Download Ready to Go on Chapter 3

Name
Lesson
3-1
Date
Class
Ready to Go On? Skills Intervention
Lines and Angles
Find these vocabulary words in the lesson and the Multilingual Glossary.
Vocabulary
parallel lines
perpendicular lines
skew lines
transversal corresponding angles
alternate interior angles
alternate exterior angles
parallel planes
same-side interior angles
Identifying Types of Lines and Planes
Identify each of the following.
O
P
A. Skew segments do not lie in the same
;
and do not
they are not
L
.
B. Perpendicular segments intersect at a
D
E
Name two segments in the figure that are skew.
N
M
A
C
B
angle. Name a pair of perpendicular
segments in the figure.
C. Parallel lines are
and do not
. Name a pair of
parallel segments in the figure.
Classifying Pairs of Angles
Give an example of each angle pair.
A. Corresponding angles lie on the same
of the
In the figure, /1 and /
r
side of the transversal
the other two lines.
are same-side interior angles.
C. Alternate exterior angles lie on
sides of the transversal and are
are alternate exterior angles.
the other two lines. In the figure, /5 and /
D. Alternate interior angles lie on
sides of the transversal and are
the other two lines. In the figure, /3 and /
Copyright © by Holt, Rinehart and Winston.
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s
are corresponding angles.
B. Same-side interior angles lie on the
and are
4
1
7
of the other
transversal and are on the same
two lines. In the figure, /3 and /
8
5 t
3 6
2
29
are alternate interior angles.
Holt McDougal Geometry
Name
Lesson
3-1
Date
Class
Ready to Go On? Skills Intervention
Lines and Angles
Find these vocabulary words in the lesson and the Multilingual Glossary.
Vocabulary
parallel lines
perpendicular lines
skew lines
transversal corresponding angles
alternate interior angles
alternate exterior angles
parallel planes
same-side interior angles
Identifying Types of Lines and Planes
Identify each of the following.
O
P
plane
A. Skew segments do not lie in the same
they are not
parallel
_
;
intersect
and do not
.
_
Name two segments in the figure that are skew.
coplanar
parallel segments in the figure.
A
intersect
. Name a pair of
_
Sample answer: MN
​  
​and ​BC  ​
_
and do not
A. Corresponding angles lie on the same
two lines. In the figure, /3 and /
8
B. Same-side interior angles lie on the
between
side
side
transversal and are on the same
In the figure, /1 and /
of the
8
4
1
7
of the other
5 t
3 6
2
s
r
are corresponding angles.
same
side of the transversal
the other two lines.
2
are same-side interior angles.
C. Alternate exterior angles lie on
opposite
the other two lines. In the figure, /5 and /
D. Alternate interior angles lie on
opposite
sides of the transversal and are
7
1
29
outside
are alternate exterior angles.
sides of the transversal and are
the other two lines. In the figure, /3 and /
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
B
right
Classifying Pairs of Angles
Give an example of each angle pair.
and are
C
angle. Name a pair of perpendicular
_
_
Sample answer: AB
​  ​and ​LA  ​
B. Perpendicular segments intersect at a
C. Parallel lines are
N
M
D
E
Sample answer: MN
​  
​and AB
​  ​
segments in the figure.
L
between
are alternate interior angles.
Holt McDougal Geometry
Name
Lesson
3-2
Date
Class
Ready To Go On? Skills Intervention
Angles Formed by Parallel Lines and Transversals
Using the Corresponding Angles Postulate
Find m/RST.
Since two lines in the figure are
R (5x + 27)°
and cut
(8x + 6)°
S
by a transversal, the pairs of corresponding
T
.
are
Write an equation relating the measures of the given angles.
(5x 1 27)8 5
Solve the equation. x 5
the value of x into the expression
To find m/RST,
.
Find the measure of /RST.
Finding Angle Measures
Find each angle measure.
D
A. m/DEC
Since the two labeled angles are on
sides
(24x – 13)° E
F
C
the other
of the transversal and are
(20x + 7)°
angles.
two lines, they are
.
Since the lines in the figure are parallel, the labeled angles are
Write an equation relating the measures of the angles.
Solve the equation. x 5
the value of x into the expression
To find m/DEC,
.
Find the measure of /DEC.
B. m/DEF
so the sum of their
m/DEC and m/DEF form a
measures is
8.
from
Subtract
m/DEF 5
to find m/DEF.
2 107 5
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30
Holt McDougal Geometry
Name
Lesson
3-2
Date
Class
Ready To Go On? Skills Intervention
Angles Formed by Parallel Lines and Transversals
Using the Corresponding Angles Postulate
Find m/RST.
parallel
Since two lines in the figure are
are
and cut
(8x + 6)°
S
angles
by a transversal, the pairs of corresponding
congruent
R (5x + 27)°
T
.
Write an equation relating the measures of the given angles.
(5x 1 27)8 5
(8x 1 6)8
7
Solve the equation. x 5
To find m/RST,
substitute
the value of x into the expression
(8x 1 6) or (5x 1 27) .
628
Find the measure of /RST.
Finding Angle Measures
Find each angle measure.
D
A. m/DEC
opposite
Since the two labeled angles are on
inside
of the transversal and are
(24x – 13)° E
angles.
Since the lines in the figure are parallel, the labeled angles are
Write an equation relating the measures of the angles.
congruent
.
20x 1 7 5 24x 2 13
5
Solve the equation. x 5
To find m/DEC,
F
C
the other
alternate interior
two lines, they are
sides
(20x + 7)°
substitute
the value of x into the expression
24x 2 13 or 20x 1 7 .
1078
Find the measure of /DEC.
B. m/DEF
m/DEC and m/DEF form a
measures is
Subtract
m/DEF 5
linear pair
so the sum of their
180 8.
107
180
from
2 107 5
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180
to find m/DEF.
738
30
Holt McDougal Geometry
Name
Lesson
3-3
Date
Class
Ready To Go On? Skills Intervention
Proving Lines Parallel
Using the Converse of the Corresponding
Angles Postulate
Use the given information to show that p i q.
14
5
2 3
8 76
Given: m/2 5 (12x 2 25)8 and m/8 5 (9x 1 2)8; x 5 9
p
Substitute the value of x into each expression.
m/2 5 12(
) 2 25 5
; m/8 5 9(
q
)125
Does m/2 5 m/8?
Since m/2

m/8,
by the Property of Congruence.
angles formed by two coplanar lines cut by a
Since the
, p i q.
transversal are
14
5
2 3
8 76
Determining Whether Lines are Parallel
Using the given information and the diagram to show that p i q.
p
A. /1  /7
What type of angles are /1 and /7?
If two coplanar lines are cut by a transversal so that a pair of
q
. Since /1  /7,
angles are congruent, then the two lines are
i
.
B. m/3 5 m/5
What type of angles are /3 and /5?
Since m/3 5 m/5,


. Since
, p i q by the Converse of
.
the
R
Proving Lines Parallel
_ _
Write a paragraph proof to show that ​RS​ i ​QT​ .
Given: m/R
5 1318, m/Q 5 498
_ _
Prove: ​ RS ​ i ​ QT ​ 
Q
49°
S
T
angles by
Since m/R 5 1318 and m/Q 5 498, /R and /Q are
angles. Since /R and /Q lie on the same side
the definition of
angles.
of two coplanar lines cut by a transversal, they are
Angles Theorem, when same-side angles
By the Converse of the
are
131°
_
_
, then the two lines are parallel, so ​ RS ​ i ​ QT ​ 
.
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31
Holt McDougal Geometry
Name
Lesson
3-3
Date
Class
Ready To Go On? Skills Intervention
Proving Lines Parallel
Using the Converse of the Corresponding
Angles Postulate
Use the given information to show that p i q.
14
5
2 3
8 76
Given: m/2 5 (12x 2 25)8 and m/8 5 (9x 1 2)8; x 5 9
p
Substitute the value of x into each expression.
m/2 5 12(
9
) 2 25 5
Does m/2 5 m/8?
Since m/2
Since the
5
838 ; m/8 5 9( 9
838
Yes
m/8,
/2

corresponding
/8
by the Property of Congruence.
angles formed by two coplanar lines cut by a
congruent
transversal are
)125
q
, p i q.
14
5
2 3
8 76
Determining Whether Lines are Parallel
Using the given information and the diagram to show that p i q.
p
A. /1  /7
q
Alternate exterior angles
What type of angles are /1 and /7?
If two coplanar lines are cut by a transversal so that a pair of
angles are congruent, then the two lines are
parallel
alternate exterior
. Since /1  /7,
p
i
q
.
B. m/3 5 m/5
What type of angles are /3 and /5?
/5 . Since /3

Alternate Interior Angles Theorem
.
Since m/3 5 m/5,
the
/3
Alternate interior angles

/5 , p i q by the Converse of
R
Proving Lines Parallel
_ _
Write a paragraph proof to show that ​RS​ i ​QT​ .
Given: m/R
5 1318, m/Q 5 498
_ _
Prove: ​ RS ​ i ​ QT ​ 
the definition of
supplementary
are
supplementary
S
T
angles by
angles. Since /R and /Q lie on the same side
of two coplanar lines cut by a transversal, they are
By the Converse of the
49°
Q
Since m/R 5 1318 and m/Q 5 498, /R and /Q are
131°
Same-Side
same-side interior
angles.
Angles Theorem, when same-side angles
_
_
supplementary , then the two lines are parallel, so ​ RS ​ i ​ QT ​ .
Copyright © by Holt, Rinehart and Winston.
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31
Holt McDougal Geometry
Name
Lesson
3-4
Date
Class
Ready To Go On? Skills Intervention
Perpendicular Lines
Find these vocabulary words in the lesson and the Multilingual Glossary.
Vocabulary
perpendicular bisector
distance from a point to a line
Proving Properties of Lines
Write a two-column proof.
Given: m/1 5 m/2, b i c
Prove: d ' b
d
b
12
Plan your proof:
c
Step 1: Write the given information in the two-column proof.
Step 2: Since it is given that m/1 5 m/2, you know that /1 is
.
to /2 by the definition of
Put this information in the two-column proof.
Step 3: If two intersecting lines form a linear pair of
. So you know that d
the lines are
angles, then
c.
Put this information in Step 3 of the two-column proof.
Step 4: It is given that b i c. In Step 3, you proved that d
c. You can conclude
Theorem.
that d ' b because of the
Complete Step 4 of the two-column proof.
Statements
Reasons
1.
1. Given
2.
2.
3.
3. 4. d ' b
4.
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32
Holt McDougal Geometry
Name
Lesson
3-4
Date
Class
Ready To Go On? Skills Intervention
Perpendicular Lines
Find these vocabulary words in the lesson and the Multilingual Glossary.
Vocabulary
perpendicular bisector
distance from a point to a line
Proving Properties of Lines
Write a two-column proof.
Given: m/1 5 m/2, b i c
Prove: d ' b
d
b
12
Plan your proof:
c
Step 1: Write the given information in the two-column proof.
congruent
Step 2: Since it is given that m/1 5 m/2, you know that /1 is
to /2 by the definition of
congruent angles
.
Put this information in the two-column proof.
Step 3: If two intersecting lines form a linear pair of
the lines are
perpendicular
congruent
'
. So you know that d
angles, then
c.
Put this information in Step 3 of the two-column proof.
Step 4: It is given that b i c. In Step 3, you proved that d
that d ' b because of the
'
Perpendicular Transversal
c. You can conclude
Theorem.
Complete Step 4 of the two-column proof.
Statements
Reasons
1.
m/1 5 m/2, b i c
1. Given
2.
/1 > /2
2.
3.
d'c
3. Two
4. d ' b
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Def. of > /s
intersecting lines form a linear
pair of > /s, lines are '.
4.
32
Perpendicular Transversal Theorem
Holt McDougal Geometry
Name
Section
3A
Date
Class
Ready to Go On? Quiz
E
3-1 Lines and Angles
D
Identify each of the following.
F
1. a pair of parallel segments
A
2. a pair of perpendicular segments
B
3. a pair of skew segments
C
4. a pair of parallel planes
Give an example of each angle pair.
1 2
8 7
5. same-side interior angles
3 4
6 5
6. alternate exterior angles
7. corresponding angles
8. alternate interior angles
3-2 Angles Formed by Parallel Lines and Transversals
Find each angle measure.
9. 10.
58°
(9x – 8)°
(7x + 6)°
x°
11.
(11x + 4)°
(15x – 40)°
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All rights reserved.
33
Holt McDougal Geometry
Name
Section
3A
Date
Class
Ready to Go On? Quiz
E
3-1 Lines and Angles
Identify each of the following. Sample answers given for 1–3.
1. a pair of parallel segments
2. a pair of perpendicular segments
_ _
​  ​
​DE  ​and BA
_ _
​  ​
​DE  ​and EF
3. a pair of skew segments
​  ​and BC
EA
​  ​
F
A
B
DEF and BAC
Give an example of each angle pair.
5. same-side interior angles
6. alternate exterior angles
C
4. a pair of parallel planes
_
_
D
1 2
8 7
/2 and /3 or /7 and /6
3 4
6 5
/1 and /5 or /8 and /4
7. corresponding angles
/7 and /5; /2 and /4;
/1 and /3; /8 and /6
8. alternate interior angles
/2 and /6 or /7 and /3
3-2 Angles Formed by Parallel Lines and Transversals
Find each angle measure.
9. 10.
58°
(9x – 8)°
(7x + 6)°
x°
588
558
11.
(11x + 4)°
(15x – 40)°
1258
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
33
Holt McDougal Geometry
Name
Section
3A
Date
Class
Ready to Go On? Quiz continued
3-3 Proving Lines Parallel
Use the given information and the theorems and postulates
you have learned to show that a i b.
12. m/3 1 m/6 5 1808
a
b
1 2
4 3
7 8
6 5
13. /1  /7
14. m/4 5 (7x 2 1)8, m/8 5 (5x 1 31)8, x 5 16
15. m/7 5 m/3
_
_
16. Write a paragraph proof to show that ​ DC ​ i ​ AB ​ 
.
D
a
1 2
A4 3
C
72°
b
7 8 108°
6 5
B
t
1
3-4 Perpendicular Lines
m
17. Complete the two-column proof below.
Given: t ' m, m/1 5 m/2
Prove: n ' t
2
Statements
Reasons
1. t ' m, m/1 5 m/2
1. Given
2. /1  /2
2.
3.
3. C
onverse of the Alternate
Exterior Angles Theorem
4. n ' t
4.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
n
34
Holt McDougal Geometry
Name
Date
Class
Ready to Go On? Quiz continued
Section
3A
3-3 Proving Lines Parallel
Use the given information and the theorems and postulates
you have learned to show that a i b.
12. m/3 1 m/6 5 1808
Interior
Converse of the Same-Side
Angles Theorem
14. m/4 5 (7x 2 1)8, m/8 5 (5x 1 31)8, x 5 16
1 2
4 3
7 8
6 5
m/4 5 m/8 5 1118,
/4 > /1; Alternate Exterior Angles Theorem
15. m/7 5 m/3
Alternate
/7 > /3 by the def. of >/s;
Interior Angles Theorem
_
_
16. Write a paragraph proof to show that ​ DC ​ i ​ AB ​ 
.
D
Sample answer: 1088 1 728 5 1808; so /C and
b
Converse of the Corresponding Angles Postulate
13. /1  /7
so
a
/B are supplementary by the definition of
a
supplementary angles. Since /C and /B are on
the same side of the transversal and between the
other two lines; they are same-side interior
angles. When same-side interior angles are
supplementary the lines are parallel.
1 2
A4 3
C
72°
b
7 8 108°
6 5
B
t
1
3-4 Perpendicular Lines
m
17. Complete the two-column proof below.
Given: t ' m, m/1 5 m/2
Prove: n ' t
2
Statements
Reasons
1. t ' m, m/1 5 m/2
1. Given
2. /1  /2
2.
3.
min
4. n ' t
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
n
Def. > /s
3. C
onverse of the Alternate
Exterior Angles Theorem
4.
34
Perpendicular Transversal Theorem
Holt McDougal Geometry
Name
Section
3A
Date
Class
Ready to Go On? Enrichment
Finding Angle Measures
Use the figure at the right and the given information to answer the
questions below. s i t, s i r, l i m, n i m
r
s
m/1 5 (7x)8
m/2 5 (4x 1 18)8
m/3 5 (11a 1 10b)8

m/4 5 (6a 1 18b)8
m
m/5 5 (3y )8
n
6
t
1 35
2
4
7
m/6 5 (5a 1 2)8
m/7 5 (28b 2 5)8
1. Find the value of x.
2. Find m/1.
3. Find m/2.
4. How are /1 and /3 related?
6. What is m/4?
5. What is m/3?
7. What is the value of a?
9. Find m/5.
8. What is the value of b?
10. Find the value of y.
11. Is n i m? Explain your answer.
12. Write a paragraph proof to show that s i r.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
35
Holt McDougal Geometry
Name
Section
3A
Date
Class
Ready to Go On? Enrichment
Finding Angle Measures
Use the figure at the right and the given information to answer the
questions below. s i t, s i r, l i m, n i m
r
s
m/1 5 (7x)8
m/2 5 (4x 1 18)8
m/3 5 (11a 1 10b)8

m/4 5 (6a 1 18b)8
m
m/5 5 (3y )8
n
6
t
1 35
2
4
7
m/6 5 (5a 1 2)8
m/7 5 (28b 2 5)8
1. Find the value of x.
6
2. Find m/1.
3. Find m/2.
428
4. How are /1 and /3 related?
Same-side interior angles
6. What is m/4?
5. What is m/3?
8
9. Find m/5.
1388
7. What is the value of a?
1388
428
8. What is the value of b?
5
10. Find the value of y.
428
14
11. Is n i m? Explain your answer.
Sample answer: No, because m/7 5 28(5) 2 5 5 1358, and m/4  m/7.
12. Write a paragraph proof to show that s i r.
Sample
answer: m/6 5 5(8) 1 2 5 428; and m/6 5 m/1. By the
definition
Corresponding
of congruent angles; /6 > /1. By the Converse of the
Angles Postulate s i r
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35
Holt McDougal Geometry
Name
Date
Class
Ready to Go On? Skills Intervention
Lesson
3-5
Slopes of Lines
Find these vocabulary words in the lesson and the Multilingual Glossary.
Vocabulary
rise
run
slope
y
Finding the Slope of a Line
Use the slope formula to determine the slope of each line.
‹__›
B
A.​AB​
    
y​ 2​2 ​ 
 ​
What is the slope formula? m 5  _________ 
 
​  2 ​x​ 1​
2
–4 –2 O
What are the coordinates of A?
Substitute the coordinates
of A and B into the slope formula
‹__›
to find the slope of ​AB​
    
.
of B?
C
–4
A
2
4
x
6
D
32
 
 
 ​
m 5  _________ 
 ​5 2  _____ 
26
__
‹
›
    
B.​BD​
What are the coordinates of D?
    
Substitute the coordinates of B and D into the slope formula to find the slope of ​BD​
.
23 2
2
5  ______ ​
    ​
 
 
 
m 5   __________
21
The slope is
‹__›
‹__›
    
. What kind of line is ​BD​
?
Determining Whether Lines are Parallel,
Perpendicular,
or Neither
‹___›
‹___›
​LM
    
    
​passes through L(4, 2) and M(0, 24), and ​XY
​passes
through X(22, 5) and Y(2, 21). Use slopes to determine
whether the lines are parallel, perpendicular, or neither.
Graph the coordinates and draw each line on the grid at the right.
Find the slope of each line by substituting the coordinates into
the slope formula.
​ 2​2 ​y​ 1​
‹__›
y_______
24 2
Slope of ​LM​
    5   ​x​  ​ 2 ​x​  
 ​5  __________ 
 
 ​
 
5
2
1​
02
4
2
–4 –2 O
–2
2
4
–4
__
‹ ›
25
___________
    5  
Slope of ​XY​
  
 
 ​5  _____ 
 
 ​5  _____ 
 
 ​
2
Are they parallel?
Do the lines have the same slope?
Are the lines perpendicular?
Is the product of the slopes 21?
The lines are neither
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
nor
.
36
Holt McDougal Geometry
Name
Date
Class
Ready to Go On? Skills Intervention
Lesson
3-5
Slopes of Lines
Find these vocabulary words in the lesson and the Multilingual Glossary.
Vocabulary
rise
run
slope
y
Finding the Slope of a Line
Use the slope formula to determine the slope of each line.
‹__›
B
A.​AB​
    
y​ 2​2 ​y​ 1​
 ​
What is the slope formula? m 5  _________ 
 
​x​ 2​ 2 ​x​ 1​
(6, 1)
2
(1, 3)
What are the coordinates of A?
Substitute the coordinates
of A and B into the slope formula
‹__›
to find the slope of ​AB​
    
.
of B?
–4 –2 O
C
–4
A
2
4
x
6
D
2
32 1
 
 
 ​
m 5  _________ 
 ​5 2  _____ 
1
5
26
__
‹
›
    
B.​BD​
(1, 23)
What are the coordinates of D?
    
Substitute the coordinates of B and D into the slope formula to find the slope of ​BD​
.
23 2 3
2 6
5  ______ ​
    ​
 
 
 
m 5   __________
1 21
0
The slope is
undefined
‹__›
‹__›
    
. What kind of line is ​BD​
?
Vertical
Determining Whether Lines are Parallel,
Perpendicular,
or Neither
‹___›
‹___›
​LM
    
    
​passes through L(4, 2) and M(0, 24), and ​XY
​passes
through X(22, 5) and Y(2, 21). Use slopes to determine
whether the lines are parallel, perpendicular, or neither.
4
2
Graph the coordinates and draw each line on the grid at the right.
Find the slope of each line by substituting the coordinates into
the slope formula.
6  ​or __
​ 3 ​ 
​ __
​ 2​2 y​ ​ 1​
‹__›
y_______
24
2 2
__________
4 2
Slope of ​LM​
    5   ​x​  ​ 2 x
 ​​ 5  
 
 
 ​
 
5
​ ​ 1 
2
4
02
–4 –2 O
–2
2
4
–4
__
‹ ›
21 2 5
26 _____
23
___________
    5  
Slope of ​XY​
  
 
 ​5  _____ 
 
 ​5  
 
 
 ​
4
2
2 2 22
Do the lines have the same slope?
Is the product of the slopes 21?
The lines are neither
Copyright © by Holt, Rinehart and Winston.
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parallel
No
No
nor
Are they parallel?
No
Are the lines perpendicular?
No
perpendicular .
36
Holt McDougal Geometry
Name
Lesson
3-6
Date
Class
Ready to Go On? Skills Intervention
Lines in the Coordinate Plane
Find these vocabulary words in the lesson and the Multilingual Glossary.
Vocabulary
point-slope form
slope-intercept form
Writing Equations of Lines
A. Write the equation of the line with slope 3 through (21, 4) in point-slope form.
​ 
What is point-slope form? y 2 ​y​ 1​5
Substitute
for m,
for ​x​ 1​and
for ​y​ 1​:
B. Write the equation of the line through points (26, 2) and (3, 24) in slope-intercept form.
​
y​ 2​2  
2
2
24 2   
 ​ 
5  ______ 
 ​ 
 ​5 ​  ___________
Use the slope formula to find the slope. m 5  _________ 
 
    ​ 5  ______ 
​  2 ​x​ 1​
2 (26)
What is slope-intercept form? y 5
___ ​
Substitute ​ 22
 for m, 26 for x, and 2 for y, and then simplify to find b.
3
b 5
5  _____ 
 
 ​(26) 1 b
Write the equation in slope-intercept form.
Graphing Lines
1
Graph the line. y 1 2 5 2 __  ​(x 2 3)
2
The equation is given in
is
. The line goes through the point
point and then rise
and run
a line connecting the two points.
4
form. The slope of the line
2
. Plot the
–4 –2 O
–2
to find another point. Draw
2
4
–4
Classifying Pairs of Lines
Determine whether the lines are parallel, intersect, or coincide.
3  ​x 1 4 and 3x 2 2y 5 6
y 5 ​ __
2
and the y-intercept is
.
The slope of the first line is
Solve the second equation for y to rewrite it in slope-intercept form.
The slope of the second line is
and the y-intercepts are
The slopes lines are
are
and the y-intercept is
.
. The lines
.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
37
Holt McDougal Geometry
Name
Date
Class
Ready to Go On? Skills Intervention
Lesson
3-6
Lines in the Coordinate Plane
Find these vocabulary words in the lesson and the Multilingual Glossary.
Vocabulary
point-slope form
slope-intercept form
Writing Equations of Lines
A. Write the equation of the line with slope 3 through (21, 4) in point-slope form.
m(x 2 ​x​ 1​)
What is point-slope form? y 2 ​y​ 1​5
Substitute
3
for m,
21
for ​x​ 1​and
4
for ​y​ 1​:
y 2 4 5 3(x 1 1)
B. Write the equation of the line through points (26, 2) and (3, 24) in slope-intercept form.
​
y​ 2​2 ​y 1​
2 6
2 2
24 2 2
 ​ 
5  ______ 
 ​ 
 ​5 ​  ___________
Use the slope formula to find the slope. m 5  _________ 
 
  
    ​ 5  ______ 
​x​ 2​ 2 ​x​ 1​
9
3
3 2 (26)
What is slope-intercept form? y 5
mx 1 b
___ ​
Substitute ​ 22
 for m, 26 for x, and 2 for y, and then simplify to find b.
3
2
22
3
y 5 ___
​ 22 ​ x  1 (22)
22 Write the equation in slope-intercept form.
3
5  _____ 
 
 ​(26) 1 b
b 5
Graphing Lines
1
Graph the line. y 1 2 5 2 __  ​(x 2 3)
2
The equation is given in
point-slope
Sample answers:
form. The slope of the line
1
2 ​ __ ​
is   2 . The line goes through the point
21
point and then rise
and run
a line connecting the two points.
2
(3, 22)
. Plot the
to find another point. Draw
The slopes lines are
the same
are
.
parallel
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
2
–4 –2 O
–2
2
4
–4
Classifying Pairs of Lines
Determine whether the lines are parallel, intersect, or coincide.
3  ​x 1 4 and 3x 2 2y 5 6
y 5 ​ __
​ _32 ​ 
2
4 .
and the y-intercept is
The slope of the first line is
Solve the second equation for y to rewrite it in slope-intercept form.
The slope of the second line is
4
​ _32  ​
23 .
different
y 5 __
​ 3 ​ x 2 3
2
and the y-intercept is
and the y-intercepts are
37
. The lines
Holt McDougal Geometry
Name
Date
Class
Ready to Go On? Quiz
Section
3B
3-5 Slopes of Lines
Use the slope formula to determine the slope of each line.
‹__›
1.​AD​
    
‹__›
    
2.​AB​
y
6
D
A
2
C
–2 O
–2
‹__›
3.​AC​
    
–4
x
2
4
6
B
‹__›
    
4.​DB​
Find the slope of the line through the given points.
5. R(4, 7) and S(22, 0)
6. C(0, 24) and D(5, 9)
7. H(3, 5) and I(24, 2)
8. S(26, 1) and T(3, 26)
 
Graph each pair of lines and use their slopes to determine if they are
parallel, perpendicular, or neither.
‹___›
‹__›
‹__›
‹___›
9.​CD​
    for A(21, 0), B(1, 5),
    and ​MN​
    for L(23, 2), M(21, 5),
10.​LM​
    and ​AB​
C(4, 5), and D(22, 4) N(2, 3), and P(1, 25)
4
2
2
–4 –2 O
–2
2
–4 –2 O
–2
4
2
4
–4
–4
4
11.​‹PR​
__ 
  › and ‹​PS​
__ 
  › for P(2, 21), Q(2, 1),
12.​‹GH​
___ 
  › and ‹​FJ
__ 
  › 
​for F(23, 2), G(22, 5)
R(23, 1), and S(22, 22) H(2, 4), and J(2, 1)
4
2
2
–4 –2 O
–2
–4 –2 O
–2
4
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
2
4
–4
–4
4
38
Holt McDougal Geometry
Name
Date
Class
Ready to Go On? Quiz
Section
3B
3-5 Slopes of Lines
Use the slope formula to determine the slope of each line.
‹__›
1.​AD​
    
‹__›
    
2.​AB​
y
1
2 __ ​ 
2
__
​ 7 ​ 
2
__
​ 2 ​  or __
​ 1 ​ 
4 2
10
5
2 ___ ​ or 2 __ ​ 
4
2
‹__›
3.​AC​
    
‹__›
    
4.​DB​
6
D
A
2
C
–2 O
–2
–4
x
2
4
6
B
Find the slope of the line through the given points.
5. R(4, 7) and S(22, 0)
6. C(0, 24) and D(5, 9)
__
​ 7 ​ 
6
8. S(26, 1) and T(3, 26)
3  ​
​ __
7
5
7. H(3, 5) and I(24, 2)
___
​ 13 ​ 
7 ​ 
2 __
9
Graph each pair of lines and use their slopes to determine if they are
parallel, perpendicular, or neither.
‹___›
‹__›
‹__›
‹___›
9.​CD​
    for A(21, 0), B(1, 5),
    and ​MN​
    for L(23, 2), M(21, 5),
10.​LM​
    and ​AB​
C(4, 5), and D(22, 4) N(2, 3), and P(1, 25)
4
2
2
–4 –2 O
–2
2
–4 –2 O
–2
4
2
4
–4
–4
4
Neither
Perpendicular
11.​‹PR​
__ 
  › and ‹​PS​
__ 
  › for P(2, 21), Q(2, 1),
12.​‹GH​
___ 
  › and ‹​FJ
__ 
  › 
​for F(23, 2), G(22, 5)
R(23, 1), and S(22, 22) H(2, 4), and J(2, 1)
4
2
2
–4 –2 O
–2
–4 –2 O
–2
4
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
2
4
–4
–4
4
Neither
38
Parallel
Holt McDougal Geometry
Name
Section
3B
Date
Class
Ready to Go On? Quiz continued
3-6 Lines in the Coordinate Plane
Write the equation of each line in the given form.
13. the line through (23, 21) and (3, 23) in slope-intercept form
3
14. the line through (6, 22) with slope 2 __  ​in point-slope form
4
15. the line with y-intercept 23 through the point (2, 5) in point-slope form
16. the line with x-intercept 24 and y-intercept 2 in slope-intercept form
Graph each line.
3  ​(x 1 2)
18. y 2 1 5 ​ __
5
17. y 5 3x 2 1
19. y 5 25
4
4
4
2
2
–4 –2
2
–4 –2 O
–2
4
–4
2
–4 –2 O
–2
4
–4
2
4
–4
Write the equation of each line.
20.
21.
22.
y
y
4
2
–4 –2 O
–2
y
4
2
x
2
–4 –2 O
–2
2
2
–4 –2 O
–2
x
–4
4
2
4
x
–4
Determine whether the lines are parallel, intersect, or coincide.
1  ​x 1 2
23. y 5 2​ __
5
24. 2x 1 3y 5 9
25. y 5 5x 2 3
2  ​x 2 1 y 5 5x 1 1
x 1 5y 5 10 y 5 ​ __
3
Copyright © by Holt, Rinehart and Winston.
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39
Holt McDougal Geometry
Name
Section
3B
Date
Class
Ready to Go On? Quiz continued
3-6 Lines in the Coordinate Plane
Write the equation of each line in the given form.
1
y 5 2 __ ​ x 2 2
3
13. the line through (23, 21) and (3, 23) in slope-intercept form
y 1 2 5 2__
​ 3 ​ (x 2 6)
3
__
14. the line through (6, 22) with slope 2    ​in point-slope form
4
4
15. the line with y-intercept 23 through the point (2, 5) in point-slope form y 2 5 5 4(x 2 2)
1  ​x 1 2
y 5 ​ __
2
16. the line with x-intercept 24 and y-intercept 2 in slope-intercept form
Graph each line.
3  ​(x 1 2)
18. y 2 1 5 ​ __
5
17. y 5 3x 2 1
19. y 5 25
4
4
4
2
2
–4 –2
2
–4 –2 O
–2
4
–4
2
–4 –2 O
–2
4
–4
2
4
–4
Write the equation of each line.
20.
21.
22.
y
y
4
2
–4 –2 O
–2
y
4
2
x
2
–4 –2 O
–2
2
2
–4 –2 O
–2
x
–4
x54
4
2
4
x
–4
3  ​x 2 3
y 5 ​ __
5
y 5 23
Determine whether the lines are parallel, intersect, or coincide.
1  ​x 1 2
23. y 5 2​ __
5
24. 2x 1 3y 5 9
25. y 5 5x 2 3
2  ​x 2 1 y 5 5x 1 1
x 1 5y 5 10 y 5 ​ __
3
Coincide
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Intersect
39
Parallel
Holt McDougal Geometry
Name
Section
3B
Date
Class
Ready to Go On? Enrichment
Slopes and Lengths of Segments
Quadrilateral ABCD has vertices A(25, 3), B(21, 4), C(5, 23) and
D(24, 21).
4
2
1. Sketch and label the quadrilateral using the grid at the right.
_
–4 –2 O
–2
_
2
4
2
4
–4
2. Find the slopes of ​ AC ​ and ​ BD ​ 
.
3. How are the segments related?
Quadrilateral PQRS has vertices P(2, 3), Q(2, 22), R(22, 25), S(22, 0).
Use the information to answer the following questions:
4. Sketch and label the quadrilateral using the grid at the right.
4
2
–4 –2 O
–2
–4
Find the length of each segment.
5. Find PQ.
6. Find QR.
7. Find RS.
8. Find PS.
9. What can you conclude about the side lengths of the quadrilateral?
_
10. What is the slope of ​ PR ​ 
?
_
11. What is the slope of ​ QS ​ 
?
12. What can you conclude about the diagonals of the quadrilateral?
13. Is the quadrilateral a square? Explain your answer.
_
14. A triangle has vertices L(2, 8), M(5, 9), and N(4, 2). Write a paragraph proof to show that
triangle LMN is a right triangle.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
40
Holt McDougal Geometry
Name
Section
3B
Date
Class
Ready to Go On? Enrichment
Slopes and Lengths of Segments
4• •
Quadrilateral ABCD has vertices A(25, 3), B(21, 4), C(5, 23) and
D(24, 21).
••
1. Sketch and label the quadrilateral using the grid at the right.
–4 –2 O
• • –2
_
_
2. Find the slopes of ​ AC ​ and ​ BD ​ 
.
3. How are the segments related?
2
3
 ​  and __
​ 5 ​ 
2 __
5
3
4
2
••
–4 –2 O
–2
5. Find PQ.
8. Find PS.
5 units
5 units
9. What can you conclude about the side lengths of the quadrilateral?
_
10. What is the slope of ​ PR ​ 
?
2
4
• •
5 units
7. Find RS.
2
6. Find QR.
5 units
••
–4
••
Find the length of each segment.
••
They are perpendicular.
4. Sketch and label the quadrilateral using the grid at the right.
4
–4
Quadrilateral PQRS has vertices P(2, 3), Q(2, 22), R(22, 25), S(22, 0).
Use the information to answer the following questions:
2
They are congruent.
_
11. What is the slope of ​ QS ​ 
?
1
2 __ ​ 
2
12. What can you conclude about the diagonals of the quadrilateral? They are perpendicular.
13. Is the quadrilateral a square? Explain your answer.
right
_
No; the sides do not meet at
angles.
14. A triangle has vertices L(2, 8), M(5, 9), and N(4, 2). Write a paragraph proof to show that
triangle LMN is a right triangle.
_ __1
​is ​   ​and the slope of
Sample answer: Using the slope formula; the slope of ​LM  
3
_
​is 23. The product of the slopes is 21, so by the Perpendicular Lines Theorem,
​LN  
the segments are perpendicular. Perpendicular lines form right angles, and by
definition, a right triangle is a triangle that has one right angle.
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All rights reserved.
40
Holt McDougal Geometry