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Transcript
!
!
!
3-1
Lines and Angles
Vocabulary
Review
Write T for true or F for false.
1. You can name a plane by a capital letter, such as A.
2. A plane contains a finite number of lines.
3. Two points lying on the same plane are coplanar.
4. If two distinct planes intersect, then they intersect in exactly one line.
Vocabulary Builder
PA
The symbol for
parallel is .
ruh lel
Definition: Parallel lines lie in the same plane but never intersect,
no matter how far they extend.
Use Your Vocabulary
5. Circle the segment(s) that are parallel to the x-axis.
AB
BC
CD
AD
6. Circle the segment(s) that are parallel to the y-axis.
AB
BC
CD
parallelogram
AD
square
trapezoid
Complete each statement below with line or segment.
8. A 9 consist of two endpoints and all the
points between them.
9. A 9 is made up of an infinite number of points.
Chapter 3
58
3
2
1
Ľ5 Ľ4 Ľ3 Ľ2 Ľ1 O
7. Circle the polygon(s) that have two pairs of parallel sides.
rectangle
A
D
Ľ2
Ľ3
y
B
x
1 2 3 4 5
C
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
parallel (noun)
Key Concept Parallel and Skew
Parallel lines are coplanar lines that do not intersect.
Parallel planes are planes that do not intersect.
A
coplanar
* )
* )
* )
* )
CB and AE
Parallel
G
F
E
do not intersect
AE and CG
intersect
B
H
10. Write each word, phrase, or symbol in the correct oval.
noncoplanar
C
D
Skew lines are noncoplanar; they are not parallel and do not intersect.
Use arrows to show
X X
X X
AE I BF and AD I BC.
Skew
Problem 1 Identifying Nonintersecting Lines and Planes
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Got It? Use the figure at the right. Which segments are parallel to AD?
11. In plane ADHE,
is parallel to AD.
12. In plane ADBC,
is parallel to AD .
13. In plane ADGF,
is parallel to AD .
C
B
D
A
G
F
Got It? Reasoning Explain why FE and CD are not skew.
E
H
14. Cross out the words or phrases below that do NOT describe skew lines.
coplanar
do not intersect
parallel
intersect
noncoplanar
not parallel
15. Circle the correct statement below.
Segments and rays can be skew if they lie in skew lines.
Segments and rays are never skew.
16. Underline the correct words to complete the sentence.
FE and CD are in a plane that slopes from the bottom / top left edge to the
bottom / top right edge of the figure.
17. Why are FE and CD NOT skew?
_______________________________________________________________________
_______________________________________________________________________
59
Lesson 3-1
Key Concept Angle Pairs Formed by Transversals
Alternate interior angles are nonadjacent interior angles
that lie on opposite sides of the transversal.
t
Exterior
1 2
4 3
Same-side interior angles are interior angles that lie on
the same side of the transversal.
Interior
Corresponding angles lie on the same side of a
transversal t and in corresponding positions.
5 6
8 7
Alternate exterior angles are nonadjacent exterior angles
that lie on opposite sides of the transversal.
m
Exterior
Use the diagram above. Draw a line from each angle pair in Column A to its
description in Column B.
Column A
Column B
18. /4 and /6
alternate exterior angles
19. /3 and /6
same-side interior angles
20. /2 and /6
alternate interior angles
21. /2 and /8
corresponding angles
Identifying an Angle Pair
Got It? What are three pairs of corresponding angles in the diagram at
m
the right?
n
1 2
8 7
Underline the correct word(s) or letter(s) to complete each sentence.
3 4
6 5
22. The transversal is line m / n / r .
r
23. Corresponding angles are on the same side / different sides of the
transversal.
24. Name three pairs of corresponding angles.
/
and /
/
and /
/
and /
/
and /
Problem 3 Classifying an Angle Pair
Got It? Are angles 1 and 3 alternate interior angles, same-side interior
1 2
angles, corresponding angles, or alternate exterior angles?
25. Are /1 and /3 on the same side of the transversal?
Yes / No
26. Cross out the angle types that do NOT describe /1 and /3.
alternate exterior
alternate interior
corresponding
27. /1 and /3 are 9 angles.
Chapter 3
60
same-side interior
3
4
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Problem 2
Lesson Check • Do you know HOW?
Name one pair each of the segments or planes.
28. parallel segments
29. skew segments
AB 6
30. parallel planes
ABCD 6
HD and
F
B
E
H
C
A
D
Name one pair each of the angles.
31. alternate interior
32. same-side interior
/8 and /
8
/8 and /
33. corresponding
5
34. alternate exterior
/1 and /
3
4
6
G
1
7
2
/7 and /
Lesson Check • Do you UNDERSTAND?
Error Analysis Carly and Juan examine the figure at the right.
Carly says AB 6 HG. Juan says AB and HG are skew. Who is
correct? Explain.
D
C
A
B
G
Write T for true or F for false.
H
F
35. Parallel segments are coplanar.
E
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
36. There are only six planes in a cube.
37. No plane contains AB and HG .
38. Who is correct? Explain.
_______________________________________________________________________
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
angle
parallel
skew
transversal
Rate how well you can classify angle pairs.
Need to
review
0
2
4
6
8
Now I
get it!
10
61
Lesson 3-1
Name
3-1
Class
Date
Additional Problems
Lines and Angles
Problem 1
a. Which segments are parallel to RU ?
Y
b. Which segments are skew to SW ?
V
c. What are two pairs of parallel planes?
R
X
W
U
T
S
d. What are two segments parallel to plane RUYV?
Problem 2
Which is a pair of alternate interior angles?
5
A. /1 and /2
B. /3 and /6
1
7 8
6
3 4
2
C. /2 and /7
D. /4 and /8
Problem 3
In the figure above, are /3 and /7 alternate interior angles,
same-side interior angles, corresponding angles, or alternate
exterior angles?
Prentice Hall Geometry • Additional Problems
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
29
Name
3-1
Class
Date
ELL Support
Lines and Angles
Concept List
alternate exterior angles alternate interior angles
corresponding
angles
parallel lines
parallel planes
plane
sam e-side interior angles skew lines
transversal
Choose the concept from the list above that best represents
the item in each box.
1.
2.
3.
4.
5.
6.
7.
8.
9.
Prentice Hall Geometry • Teaching Resources
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
1
Properties of Parallel Lines
3-2
Vocabulary
Review
1. Circle the symbol for congruent.
>
5
6
Identify each angle below as acute, obtuse, or right.
2.
3.
4.
72í
125í
Vocabulary Builder
interior (noun) in TEER ee ur
m
E
interior
Related Words: inside (noun), exterior (noun, antonym)
Definition: The interior of a pair of lines is the region between the two lines.
Example: A painter uses interior paint for the inside of a house.
Use Your Vocabulary
Use the diagram at the right for Exercises 5 and 6. Underline the
correct point to complete each sentence.
B
A
5. The interior of the circle contains point A / B / C .
C
6. The interior of the angle contains point A / B / C .
7. Underline the correct word to complete the sentence.
The endpoint of an angle is called its ray / vertex .
A
8. Write two other names for /ABC in the diagram at the right.
B
Chapter 3
62
1
C
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Main Idea: The interior is the inside of a figure.
Postulate 3-1, Theorems 3-1, 3-2, 3-3
Then...
Postulate 3-1
corresponding
angles are congruent.
If...
Postulate
Theorem 3-1
alternate interior
a transversal
Corresponding Angles
Alternate Interior
angles are congruent.
Angles Theorem
intersects two
parallel lines,
same-side interior angles
are supplementary.
Theorem 3-2
Same-Side Interior
Angles Theorem
Theorem 3-3
alternate exterior
angles are congruent.
Alternate Exterior
Angles Theorem
Use the graphic organizer and the diagram to find each congruent angle.
9. Postulate 3-1
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
/3 >
10. Theorem 3-1
4 5
3 6
2 7
1 8
11. Theorem 3-3
/3 >
/1 >
r
s
Problem 1 Identifying Congruent Angles
Got It? Reasoning One way to justify ml5 5 55 is shown below.
Can you find another way to justify ml5 5 55? Explain.
1
m/1 5 55 by the Vertical Angles Theorem.
m/5 5 55 by the Corresponding Angles Postulate because /1
and /5 are corresponding angles.
5
2
4
55
6
8 7
12. Write a reason for each statement.
m/7 5 55
m/5 5 m/7
m/5 5 55
63
Lesson 3-2
Name
3-2
Class
Date
Additional Problems
Properties of Parallel Lines
Problem 1
Which angles measure 130?
1
5
2
3
6
130
4 8
Problem 2
Given: a 6 b
1
Prove: /1 and /3 are supplementary.
a
3
2
Prentice Hall Geometry • Additional Problems
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
30
b
Name
3-2
Class
Date
Additional Problems (continued)
Properties of Parallel Lines
Problem 3
What are the measures of /8 and /4?
Explain.
5
2 6
4
3
m
1
8
115
n
Problem 4
What is the value of y?
y
Prentice Hall Geometry • Additional Problems
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
31
60 50
a
b
Name
3-2
Class
Date
ELL Support
Properties of Parallel Lines
Complete the vocabulary chart by filling in the missing information.
Word or
Word Phrase
Definition
Picture or Example
angle pairs
Angle pairs are two angles that are
related in some way.
parallel lines
1.
transversal
2.
congruent
Two angles are congruent if the angles
have the same measure.
3.
supplementary
angles
The sum of the measures of two
supplementary angles is 180.
4.
complementary
angles
5.
corresponding
angles
6.
congruent angles,
supplementary angles,
corresponding angles,
exterior angles,
interior angles, vertical
angles
Prentice Hall Geometry • Teaching Resources
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
11
3-3
Proving Lines Parallel
Vocabulary
Review
Write the converse of each statement.
1. Statement: If you are cold, then you wear a sweater.
Converse: If 9, then 9.
If
, then
.
2. Statement: If an angle is a right angle, then it measures 90°.
Converse:
3. The converse of a true statement is always / sometimes / never true .
Vocabulary Builder
E
m
Related Words: exterior (noun), external, interior (antonym)
exterior
Definition: Exterior means on the outside or in an outer region.
Example: Two lines crossed by a transversal form four exterior angles.
Use Your Vocabulary
Underline the correct word to complete each sentence.
4. To paint the outside of your house, buy interior / exterior paint.
5. The protective cover prevents the interior / exterior of the book
from being damaged.
6. In the diagram at the right, angles 1 and 7 are alternate interior / exterior angles.
7. In the diagram at the right, angles 4 and 5 are same-side interior / exterior angles.
Underline the hypothesis and circle the conclusion in the following statements.
8. If the lines do not intersect, then they are parallel lines.
9. If the angle measures 180˚, then it is a straight angle.
Chapter 3
66
1 2
4 3
5 6
8 7
E
m
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
exterior
exterior (adjective) ek STEER ee ur
Postulates 3-1 and 3-2 Corresponding Angles Postulate and Its Converse
Postulate 3-1 Corresponding Angles Postulate
If a transversal intersects two parallel lines, then corresponding angles are congruent.
10. Complete the statement of Postulate 3-2.
Postulate 3-2 Converse of the Corresponding Angles Postulate
If two lines on a transversal form corresponding angles that are congruent,
then the lines are 9.
11. Use the diagram below. Place appropriate marking(s) to show that /1 and
/2 are congruent.
r
s
1
2
t
12. Circle the diagram that models Postulate 3-2.
1
2
E
1
m
E
m
2
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Theorems 3-4, 3-5, and 3-6
Theorem 3-4 Converse of the Alternate Interior Angle Theorem
If two lines and a transversal form alternate interior angles that are congruent, then
the two lines are parallel.
Theorem 3-5 Converse of the Same-Side Interior Angles Theorem
If two lines and a transversal form same-side interior angles that are supplementary,
then the two lines are parallel.
Theorem 3-6 Converse of the Alternate Exterior Angles Theorem
If two lines and a transversal form alternate exterior angles that are congruent, then
the two lines are parallel.
13. Use the diagram at the right to complete each example.
Theorem 3-4
Theorem 3-5
Theorem 3-6
If /4 >
then b6c.
If /3 and
are supplementary,
then b6c.
If /1 >
then b6c.
,
67
,
6 7
5 8
4 3
1 2
b
c
Lesson 3-3
Problem 1
Identifying Parallel Lines
Got It? Which lines are parallel if l6 O l7? Justify your answer.
14. Underline the correct word(s) to complete each sentence.
/6 > /7 is given / to prove .
a
3
b
4 5
m
6
1
8
/6 and /7 are alternate / same-side angles.
2
7
/6 and /7 are corresponding / exterior / interior angles.
I can use Postulate 3-1 / Postulate 3-2 to prove the lines parallel.
Using /6 > /7, lines a and b / ! and m are parallel and the transversal is a/ b / !/ m .
Problem 2
Writing a Flow Proof of Theorem 3-6
Got It? Given that l1 O l7. Prove that l3 O l5 using a flow proof.
15. Use the diagram at the right to complete the flow proof below.
1
m
5 6
3
7
Given
Ƌ7 ƹ
Ƌ3 ƹƋ7
Ƌ1 ƹƋ3
Ƌ3 ƹƋ5
Vertical angles
Problem 3
Determining Whether Lines Are Parallel
Got It? Given that l1 O l2, you can use the Converse of the Alternate Exterior
Angles Theorem to prove that lines r and s are parallel. What is another way to
explain why r ns? Justify your answer.
16. Justify each step.
3
1
/1 > /2
/2 > /3
/1 > /3
17. Angles 1 and 3 are alternate / corresponding .
18. What postulate or theorem can you now use to explain why r6s?
_______________________________________________________________________
Chapter 3
68
s
r
2
t
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
are ƹ.
Problem 4 Using Algebra
Got It? What is the value of w for which c n d?
c
55
Underline the correct word to complete each sentence.
d
(3w 2)
19. The marked angles are on opposite sides / the same side of the transversal.
20. By the Corresponding Angles Postulate, if c 6 d then corresponding angles are
complementary / congruent / supplementary .
21. Use the theorem to solve for w.
Lesson Check • Do you UNDERSTAND?
* ) * )
Error Analysis A classmate says that AB n DC based on the diagram at right. Explain
your classmate's error.
A
22. Circle the segments that are sides of /D and /C. Underline the transversal.
AB
BC
DC
DA
D
B
83
97
C
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
23. Explain your classmate’s error.
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
flow proof
two-step proof
parallel lines
Rate how well you can prove that lines are parallel.
Need to
review
0
2
4
6
8
Now I
get it!
10
69
Lesson 3-3
Name
Class
Date
Additional Problems
3-3
Proving Lines Parallel
Problem 1
Which lines are parallel if /2 > /3? Justify
your answer.
a
1
2 8
3
4
Problem 2
Use the diagram below. Write a flow proof.
1
2
a
5
4
b
Given: m/5 5 40; m/2 5 140
Prove: a 6 b
Prentice Hall Geometry • Additional Problems
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
32
b
n
5
6
7
m
Name
3-3
Class
Date
Additional Problems (continued)
Proving Lines Parallel
Problem 3
If m/1 5 65 and m/2 5 115, are lines
a and b parallel? Explain.
a
1
2
b
Problem 4
What is the value of x for which a 6 b?
a
48
(3x 3)
Prentice Hall Geometry • Additional Problems
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
33
b
Name
3-3
Class
Date
ELL Support
Proving Lines Parallel
The column on the left shows the steps for proving lines parallel. Use the
column on the left to answer each question in the column on the right.
1. What does converse mean?
Proving Lines Parallel
How can you prove the Converse of the
Same-Side Interior Angles Theorem?
Given: ∠ 3 and ∠ 5
are supplementary.
Prove: a ║ b
__________________________________
__________________________________
__________________________________
__________________________________
Start with the given information.
2. What does given mean?
∠3 and ∠5 are supplementary.
__________________________________
__________________________________
Use the definition of supplementary angles.
3. What are supplementary angles?
__________________________________
∠3 and ∠1 are supplementary.
__________________________________
Recognize that supplements of the same angle
are congruent.
4. What does the symbol ≅ stand for?
__________________________________
∠5 ≅ ∠1
__________________________________
Recall that if corresponding
lines are parallel.
are congruent,
5. How do corresponding angles
correspond with each other?
a and b are parallel.
__________________________________
__________________________________
__________________________________
Prentice Hall Geometry • Teaching Resources
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
21
Parallel and
Perpendicular Lines
3-4
Vocabulary
Review
Complete each statement with always, sometimes or never.
1. A transversal 9 intersects at least two lines.
2. A transversal 9 intersects two lines at more than two points.
3. A transversal 9 intersects two parallel lines.
4. A transversal 9 forms angles with two other lines.
Vocabulary Builder
transitive (adjective)
TRAN
Transitive
si tiv
If A B
and B C
then A C.
Related Words: transition, transit, transitivity
Main Idea: You use the Transitive Property in proofs when what
you know implies a statement that, in turn, implies what you want to prove.
Definition: Transitive describes the property where one element in relation to a
second element and the second in relation to the third implies the first element is
in relation to the third element.
Use Your Vocabulary
Complete each example of the Transitive Property.
6. If Joe is younger than Ann
5. If a . b
and b . c,
then
.
7. If you travel from
and Ann is younger than
Station 2 to Station 3
Sam, then
and you travel from
.
,
then you travel from
Station 2 to Station 4.
Chapter 3
70
Theorem 3-7 Transitive Property of Parallel Lines and Theorem 3-8
8. Complete the table below.
Theorem 3-7
Theorem 3-8
Transitive Property of Parallel Lines
In a plane, if two line are perpendicular to the
same line, then they are parallel to each other.
If two lines are parallel to the same line,
then they are parallel to each other.
If
a b
m ȓt
and
n ȓt
then
a c
m
n
Problem 1 Solving a Problem With Parallel Lines
Got It? Can you assemble the pieces at the right to form a
picture frame with opposite sides parallel? Explain.
9. Circle the correct phrase to complete the sentence.
60
60
60
60
30˚
30˚
To make the picture frame, you will glue 9.
the same angle to the same angle
two different angles together
10. The angles at each connecting end measure
8 and
8.
8.
11. When the pieces are glued together, each angle of the frame will measure
12. Complete the flow chart below with parallel or perpendicular.
The left piece will be
The top and bottom
to the right piece.
Opposite sides are
pieces will be
.
to the side pieces.
The top piece will be
to the bottom piece.
13. Underline the correct words to complete the sentence.
Yes / No , I can / cannot assemble the pieces to form a picture frame with opposite
sides parallel.
71
Lesson 3-4
Theorem 3-9 Perpendicular Transversal Theorem
n
In a plane, if a line is perpendicular to one of two parallel lines,
then it is also perpendicular to the other.
14. Place a right angle symbol in the diagram
at the right to illustrate Theorem 3-9.
E
m
Use the information in each diagram to complete each statement.
15. a
g
n
16.
t
p
c
a6
and a '
, so
and n 6
c'
.
'
, so
.
'
Problem 2 Proving a Relationship Between Two lines
Got It? Use the diagram at the right. In a plane, c ' b, b ' d,
and d ' a. Can you conclude that anb? Explain.
c
17. Circle the line(s) perpendicular to a. Underline the line(s)
perpendicular to b.
d
a
b
c
d
a
18. Lines that are perpendicular to the same line are parallel / perpendicular .
19. Can you conclude that a6b ? Explain.
_______________________________________________________________________
_______________________________________________________________________
Lesson Check • Do you know HOW?
In one town, Avenue A is parallel to Avenue B. Avenue A is also perpendicular
to Main Street. How are Avenue B and Main Street related? Explain.
20. Label the streets in the diagram A for Avenue A, B for Avenue B, and M for
Main Street.
21. Underline the correct word(s) to complete each sentence.
The Perpendicular Transversal Theorem states that, in a plane, if a line is
parallel / perpendicular to one of two parallel / perpendicular lines, then it is
also parallel / perpendicular to the other.
Avenue B and Main Street are parallel / perpendicular streets.
Chapter 3
72
b
Name
3-4
Class
Date
Additional Problems
Parallel and Perpendicular Lines
Problem 1
What is the relationship between segments
AB and CD? Explain.
B
68
22
35
55
C
D
A
Problem 2
In a plane, c ' b, b ' d, and d ' a.
Prove that c 6 d.
c
d
a
Prentice Hall Geometry • Additional Problems
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
34
b
Name
Class
3-4
Date
ELL Support
Parallel and Perpendicular Lines
Choose the word from the list below that best matches each sentence.
intersect
parallel
perpendicular
transversal
1.
two lines in the same plane that never intersect
2.
a line that crosses two lines in the same plane at two different points
3.
what two lines do that cross at a point
4.
two lines that cross at a 90° angle
Use a word from the list above to complete each sentence.
5. The symbol ┴ is used to show that lines are ______________ .
6. If two lines are parallel to the same line, they are ____________ to each other.
7. The symbol || is used to show that lines are _____________ .
8. Parallel lines are lines that never ________________ .
Draw the lines described below.
9.
Draw parallel lines a and
b and transversal c.
10. Draw line d
11.
perpendicular to line e.
Draw line f intersecting
line g.
Multiple Choice
12. What type of angle is formed by the intersection of two perpendicular lines?
acute
obtuse
right
straight
13. If lines a and b are perpendicular to line c, what word best describes how lines a and b
relate to each other?
intersecting
parallel
perpendicular
Prentice Hall Geometry • Teaching Resources
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31
skew
3-5
Parallel Lines and
Triangles
Vocabulary
Review
Identify the part of speech for the word alternate in each sentence below.
1. You vote for one winner and one alternate.
2. Your two friends alternate serves during tennis.
3. You and your sister babysit on alternate nights.
4. Write the converse of the statement.
Statement: If it is raining,
raining then I need an umbrella.
Converse:
tri- (prefix) try
Related Word: triple
Main Idea: Tri- is a prefix meaning three that is used to form compound words.
Examples: triangle, tricycle, tripod
Use Your Vocabulary
Write T for true or F for false.
5. A tripod is a stand that has three legs.
6. A triangle is a polygon with three or more sides.
7. A triatholon is a race with two events — swimming and bicycling.
8. In order to triple an amount, multiply it by three.
Chapter 3
74
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Vocabulary Builder
Theorem 3-11 Triangle Exterior Angle Theorem
An exterior angle of a polygon is an angle formed by a side and an extension
of an adjacent side. For each exterior angle of a triangle, the two nonadjacent
interior angles are its remote interior angles.
2
The measure of each exterior angle of a triangle equals the sum of the measures
of its two remote interior angles.
18.
1
5 m/2 1 m/3
3
Circle the number of each exterior angle and draw a box around the number of
each remote interior angle.
19.
20.
5
3
6
4
1
2
Problem 2 Using the Triangle Exterior Angle Theorem
Got It? Two angles of a triangle measure 53. What is the
measure of an exterior angle at each vertex of the triangle?
Label the interior angles 538, 538, and a.
Label the exterior angles adjacent to the 538 angles as x and y.
Label the third exterior angle z.
22. Complete the flow chart.
Triangle Angle-Sum
53 à 53 à a â
aâ
Ľ
â
Exterior Angle
x âa à
â
â
Chapter 3
Exterior Angle
zâ
y âa à
à
à
â
â
76
Exterior Angle
â
à
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21. Use the diagram at the right.
Problem 3 Applying the Triangle Theorems
B
30í
xí
Got It? Reasoning Can you find mlA without using the
A
Triangle Exterior Angle Theorem? Explain.
80í
23. /ACB and /DCB are complementary / supplementary angles.
24. Find m/ACB.
C
D
25. Can you find m/A if you know two of the angle measures? Explain.
__________________________________________________________________________________
Lesson Check • Do you UNDERSTAND?
3
Explain how the Triangle Exterior Angle Theorem makes sense based on the
Triangle Angle-Sum Theorem.
1
26. Use the triangle at the right to complete the diagram below.
2 4
à mƋ2 â180
Triangle Angle-Sum Theorem
mƋ1 à mƋ3 â mƋ4
à mƋ2 â180
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Linear Pair Postulate
27. Explain how the Triangle Exterior Angle Theorem makes sense based on the
Triangle Angle-Sum Theorem.
______________________________________________________________________________________
______________________________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
exterior angle
remote interior angles
Rate how well you can use the triangle theorems.
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review
0
2
4
6
8
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get it!
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Lesson 3-5
Name
Class
3-5
Date
Additional Problems
Parallel Lines and Triangles
Problem 1
Solve for x, y, and z in the figure below.
36 ° 32 °
z°
y° x°
46 °
Problem 2
a. What is the measure of /1?
1
46 °
Prentice Hall Geometry • Additional Problems
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35
41 °
Name
3-5
Class
Date
Additional Problems (continued)
Parallel Lines and Triangles
Problem 2
b. What is the measure of /2?
2
53 °
119 °
Problem 3
What is the value of x?
x°
A. 170
B. 110
110 °
C. 70
D. 50
Prentice Hall Geometry • Additional Problems
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36
60 °
Constructing Parallel
and Perpendicular Lines
3-6
Vocabulary
Review
Write T for true or F for false.
1. A rectangle has two pairs of parallel segments.
2. A rectangle has two pairs of perpendicular segments.
Write alternate exterior, alternate interior, or corresponding to describe each
angle pair.
4.
1
2
5.
4
5
3
6
Vocabulary Builder
construction (noun) kun STRUCK shun
Other Word Forms: construct (verb), constructive (adjective)
Main Idea: Construction means how something is built or constructed.
Math Usage: A construction is a geometric figure drawn using a straightedge
and a compass.
Use Your Vocabulary
6. Complete each statement with the correct form of the word construction.
VERB
You 9 sand castles at the beach.
NOUN
The 9 on the highway caused quite a traffic jam.
ADJECTIVE
The time you spent working on your homework was 9.
Chapter 3
78
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3.
3-7
Equations of Lines in the
Coordinate Plane
Vocabulary
Review
Write T for true or F for false.
1. An ordered pair describes the location of a point in a coordinate grid.
2. An ordered pair can be written as (x-coordinate, y-coordinate) or (y-coordinate,
x-coordinate).
3. The ordered pair for the origin is (0, 0).
Vocabulary Builder
Slope â
slope (noun, verb) slohp
rise
run
Definition: The slope of a line m between two points (x1, y1) and
(x2, y2) on a coordinate plane is the ratio of the vertical change (rise) to
rise
y 2y
m 5 run 5 x2 2 x1
2
1
Use Your Vocabulary
Complete each statement with the appropriate word from the list. Use each word
only once.
slope
sloping
sloped
4. The 9 of the hill made it difficult for bike riding.
5. The driveway 9 down to the garage.
6. The 9 lawn led to the river.
Draw a line from each word in Column A to its corresponding part of speech in
Column B.
Column A
Column B
7. linear
ADJECTIVE
8. line
NOUN
Chapter 3
82
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the horizontal change (run).
Problem 1 Finding Slopes of Lines
Got It? Use the graph at the right. What is the slope of line a?
9. Complete the table below to find the slope of line a.
Think
change in y-coordinates
change in x-coordinates
mâ
.
b
y2 Ľy1
x2 Ľx1
a
d
(1, 7)
6
Write
I know the slope is the ratio
y
c
(5, 7)
4 (2, 3)
(1, 2)
2
2
x
(4, 0)
6
O
8
(4, 2)
Ľ
â
Two points on line a are (2, 3) and (5, 7).
Ľ
Now I can simplify.
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Key Concept
â
Forms of Linear Equations
Definition
Symbols
The slope-intercept form of an equation of a
nonvertical line is y 5 mx 1 b, where m is the
slope and b is the y-intercept.
y 5 mx 1 b
c
c
slope y-intercept
The point-slope form of an equation of a
nonvertical line is y 2 y1 5 m(x 2 x1),
where m is the slope and (x1, y1) is a point
on the line.
y 2 y1 5 m(x 2
c
c
y-coordinate slope
x1)
c
x-coordinate
Problem 2 Graphing Lines
5
Got It? Graph y 5 3x 2 4.
y
4
10. In what form is the given equation written?
3
2
_________________________________________
11. Written as a fraction, the slope is
12. One point on the graph is (
13. From that point, move
unit (s) to the right.
1
Ľ5 Ľ4 Ľ3 Ľ2 Ľ1 O
Ľ1
.
x
1
2
3
4
5
Ľ2
,24).
Ľ3
unit(s) up and
Ľ4
Ľ5
14. Graph y 5 3x 2 4 on the coordinate plane.
83
Lesson 3-7
Writing Equations of Lines
Problem 3
Got It? What is an equation of the line with slope 212 and y-intercept 2?
15. Complete the problem-solving model below.
Know
Need
Plan
slope m 5
Write an equation of
a line.
Use
y-intercept 5
,
the slope-intercept form of
a linear equation.
16. Now write the equation.
Problem 4 Using Two Points to Write an Equation
Got It? You can use the two points given on the line at the right to
y
6
show that the slope of the line is 5 . So one equation of the line is
6
y 2 5 5 5(x 2 3). What is an equation of the line if you use (22, 21)
instead of (3, 5) in the point-slope form of the equation?
(!2, !1)
17. The equation is found below. Write a justification for each step.
!3
y 2 y1 5 m(x 2 x1)
(3, 5)
4
Write in
O
x
2
4
!2
6
6
y 1 1 5 5(x 1 2)
Got It? Use the two equations for the line shown above. Rewrite the equations in
slope-intercept form and compare them. What can you conclude?
18. Write each equation in slope-intercept form.
6
6
y 2 5 5 5(x 2 3)
y 1 1 5 5(x 1 2)
19. Underline the correct word(s) to complete each sentence.
The equations are different / the same .
Choosing (22,21) gives a different / the same equation as choosing (3, 5).
6
6
The equations y 2 5 5 5 (x 2 3) and y 1 1 5 5 (x 1 2) are / are not equivalent.
Chapter 3
84
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y 2 (21) 5 5(x 2 (22))
Problem 5 Writing Equations of Horizontal and Vertical Lines
Got It? What are the equations for the horizontal and vertical
lines through (4, 23)?
5
Write T for true or F for false.
y
4
20. Every point on a horizontal line through (4,23)
has y-coordinate of 23.
3
2
1
21. The equation of a vertical line through (4,23)
is y 5 23.
Ľ5 Ľ4 Ľ3 Ľ2 Ľ1 O
Ľ1
x
1
2
3
4
5
Ľ2
22. The equation of a vertical line through (4,23)
is x 5 4.
Ľ3
Ľ4
23. Graph the horizontal and vertical lines through (4,23)
on the coordinate plane at the right.
Ľ5
Lesson Check • Do you UNDERSTAND?
Error Analysis A classmate found the slope of the line passing through (8,22)
and (8, 10) as shown at the right. Describe your classmate’s error. Then find
the correct slope of the line passing through the given points.
24. What is your classmate’s error?
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_________________________________________________________________
88
10 (2)
0
m
12
m
m0
25. Find the slope, m.
26. The run is 8 2 8 5
, so the slope is
.
Math Success
Check off the vocabulary words that you understand.
slope
slope-intercept form
point-slope form
Rate how well you can write and graph linear equations.
Need to
review
0
2
4
6
8
Now I
get it!
10
85
Lesson 3-7
Name
3-7
Class
Date
Additional Problems
Equations of Lines in the Coordinate Plane
Problem 1
5
4
3
2
1
a. What is the slope of line a?
b. What is the slope of line b?
y
a
(1, 3)
O1 2 3 4 5 x
5 4 3 2 1
1
(2, 3)
2
3
4
5
b
Problem 2
1
a. Graph y 5 2 x 2 1.
2
b. Graph y 2 2 5 23 (x 1 4).
Problem 3
a. What is an equation in slope-intercept form of the line with
slope 2 and y-intercept 4?
b. What is an equation in point-slope form of the line that
passes through (3, 27) with slope 21?
Prentice Hall Geometry • Additional Problems
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39
Name
3-7
Class
Date
Additional Problems (continued)
Equations of Lines in the Coordinate Plane
Problem 4
What is an equation in point-slope form
for the line graphed?
10
8
6
(2, 5)
4
2
y
O 2 4 6 8 10 x
108 6 4 2
2
4
6
8
10
(4, 4)
Problem 5
What are the equations for the horizontal and vertical lines
through (1, 7)?
Prentice Hall Geometry • Additional Problems
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40
3-8
Slopes of Parallel and
Perpendicular Lines
Vocabulary
Review
y
Use the graph at the right for Exercises 1–4. Write parallel or perpendicular to
complete each sentence.
a
b
1. Line b is 9 to line a.
x
O
2. Line b is 9 to the x-axis.
3. Line a is 9 to the y-axis.
4. The x-axis is 9 to the y-axis.
Write the converse, inverse, and contrapositive of the statement below.
If a polygon is a triangle, then the sum of the measures of its angles is 180.
_______________________________________________________________________
6. INVERSE If a polygon is not a triangle, then 9.
_______________________________________________________________________
7. CONTRAPOSITIVE If the sum of the measures of the angles of a polygon
is not 180, then 9.
_______________________________________________________________________
Vocabulary Builder
The reciprocal of x is
reciprocal (noun) rih SIP ruh kul
Other Word Forms: reciprocate (verb)
Definition: The reciprocal of a number is a number such that the product of the two
numerator
denominator
numbers is 1. The reciprocal of denominator is numerator .
Chapter 3
86
1
.
x
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
5. CONVERSE If the sum of the measures of the angles of a polygon is 180, then 9.
Use Your Vocabulary
Complete each statement with reciprocal or reciprocate. Use each word only once.
8. VERB
After your friend helps you with your homework,
you 9 by helping your friend with his chores.
9. NOUN
3
The 9 of 23 is 2 .
Key Concept Slopes of Parallel Lines
• If two nonvertical lines are parallel, then their slopes are equal.
• If the slopes of two distinct nonvertical lines are equal, then the lines are parallel.
• Any two vertical lines or horizontal lines are parallel.
Circle the correct statement in each exercise.
10. A vertical line is parallel to any other
vertical line.
11.
A vertical line is parallel to any
horizontal line.
Any two nonvertical lines have the
same slope.
Any two nonvertical lines that are
parallel have the same slope.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Problem 1 Checking for Parallel Lines
Got It? Line <3 contains A(213, 6) and B(21, 2). Line <4 contains C(3, 6) and
D(6, 7). Are <3 and <4 parallel? Explain.
12. To determine whether lines /3 and /4 are are parallel
check whether the lines have the same 9.
13. Find the slope of each line.
slope of <3
226
5
21 2 (213)
slope of <4
5
14. Are the slopes equal?
Yes / No
15. Are lines /3 and /4 parallel? Explain.
_______________________________________________________________________
_______________________________________________________________________
87
Lesson 3-8
Problem 2
Writing Equations of Parallel Lines
Got It? What is an equation of the line parallel to y 5 2x 2 7 that
contains (25, 3)?
16. The slope of the line y 5 2x 2 7 is
.
17. The equation of the line parallel to y 5 2x 2 7 will have slope m 5
.
18. Find the equation of the line using point-slope form. Complete the steps below.
y 2 y1 5
Write in point-slope form.
y235
Substitute point and slope into equation.
y235
Simplify.
y5
Add 3 to both sides.
Key Concept Slopes of Perpendicular Lines
• If two nonvertical lines are perpendicular, then the product of their slopes is 21.
• If the slopes of two lines have a product of 21, then the lines are perpendicular.
• Any horizontal line and vertical line are perpendicular.
Write T for true or F for false.
19. The second bullet in the Take Note is the contrapositive of the first bullet.
Problem 3
Checking for Perpendicular Lines
Got It? Line <3 contains A(2, 7) and B(3,21). Line <4 contains C(22, 6)
and D(8, 7). Are <3 and <4 perpendicular? Explain.
21. Find the slopes and multiply them.
m3 5
m4 5
m3 3 m4 5
22. Underline the correct words to complete the sentence.
Lines /3 and /4 are / are not perpendicular because the product of their slopes
does / does not equal 21.
Chapter 3
88
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20. The product of the slopes of any horizontal line and any vertical line is 21.
Problem 4 Writing Equations of Perpendicular Lines
Got It? What is an equation of the line perpendicular to y 5 23x 2 5 that
contains (23, 7)?
23. Complete the reasoning model below.
Write
Think
I can identify the slope, m1, of
y 5 23x 2 5 is in point-slope form, so m1 5
the given line.
.
I know that the slope, m2, of
because
m2 is
the perpendicular line is
3
5 21.
the negative reciprocal of m1.
I can use m2 and (23, 7)
y 2 y1 5 m(x 2 x1)
to write the equation of
the perpendicular line in
point-slope form.
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Lesson Check • Do you UNDERSTAND?
Error Analysis Your classmate tries to find an equation for a line
parallel to y 5 3x 2 5 that contains (24, 2). What is your
classmate’s error?
24. Parallel lines have the same / different slopes.
25. Show a correct solution in the box below.
slope of given line 3
1
slope of parallel line 3
y y1 m(x x1)
1
y 2 (x 4)
3
Math Success
Check off the vocabulary words that you understand.
slope
reciprocal
parallel
perpendicular
Rate how well you understand perpendicular lines.
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review
0
2
4
6
8
Now I
get it!
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Lesson 3-8
Name
3-8
Class
Date
Additional Problems
Slopes of Parallel and Perpendicular Lines
Problem 1
10
8
6
4
2
Are the lines shown in the graph
parallel? Explain.
y
(1, 3)
(5, 4)
O 2 4 6 8 10 x
108 6 4 2
2
(2, 2)
4
(3, 5)
6
8
10
Problem 2
What is an equation in slope-intercept form for the line parallel
to y 5 4x 2 2 that contains (22, 22)?
Problem 3
Are the lines shown in the graph
perpendicular? Explain.
10
8
6
(2, 5)
4
2
(4, 1)
y
(2, 4)
O 2 4 6 8 10 x
10 8 6 4 2
2
4
6
8
10
Prentice Hall Geometry • Additional Problems
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41
(1, 4)
Name
3-8
Class
Date
Additional Problems (continued)
Slopes of Parallel and Perpendicular Lines
Problem 4
What is an equation in slope-intercept form for the line
perpendicular to y 5 3x 1 2 that contains (6, 2)?
Problem 5
What is an equation for a line
perpendicular to the one shown
that contains (5, 21)?
10
8
6
4
2
y
(2, 3)
O 2 4 6 8 10 x
108 6 4 2
2
4
6
(2, 7)
8
10
Prentice Hall Geometry • Additional Problems
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42
Name
Class
Date
Extra Practice
Chapter 3
Lesson 3-1
Use the cube to name each of the following.
1. all lines that are parallel to
2. a pair of parallel planes
3. two lines that are skew to
Identify all pairs of each type of angles in the diagram.
Name the two lines and the transversal that form each pair.
4. corresponding angles
5. alternate interior angles
6. same-side interior angles
7. alternate exterior angles
Lesson 3-2
Find m∠1 and m∠2. State the theorems or postulates that justify your answers.
8.
9.
10.
11.
12.
13.
Find the value of each variable. Then find the measure of each labeled angle.
14.
15.
16.
Prentice Hall Geometry • Extra Practice
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8
Name
Class
Date
Extra Practice (continued)
Chapter 3
17. Complete the proof.
Given:
l m, a b
Prove: ∠1 ≅ ∠5
Statements
1.
2.
3.
4.
5.
6.
7.
8.
Reasons
1. Given
l m, a b
∠1 ≅ ∠2
a. __?__
b. __?__
∠2 and ∠3 are supplementary.
∠3 and ∠4 are supplementary.
∠2
∠1
∠4
∠1
≅
≅
≅
≅
c. __?__
d. __?__
∠4
∠4
∠5
∠5
e. __?__
f. __?__
g. __?__
Lesson 3-3
Refer to the diagram at the right. Use the given information to
determine which lines, if any, must be parallel. If any lines are
parallel, use a theorem or postulate to tell why.
18. ∠9 ≅ ∠14
19. ∠1 ≅ ∠9
20. ∠2 is supplementary to ∠3.
21. ∠7 ≅ ∠10
22. m∠6 = 60, m∠13 = 120
23. ∠4 ≅ ∠13
24. ∠3 is supplementary to ∠10.
25. ∠10 ≅ ∠15
26. ∠1 ≅ ∠8
Find the value of x for which a || b.
27.
28.
29.
30. Complete the flow proof below.
Given: ∠1 is
supplementary
to ∠2
Prove: l m
Prentice Hall Geometry • Extra Practice
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9
Name
Class
Date
Extra Practice (continued)
Chapter 3
Lesson 3-4
31. Given: l m , a b , a ⊥ l
Prove: b ⊥ m
32 Write a paragraph proof.
Given: a b , a ⊥ l , b ⊥ m
Prove: l m
Lesson 3-5
33. Use the figure at the right. What is the relationship
between
and
? Justify your answer.
Find m∠1.
34.
35.
36.
Find the value of each variable.
37.
38.
39.
Lesson 3-6
Use the segments for each construction.
40. Construct a square with side length 2a.
41. Construct a quadrilateral with one pair of
parallel sides each of length 2b.
42. Construct a rectangle with sides b and a.
Prentice Hall Geometry • Extra Practice
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10
Name
Class
Date
Extra Practice (continued)
Chapter 3
Lesson 3-7
Use the given information to write an equation of each line.
43. slope −4, y-intercept 6
44. slope 7, passes through (1, −2)
Write an equation in point-slope form of the line that contains the given points.
45. A(4, 2), B(6, −3)
46. C(−1, −1), D(1, 1)
47. F(3, −5), G(−5, 3)
Write an equation in slope-intercept form of the line through the given points.
48. M(−2, 4), N(5, −8)
49. P(0, 2), Q(6, 8)
50. K(5, 0), L(−5, 2)
Lesson 3-8
Without graphing, tell whether the lines are parallel, perpendicular, or neither.
Explain.
51. y = 4 x − 8
52. 13 y − x = 7
y = 4x − 2
y
7− =x
2
54. 2x + 3y = 5
55. y = –2x + 7
5x – 10y = 30
53. y = −4 x + 2
3
4
y = x −1
3
56. 5x – 3y = 0
y=
x – 2y = 8
5
x+2
3
Write an equation for the line parallel to the given line through the given point.
57. y = x − 7, (0, 4)
58. y =
1
x + 3, ( 6,3)
2
59. y = −
1
x, ( 5, −8)
5
Write an equation for the line perpendicular to the given line through the given
point.
60. y = x + 2, (3, 2)
61. y = −2x, (4, 0)
62. y =
1
2
x − , (5, −1)
3
5
63. On a city map, Washington Street is straight and passes through points at (7, 13) and
(1, 5). Wellington Street is straight and passes through points at (3, 24) and (9, 32).
Do Washington Street and Wellington Street intersect? How do you know?
Prentice Hall Geometry • Extra Practice
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11