©
Parallel Lines and Transversals
Glencoe/McGraw-Hill
A2
©
Glencoe/McGraw-Hill
苶T S 苶, T 苶M 苶, N 苶Q 苶, Q 苶R 苶, T 苶O 苶, M 苶H 苶, N 苶E 苶, Q 苶X 苶
7. Name all segments skew to A 苶G 苶.
苶X A 苶, H 苶O 苶, M 苶T 苶
6. Name all segments parallel to Q 苶R 苶.
MHO, NEX, HEX, MNQ, SGO, RAX
125
5. Name all planes that intersect plane MHE.
苶A R 苶, S 苶G 苶, T 苶O 苶, M 苶H 苶, N 苶E 苶
4. Name all segments parallel to Q 苶X 苶.
For Exercises 4–7, refer to the figure at the right.
苶R M 苶, M 苶N 苶, M 苶S 苶, P 苶S 苶, P 苶O 苶
3. Name all segments that intersect M 苶P 苶.
苶T O 苶, P 苶S 苶, M 苶R 苶
2. Name all segments that are parallel to N 苶U 苶.
MNO, MPS, NOT, RST
1. Name all planes that intersect plane OPT.
For Exercises 1–3, refer to the figure at the right.
Exercises
c. Name all segments that are skew to E 苶H 苶. BF 苶 苶, 苶 CG 苶, 苶 BD 苶, 苶 CD 苶, and A 苶B 苶
b. Name all segments that are parallel to 苶 CG 苶. BF 苶 苶, 苶 DH 苶, and 苶 AE 苶
a. Name all planes that are parallel to plane ABD. plane EFH
Example
When two lines lie in the same plane and do not intersect, they are parallel. Lines that do not intersect and are not coplanar are skew lines. In the figure, ᐉ is parallel to m, or ᐉ || m. You can also write || RS . Similarly, if two planes do not intersect, they are PQ parallel planes.
T
M
R
M
E
A
F
G
R
Q
S
P
E
ᐉ
A
X
T
O
G
C
m
Glencoe Geometry
S
O
H
N
D H
U
N
S
R
B
Q
P
n
____________ PERIOD _____
Study Guide and Intervention
Relationships Between Lines and Planes
3-1
NAME ______________________________________________ DATE
exterior angles
⬔1, ⬔2, ⬔7, and ⬔8
t
5 6 8 7
1 2 4 3
2. ⬔5 and ⬔14
ᐉ
©
Glencoe/McGraw-Hill
alt. interior
10. ⬔6 and ⬔16
corresponding
7. ⬔3 and ⬔11
corresponding
4. ⬔1 and ⬔5
126
consecutive interior
11. ⬔11 and ⬔14
consecutive interior
8. ⬔12 and ⬔3
corresponding
5. ⬔6 and ⬔14
p
q
13 14 16 15
9 10 12 11
ᐉ
n
Glencoe Geometry
alt. exterior
12. ⬔10 and ⬔16
alt. interior
9. ⬔4 and ⬔6
alt. exterior
6. ⬔2 and ⬔8
Identify each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles.
q
5 6 8 7
1 2 4 3
p
3. ⬔4 and ⬔6
d. ⬔3 and ⬔9 alternate interior angles
Name the transversal that forms each pair of angles. 1. ⬔9 and ⬔13
n
m
b. ⬔4 and ⬔12 corresponding angles
Use the figure in the Example for Exercises 1–12.
Exercises
c. ⬔12 and ⬔13 consecutive interior angles
a. ⬔10 and ⬔16 alternate exterior angles
Example Identify each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles.
corresponding angles
consecutive interior angles
⬔3 and ⬔6; ⬔4 and ⬔5
alternate exterior angles
alternate interior angles
⬔3 and ⬔5; ⬔4 and ⬔6
⬔1 and ⬔5; ⬔2 and ⬔6; ⬔3 and ⬔7; ⬔4 and ⬔8
interior angles
⬔3, ⬔4, ⬔5, and ⬔6
⬔1 and ⬔7; ⬔2 and ⬔8
Name
Angle Pairs
A line that intersects two or more other lines in a plane is called a transversal. In the figure below, t is a transversal. Two lines and a transversal form eight angles. Some pairs of the angles have special names. The following chart lists the pairs of angles and their names.
Parallel Lines and Transversals
(continued)
____________ PERIOD _____
Study Guide and Intervention
Angle Relationships
3-1
NAME ______________________________________________ DATE
Answers (Lesson 3-1)
Glencoe Geometry
Lesson 3-1
©
Angles and Parallel Lines
Skills Practice
Glencoe/McGraw-Hill 4. ⬔1 110 6. ⬔6 70
3. ⬔8 110
5. ⬔4 110
A6 18. ⬔9 65
17. ⬔14 65
(3y ⫺ 1)⬚
40⬚
5x ⬚
x ⫽ 28, y ⫽ 47
©
20⬚
1
60⬚
Glencoe/McGraw-Hill
21.
80
Find m⬔1 in each figure.
19.
133
22.
20.
16. ⬔15 115
15. ⬔7 105
Find x and y in each figure.
14. ⬔5 105
13. ⬔2 105
32⬚
1
140⬚
2 6
4 7
9 10 14 13
3
s
t
12 11 16 15
y
8
5 6 8 7
10 9 11 12
s
r
Glencoe Geometry
z
u
x ⫽ 10, y ⫽ 15
5
x
1
72
w
m
1 2 3 4
q
5 6 7 8
1 2 3 4
____________ PERIOD _____
(6y ⫹ 20)⬚ 7x ⬚
(8x ⫺ 10)⬚
In the figure, m⬔3 ⫽ 75 and m⬔10 ⫽ 115. Find the measure of each angle.
12. ⬔11 100
10. ⬔2 80
9. ⬔8 80
11. ⬔5 100
8. ⬔6 80
7. ⬔9 100
In the figure, m⬔7 ⫽ 100. Find the measure of each angle.
2. ⬔5 110
1. ⬔3 70
In the figure, m⬔2 ⫽ 70. Find the measure of each angle.
3-2
NAME ______________________________________________ DATE
(Average)
Angles and Parallel Lines
Practice
4. ⬔5 106 6. ⬔13 106
3. ⬔9 88 5. ⬔11 106
x ⫽ 14, y ⫽ 37
3x ⬚ (4y ⫺ 10)⬚
(9x ⫹ 12)⬚
130
100⬚
1
50⬚
10.
8. (5y ⫺ 4)⬚ 3y ⬚
98
144⬚
©
Glencoe/McGraw-Hill
134
12. FENCING A diagonal brace strengthens the wire fence and prevents it from sagging. The brace makes a 50° angle with the wire as shown. Find y. 130
Sample proof: It is given that ᐉ || m , so ⬔1 ⬵ ⬔8 by the Alternate Exterior Angles Theorem. Since it is given that m || n , ⬔8 ⬵ ⬔12 by the Corresponding Angles Postulate. Therefore, ⬔1 ⬵ ⬔12, since congruence of angles is transitive.
62⬚ 1
m
2 8
1 2 3 4
50⬚
s
n
m
ᐉ
Glencoe Geometry
y⬚
k
12 11 13 14 10 9 15 16 r
5 6 7 8
n
7
9 10 11 12
1
4 3 5 6
____________ PERIOD _____
(2x ⫹ 13)⬚
x ⫽ 28, y ⫽ 23
11. PROOF Write a paragraph proof of Theorem 3.3. Given: ᐉ || m , m || n Prove: ⬔1 ⬵ ⬔12
9.
Find m⬔1 in each figure.
7.
Find x and y in each figure.
2. ⬔8 92
1. ⬔10 92
In the figure, m⬔2 ⫽ 92 and m⬔12 ⫽ 74. Find the measure of each angle.
3-2
NAME ______________________________________________ DATE
Answers (Lesson 3-2)
Glencoe Geometry
Lesson 3-2
©
Angles and Parallel Lines
Glencoe/McGraw-Hill
A5 4. ⬔7 102 6. ⬔14 78
3. ⬔11 102
5. ⬔15 102
©
Glencoe/McGraw-Hill
11. ⬔7 68
9. ⬔4 100
7. ⬔12 100
131
12. ⬔16 112
10. ⬔3 80
8. ⬔1 80
In the figure, m⬔9 ⫽ 80 and m⬔5 ⫽ 68. Find the measure of each angle.
2. ⬔6 78
1. ⬔5 102
In the figure, m⬔3 ⫽ 102. Find the measure of each angle.
Exercises
Example In the figure, m⬔2 ⫽ 75. Find the measures of the remaining angles. m⬔1 105 ⬔1 and ⬔2 form a linear pair. m⬔3 105 ⬔3 and ⬔2 form a linear pair. m⬔4 75 ⬔4 and ⬔2 are vertical angles. m⬔5 105 ⬔5 and ⬔3 are alternate interior angles. m⬔6 75 ⬔6 and ⬔2 are corresponding angles. m⬔7 105 ⬔7 and ⬔3 are corresponding angles. m⬔8 75 ⬔8 and ⬔6 are vertical angles.
Also, consecutive interior angles are supplementary.
• corresponding angles • alternate interior angles • alternate exterior angles
5 6 8 7
1 2 4 3
p
v
q
p
n
m
n
Glencoe Geometry
w
13 14 16 15
9 10 12 11
13 14 16 15
5 6 87
1 2 4 3
q 9 10 12 11
5 6 8 7
1 2 4 3
p
When two parallel lines are cut by a transversal, the following pairs of angles are congruent.
m
____________ PERIOD _____
Study Guide and Intervention
Parallel Lines and Angle Pairs
3-2
NAME ______________________________________________ DATE
Angles and Parallel Lines
Study Guide and Intervention
(5y ⫹ 5)⬚ (13y ⫺ 5)⬚
x ⫽ 11; y ⫽ 10
5x ⬚
(11x ⫹ 4)⬚
x ⫽ 15; y ⫽ 19
(4x ⫹ 10)⬚
(5x ⫺ 5)⬚ (6y ⫺ 4)⬚
Glencoe Geometry
(4z ⫹ 6)⬚ 106⬚ 2y ⬚
x⬚
Glencoe/McGraw-Hill
x ⫽ 74; y ⫽ 37; z ⫽ 25
Answers
©
5.
Find x, y, and z in each figure.
3.
1.
Find x and y in each figure.
Exercises
15 y
6.
4.
2.
132
75 5y 5 5
(15x ⫹ 30)⬚ 10x ⬚
3x ⬚
(5x ⫺ 20)⬚
2y ⬚
z⬚
p 1 4
x ⫽ 30; y ⫽ 15; z ⫽ 150
2x ⬚ 90⬚ x ⬚
x ⫽ 10; y ⫽ 25
4y ⬚
2y ⬚
x ⫽ 6; y ⫽ 24
(3y ⫹ 18)⬚
90⬚
m⬔2 m⬔3 75 5y
4x 5 4x 5 3x x5 x55 x
3x 15 3x 15 3x 15 15 5 20
r || s, so m⬔2 m⬔3 because they are corresponding angles.
p || q, so m⬔1 m⬔2 because they are corresponding angles.
Example If m⬔1 ⫽ 3x ⫹ 15, m⬔2 ⫽ 4x ⫺ 5, m⬔3 ⫽ 5y, and m⬔4 ⫽ 6z ⫹ 3, find x and y.
2 3
r s
Glencoe Geometry
q
(continued)
____________ PERIOD _____
Algebra and Angle Measures Algebra can be used to find unknown values in angles formed by a transversal and parallel lines.
3-2
NAME ______________________________________________ DATE
Answers (Lesson 3-2)
Lesson 3-2
©
Slopes of Lines
Skills Practice
Glencoe/McGraw-Hill 4 ⫺ᎏᎏ 3
8. a line perpendicular to NP
⫺2 N
T
A9 perpendicular
©
J (3, 3)
Glencoe/McGraw-Hill
O
D(–3, 1) Y(3, 0)
y
x
15. contains Y(3, 0), parallel to DJ with D(3, 1) and J(3, 3)
x
y
y
O
x
139
T (0, –2)
C (0, 3) O
y
X(2, –1)
x
x
Glencoe Geometry
16. contains T(0, 2), perpendicular to CX with C(0, 3) and X(2, 1)
R(–4, 5)
14. slope , contains R(4, 5)
3 2
A(0, 1)
W
P
12. A(4, 8), B(4, 6), M(3, 5), N(1, 3)
neither
13. slope 3, contains A(0, 1)
O
O
y
10. A(1, 4), B(2, 5), M(3, 2), N(3, 0)
Graph the line that satisfies each condition.
parallel
11. A(2, 7), B(4, 2), M(2, 0), N(2, 6)
parallel
9. A(0, 3), B(5, 7), M(6, 7), N(2, 1)
and MN are parallel, perpendicular, or neither. Determine whether AB
⫺2
7. a line parallel to TW
3 ᎏᎏ 4
5. NP
Find the slope of each line. 6. TW
4. J(5, 2), K(5, 4) ⫺ᎏᎏ
3. C(0, 1), D(3, 3) ᎏᎏ
2 3
2. G(2, 5), H(1, 7) ⫺4
1. S(1, 2), W(0, 4) 2
1 5
____________ PERIOD _____
Determine the slope of the line that contains the given points.
3-3
NAME ______________________________________________ DATE
(Average)
Slopes of Lines
Practice 1 2
1 ⫺ᎏᎏ 2
6. a line perpendicular to PS
2 5
⫺ᎏᎏ
4. GR
13 4
2. I(2, 9), P(2, 4) ᎏᎏ
L
G
1 2
x
U (2, –2)
O
G(4, –2)
x
P
S
perpendicular
x
Z (–3, 0)
E(–2, 4)
O
y
K(2, –2)
x
14. contains Z(3, 0), perpendicular to EK with E(2, 4) and K(2, 2)
P(–3, –3)
O
y
12. slope , contains P(3, 3)
4 3
perpendicular
10. K(3, 7), M(3, 3), S(0, 4), T(6, 5)
R
Glencoe Geometry
Glencoe/McGraw-Hill
Answers
©
140
x
Glencoe Geometry
15. PROFITS After Take Two began renting DVDs at their video store, business soared. Between 2000 and 2003, profits increased at an average rate of $12,000 per year. Total profits in 2003 were $46,000. If profits continue to increase at the same rate, what will the total profit be in 2009? $118,000
F(0, –3)
B (–4, 2)
y
13. contains B(4, 2), parallel to FG with F(0, 3) and G(4, 2)
O
y
11. slope , contains U(2, 2)
O
y
8. K(5, 2), M(5, 4), S(3, 6), T(3, 4)
Graph the line that satisfies each condition.
parallel
9. K(4, 10), M(2, 8), S(1, 2), T(4, 7)
neither
7. K(1, 8), M(1, 6), S(2, 6), T(2, 10)
are parallel, perpendicular, or neither. and ST Determine whether KM
2 ⫺ᎏᎏ 5
5. a line parallel to GR
2 ᎏᎏ 3
3. LM
Find the slope of each line.
1. B(4, 4), R(0, 2) ⫺ᎏᎏ
M
____________ PERIOD _____
Determine the slope of the line that contains the given points.
3-3
NAME ______________________________________________ DATE
Answers (Lesson 3-3)
Lesson 3-3
©
Equations of Lines
Study Guide and Intervention
____________ PERIOD _____
Glencoe/McGraw-Hill 4
3 4
m , (x1, y1) (8, 1)
Point-slope form
4
The point-slope form of the equation of the 3 line is y 1 (x 8).
y 1 (x 8)
3 4
y y1 m(x x1)
Example 2 Write an equation in point-slope form of the line with slope 3 ⫺ᎏᎏ that contains (8, 1).
A11
1 3
1 3
1 2
y ⫽ ⫺3x ⫺ 8
6. m: 3, y-intercept: 8
y ⫽ ⫺2
4. m: 0, y-intercept: 2
y ⫽ ⫺ᎏᎏx ⫹ 4
1 2
2. m: , y-intercept: 4
©
Glencoe/McGraw-Hill
5 y ⫹ 3 ⫽ ⫺ᎏᎏx 2
11. m , (0, 3)
5 2
y ⫺ 3 ⫽ ⫺(x ⫹ 1)
9. m 1, (1, 3)
1 y ⫹ 1 ⫽ ᎏᎏ (x ⫺ 3) 2
7. m , (3, 1)
1 2
143
y⫺5⫽0
12. m 0, (2, 5)
y ⫹ 2 ⫽ ᎏᎏ (x ⫹ 3)
1 4
1 10. m , (3, 2) 4
y ⫹ 2 ⫽ ⫺2(x ⫺ 4)
8. m 2, (4, 2)
Glencoe Geometry
Write an equation in point-slope form of the line having the given slope that contains the given point.
y ⫽ ⫺ᎏᎏx ⫹ ᎏᎏ
5 3
5. m: , y-intercept:
5 3
1 y ⫽ ᎏᎏx ⫹ 5 4
3. m: , y-intercept: 5
1 4
y ⫽ 2x ⫺ 3
1. m: 2, y-intercept: 3
Write an equation in slope-intercept form of the line having the given slope and y-intercept.
Exercises
The slope-intercept form of the equation of the line is y 2x 4.
Example 1 Write an equation in slope-intercept form of the line with slope ⫺2 and y-intercept 4. y mx b Slope-intercept form y 2x 4 m 2, b 4
If m is the slope of a line, b is its y-intercept, and (x1, y1) is a point on the line, then: • the slope-intercept form of the equation is y mx b, • the point-slope form of the equation is y y1 m(x x1).
Write Equations of Lines You can write an equation of a line if you are given any of the following: • the slope and the y-intercept, • the slope and the coordinates of a point on the line, or • the coordinates of two points on the line.
3-4
NAME ______________________________________________ DATE
Equations of Lines Many real-world situations can be modeled
137.5 125 or $262.50 Donna would earn more with the first plan.
C 25h 125 25(5.5) 125
For 5 hours of service Donna would earn
1 2
Second Plan
247.5 55 or $302.50
C 45h 55 45 5 55
冢 12 冣
For 5 hours of service Donna would earn
1 2
First plan
which plan would Donna earn more?
if a company has 5 ᎏᎏ hours of service calls. Under
1 2
b. Donna may change her costs to represent them by the equation C ⫽ 25h ⫹ 125, where $125 is the fixed monthly fee for a web site and the cost per hour is $25. Compare her new plan to the old one
Glencoe Geometry
Glencoe/McGraw-Hill
Answers
©
the third company 144
3. A third satellite company charges a flat rate of $69 for all channels, including the premium channels. If Jerri wants to add a fourth premium channel, which service would be least expensive?
Current service: C ⫽ 10p ⫹ 34.95 Competing service: C ⫽ 8p ⫹ 39.99
1. Write an equation in slope-intercept form that models the total monthly cost for each satellite service, where p is the number of premium channels.
Glencoe Geometry
number of premium channels represents the rate of change, or slope, of the equation.
4. Write a description of how the fee for the number of premium channels is reflected in the equation. The fee for the
competing service
2. If Jerri wants to include three premium channels in her package, which service would be less, her current service or the competing service?
For Exercises 1–4, use the following information. Jerri’s current satellite television service charges a flat rate of $34.95 per month for the basic channels and an additional $10 per month for each premium channel. A competing satellite television service charges a flat rate of $39.99 per month for the basic channels and an additional $8 per month for each premium channel.
Exercises
For each hour, the cost increases $45. So the rate of change, or slope, is 45. The y-intercept is located where there are 0 hours, or $55. C mh b 45h 55
a. Write an equation to represent the total monthly cost C for maintaining a web site and for h hours of service calls.
Example Donna offers computer services to small companies in her city. She charges $55 per month for maintaining a web site and $45 per hour for each service call.
using linear equations.
(continued)
____________ PERIOD _____
Study Guide and Intervention
Write Equations to Solve Problems
3-4
NAME ______________________________________________ DATE
Answers (Lesson 3-4)
Lesson 3-4
©
Proving Lines Parallel
Skills Practice
Glencoe/McGraw-Hill
130⬚
(2x ⫹ 6)⬚
ᐉ
m
|| m .
22 6.
k
(3x ⫹ 10)⬚
m
ᐉ
(4x ⫺ 10)⬚
20
ᐉ || m ; cons. int. ⭄
4. m⬔5 m⬔12 180
a || b ; corr. ⭄
2. ⬔9 ⬵ ⬔11
7. (6x ⫹ 4)⬚ (8x ⫺ 8)⬚
9 10 16 15
1 2 8 7
a
A15 4. Definition of complementary angles 5. Transitive Property of Equality 6. Definition of perpendicular 7. If 2 lines are ⊥ to the same line, then
4. m⬔1 m⬔2 90
5. m⬔ABC 90
6. B 苶A 苶⊥苶 BC 苶
7. B 苶A 苶
©
E(0, –3)
O
F (2, 1)
x
Yes; the slopes are the same.
B(–2, –3)
y
A(1, 3)
Glencoe/McGraw-Hill
9.
151
10.
S (5, –3)
U (4, 2) x
m
ᐉ
Glencoe Geometry
No; the slopes are not the same.
R (0, –4)
T (–4, 0) O
y
Determine whether each pair of lines is parallel. Explain why or why not.
lines are ||.
3. Given
3. ⬔1 and ⬔2 are complementary.
|| C 苶D 苶
2. Angle Addition Postulate
2. m⬔ABC m⬔1 m⬔2
6
1. Given
m
ᐉ
1. B 苶C 苶⊥苶 CD 苶
k
11 12 14 13
3 4 6 5
b
____________ PERIOD _____
8. PROOF Provide a reason for each statement in the proof of Theorem 3.7. B 2 C Given:⬔1 and ⬔2 are complementary. 1 BC 苶 苶⊥苶 CD 苶 Prove: B 苶A 苶 || C 苶D 苶 A D Proof: Statements Reasons
5. k
Find x so that ᐉ
ᐉ || m ; alt. ext. ⭄
3. ⬔2 ⬵ ⬔16
a || b ; alt. int. ⭄
1. ⬔3 ⬵ ⬔7
Given the following information, determine which lines, if any, are parallel. State the postulate or theorem that justifies your answer.
3-5
NAME ______________________________________________ DATE
(Average)
Proving Lines Parallel
Practice
Glencoe Geometry
ᐉ
12 t m
6.
(7x ⫺ 24)⬚
ᐉ
t
|| C 苶D 苶 3. A 苶B 苶
t
B
3
2
D 1
(2x ⫹ 12)⬚ (5x ⫺ 15)⬚
F H
B
ᐉ
m
G J
3. Segments contained in are ||.
9
D
|| lines
6
4 C 5
C
A
2. If consec. int ⭄ are suppl., then lines are ||.
7.
E
K
A
____________ PERIOD _____
Glencoe/McGraw-Hill
152
Glencoe Geometry
Sample answer: If the gardener digs each row at a 90⬚ angle to the footpath, each row will be perpendicular to the footpath. If each of the rows is perpendicular to the footpath, then the rows are parallel.
9. LANDSCAPING The head gardener at a botanical garden wants to plant rosebushes in parallel rows on either side of an existing footpath. How can the gardener ensure that the rows are parallel?
|| CD
2. AB
1. Given
1. ⬔2 and ⬔3 are supplementary.
21
Reasons
m
(5x ⫹ 18)⬚
Proof: Statements
Answers
©
(3x ⫹ 6)⬚
(4x ⫺ 6)⬚
|| m .
; AJ || BH alt. ext. ⭄
4. ⬔ACD ⬵ ⬔KBF
|| EG ; BD corr. ⭄
2. ⬔CBF ⬵ ⬔GFH
8. PROOF Write a two-column proof. Given:⬔2 and ⬔3 are supplementary. Prove: 苶 AB 苶 || C 苶D 苶
5.
Find x so that ᐉ
|| EG ; BD alt. int. ⭄
3. ⬔EFB ⬵ ⬔FBC
|| EG ; BD cons. int. ⭄
1. m⬔BCG m⬔FGC 180
Given the following information, determine which lines, if any, are parallel. State the postulate or theorem that justifies your answer.
3-5
NAME ______________________________________________ DATE
Answers (Lesson 3-5)
Lesson 3-5
© ____________ PERIOD _____
Proving Lines Parallel
Study Guide and Intervention
Glencoe/McGraw-Hill
2
n
m
A14
©
7
15
ᐉ
m
(6x ⫺ 20)⬚
(8x ⫹ 8)⬚ (9x ⫹ 1)⬚
ᐉ
(5x ⫺ 5)⬚
|| m .
Glencoe/McGraw-Hill
4.
1.
Find x so that ᐉ
Exercises
m
5.
2.
20
10
ᐉ
2x ⬚
m
m
B C
(6x ⫺ 20)⬚
(3x ⫹ 10)⬚
D A
Find x and m⬔ABC
m⬔CDA 6x 20 3x 20 3x x
m
(3x ⫺ 20)⬚
ᐉ
6.
3.
10
25 ᐉ
m n
m
Glencoe Geometry
70⬚
(5x ⫹ 20)⬚
(3x ⫹ 15)⬚
ᐉ
m⬔ABC 6x 20 6(10) 20 or 40
m⬔DAB 3x 10 10 30 10
We can conclude that m || n if alternate interior angles are congruent.
n
149
6x ⬚
(4x ⫹ 20)⬚
Since m⬔1 m⬔2, then ⬔1 ⬵ ⬔2. ⬔1 and ⬔2 are congruent corresponding angles, so r || s.
1
If m⬔1 ⫽ m⬔2, determine which lines, if any, are parallel. s r
Example 2 so that m || n .
the lines are parallel.
Example 1
then
• • • • •
corresponding angles are congruent, alternate exterior angles are congruent, consecutive interior angles are supplementary, alternate interior angles are congruent, or two lines are perpendicular to the same line,
If
If two lines in a plane are cut by a transversal and certain conditions are met, then the lines must be parallel.
Identify Parallel Lines
3-5
NAME ______________________________________________ DATE
©
3 2
C
B
3. Transitive Property of ⬵
5. Given
|| s 5. ⬔1 ⬵ ⬔5 6. ᐉ || m
冣
Glencoe/McGraw-Hill
冢
2 1 ⊥ TQ . ⫺ᎏᎏ ⭈ 2 ⫽ ⫺1, so PQ 2
: ⫺ᎏ1ᎏ; slope of TQ : 2 slope of PQ
150
⊥ TQ . Explain why or why not. 7. Determine whether PQ
6. If corr ⭄ are ⬵, then lines ||.
slope of Q 苶R 苶
T
P
O
y
r
Q
s
x
13 14 16 15
9 10 12 11
m
ᐉ
Glencoe Geometry
5 6 8 7
1 2 4 3
|| S 苶R 苶, 苶 PS 苶 || Q 苶R 苶, and P 苶R 苶⊥苶 SQ 苶.
4. If corr. ⭄ are ⬵, then lines ||.
2. Vertical ⭄ are ⬵. 3. ⬔5 ⬵ ⬔13
So 苶 PQ 苶
slope of P 苶R 苶 2
4 3
4 3 1 slope of S 苶苶 Q 2
x
slope of 苶 SR 苶0
–8
8
R(2, –4)
4
Q (8, 4)
苶Q 苶0 slope of P
2. ⬔13 ⬵ ⬔15 4. r
O –4
y
苶S 苶 slope of P
1. Given
Reasons
–4
S (–8, –4)
–8
P (–2, 4) 4
8
b. Which lines are parallel? Which lines are perpendicular?
1. ⬔15 ⬵ ⬔5
Statements
For Exercises 1–6, fill in the blanks. Given: ⬔1 ⬵ ⬔5, ⬔15 ⬵ ⬔5 Prove: ᐉ || m , r || s
Exercises
D
1
A
2. Transitive Property of ⬵ 3. If alt. int. angles are ⬵, then the lines are ||.
Reasons 1. Given
|| D 苶C 苶
Statements 1. ⬔1 ⬵ ⬔2 ⬔1 ⬵ ⬔3 2. ⬔2 ⬵ ⬔3 3. 苶 AB 苶 || D 苶C 苶
Prove: A 苶B 苶
a Given: ⬔1 ⬵ ⬔2, ⬔1 ⬵ ⬔3
Example
You can prove that lines are parallel by using postulates and theorems about pairs of angles. You also can use slopes of lines to prove that two lines are parallel or perpendicular.
Proving Lines Parallel
(continued)
____________ PERIOD _____
Study Guide and Intervention
Prove Lines Parallel
3-5
NAME ______________________________________________ DATE
Answers (Lesson 3-5)
Glencoe Geometry
Lesson 3-5
©
Perpendiculars and Distance
Glencoe/McGraw-Hill
A17
©
X
S
P
R X
Q
T
SX
B
Glencoe/McGraw-Hill
T
5. S to QR
U
R
3. T to RS
A
C
1. C to AB
S
155
R
6. S to RT
T
R
A
T
P X Q
4. S to PQ
X
D
2. D to AB
X
S
Draw the segment that represents the distance indicated.
Exercises
P
B
Draw the segment that represents the distance
. from E to AF Extend AF. Draw 苶 EG 苶 ⊥ AF. EG 苶 苶 represents the distance from E to AF.
Example
When a point is not on a line, the distance from the point to the line is the length of the segment that contains the point and is perpendicular to the line.
C
A
A
F
F
G
E
E
distance between M and PQ
Glencoe Geometry
B
B
Q
M
____________ PERIOD _____
Study Guide and Intervention
Distance From a Point to a Line
3-6
NAME ______________________________________________ DATE
Perpendiculars and Distance
Study Guide and Intervention
(continued)
____________ PERIOD _____
ᐉ
m
x
O
x
Glencoe Geometry
Glencoe/McGraw-Hill
Answers
©
11
1. y 8 y 3
兹8 苶
p and m ,
p and m is (2, 0).
m is 兹5苶 units.
兹苶 5
Glencoe Geometry
3. y 2x y 2x 5
兹5 苶 The distance between ᐉ and
兹苶 (2 0苶 )2 (苶 0 1苶 )2
(x2 苶 x1) 2 苶 ( y2 苶 y1)2 d 兹苶
Use the Distance Formula to find the distance between (0, 1) and (2, 0).
The point of intersection of
y x 1
1 2 1 (2) 1 1 1 0 2
Substitute 2 for x to find the y-coordinate.
4x 8 x 2 5x 10 x2
2x 4 x 1
1 2
Use substitution.
2
Line m : y 2x 4 1 Line p : y x 1
156
2. y x 3 yx1
m whose To find the point of intersection of solve a system of equations.
Find the distance between each pair of parallel lines.
Exercises
p has slope 12 and y-intercept 1. An 1 equation of p is y x 1. The point of 2 intersection for p and ᐉ is (0, 1). Line
(0, 1)
Draw a line p through (0, 1) that is perpendicular to ᐉ and m . y ᐉ p m
O
y
Example Find the distance between the parallel lines ᐉ and equations are y ⫽ 2x ⫹ 1 and y ⫽ 2x ⫺ 4, respectively.
Distance Between Parallel Lines The distance between parallel lines is the length of a segment that has an endpoint on each line and is perpendicular to them. Parallel lines are everywhere equidistant, which means that all such perpendicular segments have the same length.
3-6
NAME ______________________________________________ DATE
Answers (Lesson 3-6)
Lesson 3-6
