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