Reasons:
1) Given
2)
∠CDE and ∠EDF are supplementary
2) Linear Pair Postulate
3) m ∠CDE + m ∠EDF = 180
3) Definition of Supplementary Angles
4)
€
x + (3x + 20) = 180
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5) 4 x + 20 = 180
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6) 4 x = 160
4) Substitution Property of Equality
7)
7) Division Property of Equality
x = 40
Statements:
C is the midpoint of AD ,
3) AC = CD
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4) Addition Property of Equality
m∠CDE = x , m∠EDF = 3x + 20
AC = 4 x , and CD = 2x +12
2) AC ≅ CD
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€
3) Distributive Property
1)
1)
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€
2) Multiplication Property of Equality
5) Simplify / Combine Like Terms
6) Subtraction Property of Equality
Reasons:
1) Given
2) Definition of Midpoint
3) Definition of Congruent Segments
4)
€ +12
4 x = 2x
4) Substitution Property of Equality
5)
2x = 12
5) Subtraction Property of Equality
6)
x =6
6) Division Property of Equality
14)
Statements
1. m∠AOC = m∠BOD
2.
€
€
3.
m∠AOB + m∠BOC = m∠BOC + m∠COD
4.
m∠BOC = m∠BOC
5.
€
m∠AOB + m∠BOC = m∠AOC
m∠BOC + m∠COD = m∠BOD
m∠AOB = m∠COD
Reasons
1.
Given
2.
Angle Addition Postulate
3.
Substitution Property of Equality
4.
Reflexive Property of Equality
5.
Subtraction Property of Equality
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€
15)
€
€
€
Statements:
Reasons:
1.
∠1 and ∠2 are complementary
∠1 and ∠3 are complementary
1.
2.
∠2 ≅ ∠3
2. Complements of the same angle are
congruent.
3.
∠3 ≅ ∠4
3. Vertical angles are congruent.
4.
∠2 ≅ ∠4
4. Transitive Property of Congruence
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€
Given
€
€
16)
Statements:
1.
MI = LD
Reasons:
1.
Given
2. IL = IL
2. Reflexive Property of Equality
3. MI+IL = LD + IL
3. Addition Property of Equality
4. MI + IL =ML
4. Segment Addition Postulate
LD + IL = ID
5. ML = ID
5. Substitution Property of Equality
17)
Statements:
Reasons:
1.
∠2 ≅ ∠6
1. Given
2.
∠2 ≅ ∠4
2. Vertical angles are congruent.
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3.
∠4 ≅ ∠6
3. Transitive Property of Congruence
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4.
∠6 ≅ ∠7
4. Vertical angles are congruent.
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5.
∠4 ≅ ∠7
5. Transitive Property of Congruence.
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€
18)
Statements:
Reasons:
1.
1. Given
∠3 and ∠1 are supplementary
∠3 and ∠2 are supplementary
2.
€
€
€
€
m∠3 + m∠1 = 180
€ m∠3 + m∠2 = 180
€
3.
m∠3 + m∠1 = m∠3 + m∠2
4.
m∠3 = m∠3
5.
m∠1 = m∠2
6.
∠1 ≅ ∠2
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€
€
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2. Definition of Supplementary Angles
3. Substitution Property of Equality
4. Reflexive Property of Equality
5. Subtraction Property of Equality
6. Definition of Congruent Angles