Name ___________________________

UNIT 2 REVIEW: Proofs

DIRECTIONS: Justify each statement with a definition, property, postulate, or theorem.

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

€

∠EBC . 4)

5)

€

∠ABD and

€

∠5 are supplementary. 5)

6) If

€

m∠7 +m∠EBC = 90 , then

€

∠7 and

€

∠EBC are complementary. 6)

7) If AB=BC, then BC=AB. 7)

8) If

€

m∠5 = m∠6 , then

€

2 • m∠5 = 2 • m∠6 8)

9) AB + BC = AC 9)

10)

€

m∠8 +m∠7 = m∠ABE 10)

11) Given:

€

10 =12x − 5

Prove:

€

x = 30

Statements: Reasons:

1)

€

10 =12x − 5

2)

€

20 = 2 12x − 5

#

$ %

&

' (

3)

€

20 = x −10

4)

€

30 = x

5)

€

x = 30

1) ______________________________________

2) ______________________________________

3) ______________________________________

4) ______________________________________

5) ______________________________________

12) Given:

€

m∠CDE = x ,

€

m∠EDF = 3x + 20 Prove:

€

x = 40

Statements: Reasons:

1)

€

m∠CDE = x ,

€

m∠EDF = 3x + 20

2)

€

∠CDE and

€

∠EDF are supplementary

3) m

€

∠CDE + m

€

∠EDF = 180

4)

€

x + (3x + 20) =180

5)

€

4x + 20 =180

6)

€

4x =160

7)

€

x = 40

1) ______________________________________

2) ______________________________________

3) ______________________________________

4) ______________________________________

5) ______________________________________

6) ______________________________________

7) ______________________________________

13) Given: C is the midpoint of

€

AD

€

AC = 4x , and

€

CD = 2x +12 Prove:

€

x = 6

Statements: Reasons:

1) C is the midpoint of

€

AD,

€

AC = 4x, and

€

CD = 2x +12 2)

€

AC ≅CD

3) AC = CD

4)

€

4x = 2x +12

5)

€

2x =12

6)

€

x = 6

1) ______________________________________

2) ______________________________________

3) ______________________________________

4) ______________________________________

5) ______________________________________

6) ______________________________________

14) Given:

€

m∠AOC = m∠BOD Prove:

€

m∠AOB = m∠COD

Statements Reasons 1.

€

m∠AOC = m∠BOD

2.

€

m∠AOB +m∠BOC = m∠AOCm∠BOC +m∠COD = m∠BOD

3.

4.

€

m∠BOC = m∠BOC

5.

€

m∠AOB = m∠COD

1. ______________________________

2. ______________________________

3. Substitution Property of Equality

4. ______________________________

5. ______________________________

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: 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

Statements: Reasons:

1.

€

∠3 and

€

∠1 are supplementary

€

∠3 and

€

∠2 are supplementary

2.

€

m∠3+m∠1 =180

€

m∠3+m∠2 =180

3.

4.

5.

6.

1.

2. 3.

4.

5.

6.

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)

€

10 =12x − 5

2)

€

20 = 2 12x − 5

#

$ %

&

' (

3)

€

20 = x −10 4)

€

30 = x

5)

€

x = 30

1) Given

2) Multiplication Property of Equality

3) Distributive Property

4) Addition Property of Equality

5) Symmetric Property of Equality

12)

Statements: Reasons:

1)

€

m∠CDE = x ,

€

m∠EDF = 3x + 20

2)

€

∠CDE and

€

∠EDF are supplementary

3) m

€

∠CDE + m

€

∠EDF = 180

4)

€

x + (3x + 20) =180

5)

€

4x + 20 =180

6)

€

4x =160

7)

€

x = 40

1) Given

2) Linear Pair Postulate

3) Definition of Supplementary Angles

4) Substitution Property of Equality

5) Simplify / Combine Like Terms

6) Subtraction Property of Equality

7) Division Property of Equality

13) Statements: Reasons:

1) C is the midpoint of

€

AD,

€

AC = 4x, and

€

CD = 2x +12 2)

€

AC ≅CD

3) AC = CD

4)

€

4x = 2x +12

5)

€

2x =12

6)

€

x = 6

1) Given

2) Definition of Midpoint

3) Definition of Congruent Segments

4) Substitution Property of Equality

5) Subtraction Property of Equality

6) Division Property of Equality

14) Statements Reasons

1.

€

m∠AOC = m∠BOD

2.

€

m∠AOB +m∠BOC = m∠AOCm∠BOC +m∠COD = m∠BOD

3.

€

m∠AOB +m∠BOC = m∠BOC +m∠COD

4.

€

m∠BOC = m∠BOC

5.

€

m∠AOB = m∠COD

1. Given

2. Angle Addition Postulate

3. Substitution Property of Equality

4. Reflexive Property of Equality

5. Subtraction Property of Equality

15)

Statements: Reasons:

1.

€

∠1 and

€

∠2 are complementary

€

∠1 and

€

∠3 are complementary 2.

€

∠2 ≅∠3

3.

€

∠3 ≅∠4

4.

€

∠2 ≅∠4

1. Given

2. Complements of the same angle are congruent.

3. Vertical angles are congruent.

4. Transitive Property of Congruence

16)

Statements: Reasons:

1. MI = LD

2. IL = IL

3. MI+IL = LD + IL

4. MI + IL =ML

LD + IL = ID

5. ML = ID

1. Given

2. Reflexive Property of Equality

3. Addition Property of Equality

4. Segment Addition Postulate

5. Substitution Property of Equality

17)

Statements: Reasons:

1.

€

∠2 ≅∠6

2.

€

∠2 ≅∠4

3.

€

∠4 ≅∠6

4.

€

∠6 ≅∠7

5.

€

∠4 ≅∠7

1. Given

2. Vertical angles are congruent.

3. Transitive Property of Congruence

4. Vertical angles are congruent.

5. Transitive Property of Congruence.

18) Statements: Reasons:

1.

€

∠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

1. Given

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