Int. Geometry Unit 2 Test Review 1

7

11109 8

65

4321

C F

B

D

A

E

15x - 35( )°

115°

2x( )°10x - 40( )°

5x - 3( )°

3x - 3( )°

l

m5x + 15( )°

10x - 60( )°

l

m

7x + 6( )°3x + 64( )°

l

m

2x( )°3x - 10( )°

2x + 11( )°

Directions 1-5: Use the diagram to determine if the angles are vertical, adjacent, supplementary, complementary, or a linear pair. Write all that apply.

1. 1∠ and 2∠ ________________________________ 2. 5∠ and 6∠ ________________________________ 3. 8∠ and 11∠ _______________________________ 4. 3∠ and 4∠ ________________________________ 5. 9∠ and 10∠ ________________________________ Questions 6-8: Solve for x 6. 7. 8. Directions 9-11: Solve for x and determine if l is perpendicular to m. 9. 10. 11.

Int. Geometry Unit 2 Test Review 2

43

21RO

T

Y

S

V

Directions 12: C∠ and D∠ are complementary. Complete the following. 12. 2m C x∠ = ; 4 18m D x∠ = − . x = _________ _______m C∠ = m D∠ = __________ Directions 13: C∠ and D∠ are supplementary. Complete the following. 13. 5 10m C x∠ = − ; 7 34m D x∠ = + . x = _________ _______m C∠ = m D∠ = __________ 14. The supplement of an angle is 12 less than three times the angle. Find the angle,

its complement and its supplement. Questions 15-18: Given OT YS⊥ . Use the given drawing and the given information to

find the value of x for each case. 15. If 1 3m x∠ = ; 4 30m x∠ = + 16. If 3 2 20m x∠ = + ; 4 10m x∠ = + 17. If 1 1m x∠ = + ; 4 6m VOS x∠ = − 18. If 4m y∠ = , then find the measure of YOR∠ in terms of y. 19. Two times the supplement of an angle is five times the complement of the angle.

Find the measure of the angle.

Questions 15-18

Int. Geometry Unit 2 Test Review 3

21

4

3

O

A

C

Questions 20-29: Complete with sometimes, always, or never.

20. Right angles are ________________ vertical angles

21. Angles A, B, and C are ______________ complementary.

22. Vertical angles __________________ have a common supplement.

23. Perpendicular lines _______________ lie in the same plane.

24. Perpendicular lines _______________ form 60 angles.

25. If a pair of vertical angles are supplementary, then the lines forming the angles are _______________ perpendicular.

26. A theorem is _________________ a true statement.

27. Supplements of congruent angles are ________________ congruent.

28. A statement in a proof ________________ has a reason.

29. Two angles are _______________ supplements.

30. Point E is the midpoint of DF . 3 10DE x= + and 11 30DF x= − . Find EF. Questions 31-40: Write a proof for the statements. 31. Given: 3∠ supplements 1∠ 4∠ supplements 2∠ Prove: 3 4∠ ≅ ∠ 32. Given: 1∠ complements 2∠ Prove: AO CO⊥

1 2

Int. Geometry Unit 2 Test Review 4

321

P

Q R S

T

21

B

A C

D

8765

4321

m

8765

4321

lk

4321

B

A

C

D

E

F

33. Given: 1∠ complements 2∠ 1∠ complements 3∠ Prove: PS bisects RPT∠ 34. Given: 1 2m m∠ ≠ ∠ Prove: BC does not bisect ABD∠ 35. Given: 1 5m m∠ ≠ ∠ Prove: 2 7m m∠ ≠ ∠ 36. Given: 1 2m m∠ ≠ ∠ Prove: l is not perpendicular to m 37. Given: 2∠ and 4∠ are supplementary Prove: 1 3∠ ≅ ∠ 38. Given: ABD FBE∠ ≅ ∠ Prove: EBF∠ and DBC∠ are supplements

Int. Geometry Unit 2 Test Review 5

A E

D C

B

A B

C

D

39. Given: CBE∠ and ADC∠ are supplements Prove: ADC ABC∠ ≅ ∠ 40. Given: AC CB⊥ ; DAC∠ and ACD∠ are complementary Prove: BCD DAC∠ ≅ ∠ Selected Answers: 1. Complementary, Adjacent 2. Adjacent 3. Vertical 4. Adjacent, Supplementary, Linear Pair 5. Adjacent 6. 10x = 7. 55

3x = 8. 12x =

9. 15x = ; l m⊥ 10. 11x = ; is not to l m⊥ 11. 21x = ; is not to l m⊥ 12. 18; 36 ; 54x m C m D= ∠ = ∠ = 13. 13; 55 ; 125x m C m D= ∠ = ∠ = 14. Angle 48 ; Comp 42 ; Supp 132= = = 15. 15x = 16. 20x = 17. 37x = 18. ( )180 y−

19. Angle 30= 20. Sometimes 21. Never 22. Always 23. Always 24. Never 25. Always 26. Always 27. Always 28. Always 29. Sometimes 30. 40EF = Note with the proofs, there are multiple solutions to these problems.

31.

Statement Reason Relies on/uses to reach 1. 1 2∠ ≅ ∠ 1. Vertical angles are

always congruent. Diagram

2. 1∠ supplements 3∠ ; 2∠ supplements 4∠

2. Given

3. 3 4∠ ≅ ∠ 3. Supplements of congruent angles are also congruent

2 and 3

Int. Geometry Unit 2 Test Review 6

32.

33. 34. Temporarily assume BC bisects ABD∠ By the definition of an angle bisector 1 2∠ ≅ ∠ . This contradicts the given information that 1 2m m∠ ≠ ∠ , therefore our assumption

must be false and BC does not bisect ABD∠ . 35. Temporarily assume 2 7m m∠ = ∠ . Using the Angle Addition Postulate 1 2 180m m∠ + ∠ = and 5 7 180m m∠ + ∠ = . By substitution 1 2 5 7m m m m∠ + ∠ = ∠ + ∠ We are assuming 2 7m m∠ = ∠ , so by subtraction 1 5m m∠ = ∠ . This contradicts the given information 1 5m m∠ ≠ ∠ , therefore our assumption must

be false and 2 7m m∠ ≠ ∠ 36. Temporarily assume l m⊥ . This means 1∠ and 2∠ are both right angles and 90 . 1 2∠ ≅ ∠ because they have equal measures. This contradicts the given information 1 2m m∠ ≠ ∠ , therefore our assumption must

be false and l is not perpendicular to m

Statement Reason Relies on/uses to reach 1. 1∠ complements 2∠ 1. Given 2. 1 2 90m m∠ + ∠ = 2. Definition of

Complementary Angles 1

3. 1 2m m m AOC∠ + ∠ = ∠ 3. Angle Addition Postulate Diagram 4. 90m AOC∠ = 4. Substitution 2 and 3 5. AOC∠ a right angle 5. Definition of a Right

Angle 4

6. AO OC⊥ 6. Definition of Perpendicular Lines ( two lines intersect to form a right angle then they must be perpendicular)

5

Statement Reason Relies on/uses to reach 1. 1∠ complements 2∠ 1∠ complements 3∠ ,

1. Given

2. 2 3∠ ≅ ∠ 2. Angles complementary to the same angle are congruent. 1

3. PS bisects RPT∠ 3. Definition of an Angle Bisector (it creates two adjacent congruent angles)

2

Int. Geometry Unit 2 Test Review 7

37.

38. 39.

Statement Reason Relies on/uses to reach 1. 2∠ and 4∠ are supplementary

1. Given

2. 2 4 180om m∠ + ∠ = 2. Definition of Supplementary Angles 1

3. 3 4 180om m∠ + ∠ = 3. Angle Addition Postulate Diagram

4. 2 4 3 4m m m m∠ + ∠ = ∠ + ∠ 4. Substitution 2 and 3 5. 4 4m m∠ = ∠ 5. Reflexive PoE Diagram 6. 2 3m m∠ = ∠ 6. Subtraction PoE 4 and 5 7. 1 2m m∠ = ∠ 7. Vertical angles are

congruent Diagram

8. 1 3m m∠ = ∠ 8. Transitivity PoE (or substitution poe) 6 and 7

Statement Reason Relies on/uses to reach 1. 180om ABD m DBC∠ + ∠ = 1. Angle Addition

Postulate Diagram

2. m ABD m FBE∠ = ∠ 2. Given 3. 180om FBE m DBC∠ + ∠ = 3. Substitution PoE 1 and 2 4. FBE∠ and DBC∠ are supplements.

4. Definition of Supplementary Angles 3

Statement Reason Relies on/uses to reach 1. CBE∠ and ADC∠ are supplements

1. Given

2. 180om CBE m ADC∠ + ∠ = 2. Definition of Supplementary Angles 1

3. 180om CBE m ABC∠ + ∠ = 3. Angle Addition Postulate Diagram

4. m CBE m ADC m CBE m ABC∠ + ∠ = ∠ + ∠

4. Substitution PoE 2 and 3

5. m CBE m CBE∠ = ∠ 5. Reflexive PoE Diagram 6. m ADC m ABC∠ = ∠ 6. Subtraction PoE 4 and 5

Int. Geometry Unit 2 Test Review 8

40. Statement Reason Relies on/uses to reach

1. AC CB⊥ 1. Given 2. ACB∠ is a right angle 2. Definition of

Perpendicular Lines 1

3. 90om ACB∠ = 3. Definition of a Right Angle 2

4. m ACD m BCD m ACB∠ + ∠ = ∠ 4. Angle Addition Postulate Diagram

5. 90om ACD m BCD∠ + ∠ = 5. Substitution PoE (or transitive) 3 and 4

6. DAC∠ and ACD∠ are complementary

6. Given

7. 𝑚∠𝐴𝐶𝐷 +𝑚∠𝐷𝐴𝐶 = 90° 7. Definition of complementary angles 6

8. m ACD m BCD m ACD m DAC∠ + ∠ = ∠ + ∠

8. Substitution PoE 5 and 7

9. m ACD m ACD∠ = ∠ 9. Reflexive PoE Diagram 10. m BCD m DAC∠ = ∠ 10. Subtraction PoE 8 and 9