Formal Geometry – Unit 3 – Worksheet Packet

Section 2.7 – Day 1 – Naming Angles Formed by Parallel Lines and Transversals

Use the diagram for 1 – 7 to the right to identify each pair of angles as Alternate Interior, Alternate Exterior, Consecutive Interior, Corresponding, Linear Pair,

Vertical Angles, or none.

1. 1 and 7 ________________________

2. 1 and 5 ________________________

3. 8 and 6 ________________________

4. 8 and 5 ________________________

5. 4 and 8 ________________________

6. 4 and 5 ________________________

7. 6 and 7 ________________________

State the relationship between angle A and B.

8. 9. 10.

_________________________ _________________________ _________________________ 11. 12. 13.

_________________________ _________________________ _________________________

Alternate Exterior

Corresponding

Vertical Angles

Linear Pair

Alternate Interior

sec Con utive Interior

Linear Pair

Alternate Interior Alternate Exterior Corresponding

Vertical Anglessec Con utive Interior Linear Pair

Use the diagram for 8 – 15 to the right to identify each pair of angles as Alternate Interior, Alternate Exterior, Consecutive Interior, Corresponding, Linear Pair, Vertical Angles, or none.

1. 9 and 11 ________________________

2. 3 and 9 ________________________

3. 3 and 12 ________________________

4. 14 and 16 ________________________

5. 8 and 15 ________________________

6. 4 and 5 ________________________

7. 1 and 7 ________________________

8. 8 and 6 ________________________

Mixed Review:

4. Find x and y so that DG and BE are perpendicular.

5. Determine whether each statement can be assumed from the figure. Explain.

Linear Pair

sec Con utive Interior

none

Linear Pair

none

Alternate Interior

none

Linear Pair

LPLots

Lots

15

14

x

y

a. BFC and AFG are complementary.

b. DFA and AFG are a linear pair.

c. DFC and BFC are complementary.

Name: ____________________________

Date: __________ Period:_____________

No

Yes

Yes

Section 2.7 – Day 2 –

Angle Relationships of Parallel Lines and Transversals

Refer to the diagram below and identify the special angle pair names.

1) 3 and 16 _____________________________________

2) 8 and 6 ______________________________________

3) 11 and 15 ____________________________________

4) 9 and 6 _____________________________________

5) 1 and 6 ______________________________________

6) 6 and 10 _____________________________________

7) 14 and 5 _____________________________________

8. Given 𝑎||𝑏, select all of the angles that are congruent to ∠4.

Solve for the following variables.

9. 10. 11.

12. 13. 15.

a. ∠1 b. ∠2 c. ∠3 d. ∠5

e. ∠6 f. ∠7 g. ∠8

||

1 2

5 52

6

p q

m y

m y

Find m

Alternate Interior

Alternate Exterior

Corresponding

Vertical Angles

sec Con utive Interior

Linear Pair

none

76x 54

114

x

y

40

50

x

y

63x 20

30

x

y

7

131

y

x

Solve for the following variables.

16. 17. 18.

Find the value of x.

19. 20.

21. Carlos constructed 3 parallel lines as part of an art project. He also drew a line passing through each of them. Some of the angles formed by the intersection of line t, l, m, and n are numbered below. Select all of the conjectures that are correct.

a. Angles, 1, 2, and 3 are congruent.

b. Angles 1, 3, and 5 are congruent.

c. Angles 2, 4, and 6 are congruent.

d. Angles 2, and 4 are supplementary.

e. Angles 5, and 6 are supplementary.

f. Angles 2, and 3 are supplementary.

22. Given the following diagram and 𝑎||𝑏, solve for the variables.

42

14

x

y

54

12

x

y

60

10

x

x

30x 66x

118

28

31

x

y

z

Section 2.7 – Day 3 –

Parallel Lines and Shapes

For 1 – 9, solve for the variable. 1. 2. 3. 4. 5. 6. 7. 8. 9.

26x 45y 81

35

x

y

82

56

x

y

137

3

51

x

y

11

22

57

x

y

z

11

10

x

y

6

24

x

y

6, 10

100

x x

y

10. Given: 𝑆𝑇̅̅̅̅ ||𝑋𝑊̅̅ ̅̅ ̅ 𝑆𝑇̅̅̅̅ bisects ∠𝑉𝑆𝑊

Find: 𝑚∠𝑋 & 𝑚∠𝑇𝑆𝑊

11. Given: 𝐴𝐷̅̅ ̅̅ ||𝐵𝐶̅̅ ̅̅ Name all the pairs of angles that must be congruent

12. Given: 𝑎||𝑏 𝑚∠1 = 𝑥 + 3𝑦

𝑚∠2 = 2𝑥 + 30 𝑚∠3 = 5𝑦 + 20

Find: 𝑚∠1

13. If 𝑎||𝑏, find 𝑚∠2

14.

1 2

2 2 25

3 6 5

m x y

m x

m y

8 65m

5 _________

6 _________

7 _________

m

m

m

70

70

m X

m TSW

DAC BCA

1 70m

2 135m

65

115

65

Section 2.8 – Day 1 – Slope of Lines Examples #1-3: Find the slope of each line. 1. 2. 3. Examples #4-7: Determine the slope of the line that contains the given points. 4. 𝐶(3,1), 𝐷(−2, 1) 5. 𝐺(−4,3), 𝐻(−4, 7) 6. 𝐿(8, −3), 𝑀(−4, −12) 7. 𝑅(2, −6), 𝑆(−6, 5)

Examples #8-13: Determine whether 𝑨𝑩 ⃡ and 𝑪𝑫 ⃡ are parallel, perpendicular or neither. 8. 𝐴(1, 5), 𝐵(4, 4), 𝐶(9, −10), 𝐷(−6, −5) 9. 𝐴(−6, −9), 𝐵(8, 19), 𝐶(0, −4), 𝐷(2,0) 10. 𝐴(4, 2), 𝐵(−3, 1), 𝐶(6,0), 𝐷(−10, 8) 11. 𝐴(8, −2), 𝐵(4, −1), 𝐶(3,11), 𝐷(−2, −9) 12. 𝐴(8, 4), 𝐵(4, 3), 𝐶(4, −9), 𝐷(2, −1) 13. 𝐴(4, −2), 𝐵(−2, −8), 𝐶(4,6), 𝐷(8, 5)

3

7m

6

7m

0m

0m m undefined

3

4m

11

8m

parallel parallel

neither

neither

Graph the line that satisfies each condition.

14. Passes through 𝐴(2, −5), parallel to 𝐵𝐶 ⃡ with 𝐵(1, 3) and 𝐶(4,5).

15. Passes through 𝐾(3, 7), perpendicular to 𝐿𝑀 ⃡ with 𝐿(−1, −2) and 𝑀(−4,8).

Examples #16-18: Find the value of 𝒙 or 𝒚 that satisfies the given condition.

16. The line containing (4, −1) and (𝑥, −6) has a slope of 5

2

.

17. The line containing (−4, 9) and (4, 3) is parallel to the line containing (−8, 1) and (4, 𝑦). 18. The line containing (8, 7) and (7, −6) is perpendicular to the line containing (2, 4) and (𝑥, 3).

6x

8y

15x

Section 2.8 – Day 2 – Equations of Lines Name the slopes of a line parallel and perpendicular to the given line.

1. 𝑦 = 3𝑥 + 4 2. 𝑦 = −1

5𝑥 + 3 3. 𝑦 =

4

−5𝑥 4. 𝑦 = −𝑥 + 8

Parallel: ________ Parallel: ________ Parallel: ________ Parallel: ________ Perp: _________ Perp: _________ Perp: _________ Perp: _________ State if the following pairs of lines are parallel, perpendicular, the same line, or just intersecting.

5. 2 6

2 3

y x

y x

6.

16

2

2 3

y x

y x

7. 3 8

2 3

y x

y x

8.

44

4 3

xy

y x

9.

3 2 12

36

2

x y

y x

________________ _______________ _______________ _______________ ________________ Write the equation of line that passes through the given point and is parallel to each given line. 10. (1, 7); 𝑦 = 3𝑥 − 2 11. (0, – 4); 𝑦 = −2𝑥 + 1 12. 2𝑥 − 3𝑦 = 12 through the origin 13. (13, 5): 3𝑥 − 15 = 𝑦 − 2𝑥 Write the equation of the line through the given point and is perpendicular to the given line.

14. (6, 1); 𝑦 = 3𝑥 + 7 15. (– 2, 1); 𝑦 =1

2𝑥 + 10

16. (0, – 4); 𝑦 = −3

4𝑥 17. (−9, −9): 3𝑥 − 2𝑦 = 6

3m

1

3m

5m

1

5m

5

4m

4

5m

1m

1m

Parallel Neither PerpendicularNeither Same Line

3 4y x 2 4y x

2

3y x

5 60y x

44

3y x

215

3y x

13

3y x

2 3y x

18. Given the two lines below, which statement is true?

𝐿𝑖𝑛𝑒 1: 𝑥 − 3𝑦 = −15 and 𝐿𝑖𝑛𝑒 2: 𝑦 = 3(𝑥 + 2) − 1

A. The lines are parallel. B. They are the same line.

C. The lines are perpendicular. D. They are intersecting but not perpendicular.

19. Which equation of the line passes through (8, 10) and is parallel to the graph of the line

𝑦 =8

3𝑥 + 7?

A. 𝑦 =

8

3𝑥 −

34

3 C. 𝑦 = 6𝑥 −

34

3

B. 𝑦 =8

3𝑥 +

8

3 D. 𝑦 = 16𝑥 +

8

3

20. Which equation of the line passes through (4, 7) and is perpendicular to the graph of the line that

passes through the points (1, 3) and (−2, 9) ?

A. 𝑦 = 2𝑥 − 1 C. 𝑦 =

1

2𝑥 − 5

B. 𝑦 =1

2𝑥 + 5 D. 𝑦 = −2𝑥 + 15

Section 2.9 – Day 1 –

Proving Lines Parallel

Use the diagram for 1 – 7 to the right to identify each pair of angles as Alternate Interior, Alternate Exterior, Consecutive Interior, Corresponding, Linear Pair, Vertical Angles, or none.

1. 1 and 7 ________________________

2. 1 and 5 ________________________

3. 8 and 6 ________________________

4. 8 and 5 ________________________

5. 4 and 8 ________________________

6. 4 and 5 ________________________

7. 2 and 8 ________________________

In each example, determine if the lines are parallel or not. Explain why or why

not. NOTE – NOT DRAWN TO SCALE!

8. 9. 10.

____________________________ ___________________________ ____________________________

____________________________ ___________________________ ____________________________

11. 12. 13.

____________________________ ___________________________ ____________________________ ____________________________ ___________________________ ____________________________

Alternate Exterior

Corresponding

Linear Pair

Alternate Interior

Vertical Angles

sec Con utive Interior

Corresponding

. . Alt Ext Converse

, ||Yes they are

. . not Alt Int

, ' ||No they aren t

sec. . not supCon Int p

, ' ||No they aren t

. Int. Alt Converse

, ||Yes they are

. Ext. not Alt

, ' ||No they aren t

, ' ||No they aren t

14. 15. 16.

____________________________ ___________________________ ____________________________ ____________________________ ___________________________ ____________________________

17. 18.

_____________________________________ ________________________________________

_____________________________________ ________________________________________

Find the value of x that makes 𝒎 || 𝒏.

19. 20. 21.

sec. . not supCon Int p

, ' ||No they aren t , ' ||No they aren t

. Corr Converse

, ||Yes they are

. . Alt Int Converse

, ||Yes they are

. . Alt Int Converse

, ||Yes they are

40x 30x 30x

22. 23. Which two lines a

||d e

14x

||u v

Section 2.9 – Day 2 –

Parallel Lines Proofs

Write a two-column proof for each of the following: Parallel Lines: fill in the blanks about parallel lines.

1. Given: 𝑑 𝑒,

Prove: 4 6

Statements Reasons

1) 𝑑 𝑒, 1) Given

2) 4 6 2) Alternate Interior Angles Theorem

2. Given: 𝑑 𝑒,

Prove: 1 5

Statements Reasons

1) 𝑑 𝑒, 1) Given

2) 1 5 2) Corresponding Angles Postulate

3. Given: 2 8

Prove: 𝑑 𝑒,

Statements Reasons

1) 2 8 1) Given

2) 𝑑 𝑒, 2) Alternate Exterior Angles Converse

4. Given:, a b 1 2

Prove: l m

Statements Reasons

1) a b 1) Given

2) 1 2 2) Given

3) 2 3 3) Alternate Exterior Angles Theorem

4) 1 3 4) Transitive Property

5) l m 5) Corresponding Angles Converse Postulate

5. Given: 1 3, 2 4

Prove: AB CD

Statements Reasons

1) 1 3 1) Given

2) 2 4 2) Given

3) 3 4 3) Vertical Angles Theorem

4) 1 2 4) Transitive Property

5) AB CD 5) Alternate Interior Angles Converse Theorem

6. Given: n m

Prove: 1 supp 8

Statements Reasons

1) n m 1) Given

2) 3 supp 6 2) Consecutive Interior Angles Theorem

3) 1 3 3) Vertical Angle Theorem

4) 6 8 4) Vertical Angle Theorem

5) 1 3m m 5) Definition of Congruent Angles

6) 6 8m m 6) Definition of Congruent Angles

7) 3 6 180m m 7) Definition of Supplementary Angles

8) 1 8 180m m 8) Substitution Prop

9) 1 supp 8 9) Definition of Supplementary

7. Given: l m , 1 and 2 are supp.

Prove: a b

Statements Reasons

1) l m 1) Given

2) 1 and 2 are supp. 2) Given

3) 1 3 3) Alternate Interior Angles Theorem

4) 1 3m m 4) Definition of Congruent Angles

5) 1 2 180m m 5) Definition of Supplementary Angles

6) 3 2 180m m 6) Substitution

7) 2 and 3 are supp. 7) Definition of Supplementary Angles

8) a b 8) Consecutive Interior Angles Converse Theorem

8. Given: ∠1 𝑖𝑠 𝑠𝑢𝑝𝑝 𝑡𝑜 ∠4, ∠1 𝑖𝑠 𝑠𝑢𝑝𝑝 𝑡𝑜 ∠2

Prove: 𝐴𝐵̅̅ ̅̅ ∥ 𝐶𝐷̅̅ ̅̅

Statements Reasons

1) ∠1 𝑖𝑠 𝑠𝑢𝑝𝑝 𝑡𝑜 ∠4 1) Given

2) ∠1 𝑖𝑠 𝑠𝑢𝑝𝑝 𝑡𝑜 ∠2 2) Given

3) 2 4 3) Supplementary Angle Theorem

4) 𝐴𝐵̅̅ ̅̅ ∥ 𝐶𝐷̅̅ ̅̅ 4) Corresponding Angles Theorem Converse

Write a two column proof for the following proofs.

10. 11.

Statements Reasons

1 𝒑||𝒒 Given

2 ∠𝟑 𝒊𝒔 𝒔𝒖𝒑𝒑. 𝒕𝒐 ∠𝟔 Definition of

Supplementary

3 𝒎∠𝟑 + 𝒎∠𝟔= 𝟏𝟖𝟎

Linear Pair Theorem

12.

Statements Reasons

1 𝒑||𝒒 Given

2 ∠𝟏 ≅ ∠𝟑 Vertical Angles

Theorem

Statements Reasons

1 𝑾𝑿||𝒀𝒁 Given

2 ∠𝟐 ≅ ∠𝟒 Given

3 ∠𝟏 ≅ ∠𝟒 Alternate

Interior Angles Theorem

4 ∠𝟐 ≅ ∠𝟏 Transitive Prop

5 𝑾𝒀||𝑿𝒁

Corresponding Angles

Converse Postulate

Given :

Prove : 3 6 180

p q

m m

Given :

2 4

Prove :

WX YZ

WY XZ

Given :

Prove : 1 3

p q

Section 2.10 – Day 2 –

Perpendiculars and Distance

Examples #1-7: Find the distance from the line to the given point 𝑷.

1. 3 5,2 and y P 2. 4 2,5 and x P

3. 2 7, 6 and y x P 4. 1

1 2,53

and y x P

5. 1

6 6,56

and y x P 6. Line 𝑚 contains points (0, −3)

and (7,4). Point P (4 ,3)

5 5

4 5 2 10

02

7. Line 𝑚 contains points (1,5) and (4, −4). Point P has coordinates (−1 ,1)

Examples #8-11: Find the distance between each pair of parallel lines.

8. 2 4 and y y 9. 3 7 and x x

10. 2 3 2 2 and y x y x 11. 1 1

2 83 3

and y x y x

12. Line 𝑚 is represented by the equation 2

63

y x . Which equation would you use to determine the

distance between the line 𝑚 and point (6, −2)?

A. 2

3y x B.

37

2y x C.

22

3y x D.

36

2y x

10

6 4

5 3 10