Download Lines and Angles

3-1
Lines and Angles
Vocabulary
Review
Write T for true or F for false.
T
1. You can name a plane by a capital letter, such as A.
F
2. A plane contains a finite number of lines.
T
3. Two points lying on the same plane are coplanar.
T
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
A
CD
Ľ5 Ľ4 Ľ3 Ľ2 Ľ1 O
AD
D
7. Circle the polygon(s) that have two pairs of parallel sides.
rectangle
parallelogram
square
trapezoid
Complete each statement below with line or segment.
8. A 9 consist of two endpoints and all the
points between them.
segment
9. A 9 is made up of an infinite number of points.
line
Chapter 3
58
3
2
1
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.
C
D
Skew lines are noncoplanar; they are not parallel and do not intersect.
A
Parallel planes are planes that do not intersect.
H
10. Write each word, phrase, or symbol in the correct oval.
noncoplanar
coplanar
intersect
* )
do not intersect
* )
* )
AE and CG
Use arrows to show
X X
X X
AE I BF and AD I BC.
CB and AE
Parallel
Skew
coplanar
noncoplanar
do not intersect
do not intersect
AE and CG
CB and AE
* )
* )
* )
G
F
E
* )
B
* )
Problem 1 Identifying Nonintersecting Lines and Planes
Got It? Use the figure at the right. Which segments are parallel to AD?
C
B
11. In plane ADHE, EH is parallel to AD.
A
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12. In plane ADBC, BC is parallel to AD .
13. In plane ADGF, FG is parallel to AD .
Got It? Reasoning Explain why FE and CD are not skew.
D
F
E
G
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?
Answers may vary. Sample: They are not skew segments because
_______________________________________________________________________
they are part of the plane CDEF. Therefore, they are coplanar.
_______________________________________________________________________
59
Lesson 3-1
Key Concept Angle Pairs Formed by Transversals
t
Alternate interior angles are nonadjacent interior angles
that lie on opposite sides of the transversal.
Exterior
1 2
4 3
Same-side interior angles are interior angles that lie on
the same side of the transversal.
Corresponding angles lie on the same side of a
transversal t and in corresponding positions.
Interior
5 6
8 7
m
Alternate exterior angles are nonadjacent exterior angles
that lie on opposite sides of the transversal.
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?
1 2
8 7
Underline the correct word(s) or letter(s) to complete each sentence.
22. The transversal is line m / n / r .
n
3 4
6 5
r
23. Corresponding angles are on the same side / different sides of the
transversal.
24. Name three pairs of corresponding angles. Answers may vary. Accept any three of
/ 1 and / 3
/ 2 and / 4
/ 8 and / 6
/ 7 and / 5
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
corresponding
60
same-side interior
4
3
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. Answers may vary. Samples are given.
F
28. parallel segments
29. skew segments
30. parallel planes
B
E
EFGH
HD and BC
ABCD 6
AB 6 EF
A
Name one pair each of the angles.
31. alternate interior
32. same-side interior
/8 and / 6
8
/8 and / 3
33. corresponding
5
34. alternate exterior
/1 and / 3
3
4
6
H
G
C
D
1
7
2
/7 and / 5
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
A
B
G
Write T for true or F for false.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
C
H
F
T
35. Parallel segments are coplanar.
F
36. There are only six planes in a cube.
F
37. No plane contains AB and HG .
E
38. Who is correct? Explain.
Carly; Sample explanation: The segments are parallel since they do
_______________________________________________________________________
not intersect and are coplanar (plane ABGH contains AB and HG ).
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
angle
parallel
skew
transversal
Rate how well you can classify angle pairs.
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61
Lesson 3-1
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
obtuse
right
acute
Vocabulary Builder
E
interior (noun) in TEER ee ur
m
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.
l1
Chapter 3
lB
62
B
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...
corresponding
angles are congruent.
If...
Theorem 3-1
Alternate Interior
Angles Theorem
alternate interior
angles are congruent.
a transversal
Theorem 3-2
Corresponding Angles
Theorem
intersects two
parallel lines,
same-side interior angles
are supplementary.
alternate exterior
angles are congruent.
Postulate 3-1
Same-Side Interior
Angles Postulate
Theorem 3-3
Alternate Exterior
Angles Theorem
Use the graphic organizer and the diagram to find each congruent angle.
9. Theorem 3-2
10. Theorem 3-1
l1 /3 > l7 /3 > 4 5
3 6
2 7
1 8
11. Theorem 3-3
l5
/1 > r
s
HSM11_GEMC_0302_T93307
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
HSM11_GEMC_0302_T93303
Problem 1 Identifying Congruent Angles
Got It? Reasoning Can you always find the measure of all 8 angles
when two parallel lines are cut by a transversal? Explain.
1
Yes, because 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. Answers may vary. Sample:
m/7 5 55
Corresponding Angles Postulate
m/5 5 m/7
Vertical Angles Theorem
m/5 5 55
Transitive Property of Equality
m/2 5 125
Same-Side Interior Angles Postulate
m/4 5 125
Vertical Angles Theorem
m/6 and m/8 5 125
Corresponding Angles Postulate
63
hsm11gmse_0302_t00237.ai
Lesson 3-2
Problem 2 Proving an Angle Relationship
Got It? Given: a 6 b
a
1
Prove: /1 > /7
13. Use the reasons at the right to write each step of the proof.
Statements
b
Reasons
5
8 7
anb
1) 1) Given
l1 > l5
2) 2) If lines are 6, then corresp. angles are >.
ml1 5 ml5
3) hsm11gmse_0302_t00240.ai
3) Congruent angles have
equal measure.
l5 > l7
4) 4) Vertical angles are congruent.
ml5 5 ml7
5) 5) Congruent angles have equal measure.
ml1 5 ml7
6) 6) Transitive Property of >
l1 > l7
7) 7) Angles with equal measure are >.
Problem 3 Finding Measures of Angles
14. There are two sets of parallel lines.
Each parallel line also acts as a 9.
transversal
15. The steps to find m/1 are given below. Justify each step.
Statements
m

p
q
2
1 8
4
6
5
3
105
7
Reasons
1)/1 > /4
1) Alternate Interior Angles Theorem
2)m/1 5 m/4
the same measure.
2) Congruent angles have
hsm11gmse_0302_t00242.ai
3)/4 and /6 are supplementary.
3) Linear Pair Postulate
4)m/4 1 m/6 5 180
4) Definition of supplementary angles
5)m/1 1 m/6 5 180
5) Transitive Property of Equality
6)m/5 5 105
6) Corresponding angles have the same measure.
7)m/6 5 105
7) Alternate interior angles have the same measure.
8)m/1 1 105 5 180
8) Substitute into Statement 5.
9)m/1 5 75
9) Subtraction Property of Equality
Chapter 3
64 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Got It? Find the measure of l1. Justify your answer.
Problem 4 Using Algebra to Find an Angle Measure
Got It? In the figure at the right, what are the values of x and y?
2x 
3y 
16. The bases of a trapezoid are parallel / perpendicular .
17. Use the Same-Side Interior Angles Postulate to complete each statement.
(y  20)
(x  12)
y 1 20 5 180
3y 1 x 2 12 5 180
2x 1 18. Solve each equation.
2x 1 (x 2 12) 5 180,
3x 2 12 5 180
3x 5 192
x 5 64
3y 1 (y 1 20) 5 180
4y 1 20 5 180
4y 5 160
y 5 40
hsm11gmse_0302_t00244.ai
Lesson Check • Do you UNDERSTAND?
In the diagram at the right, l1 and l8 are supplementary. What is a good
name for this pair of angles? Explain.
a
19. Circle the best name for lines a and b.
b
parallel
perpendicular
skew
transversals 1
5
8 7
20. Circle the best name from the list below for /1 and /8.
alternate
congruent
corresponding
same-side Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
21. Circle the best name from the list below for /1 and /8.
exterior
interior
hsm11gmse_0302_t00240.ai
22. Use your answers to Exercises 20 and 21 to write a name for /1 and /8.
same-side exterior angles
________________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
alternate interior angles
alternate exterior angles
Rate how well you can prove angle relationships.
Need to
review
0
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65
Lesson 3-2
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 you wear a sweater
, then you are cold
.
2. Statement: If an angle is a right angle, then it measures 90°.
Converse: If an angle measures 90°, then it is a right angle.
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
Theorems 3-2 and 3-4 Corresponding Angles Theorem and Its Converse
Theorem 3-2 Corresponding Angles Theorem
If a transversal intersects two parallel lines, then corresponding angles are congruent.
10. Complete the statement of Theorem 3-2.
Theorem 3-4 Converse of the Corresponding Angles Theorem
If two lines on a transversal form corresponding angles that are congruent,
then the lines are 9. parallel
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 Theorem 3-4.
1
HSM11_GEMC_0303_T93477
ℓ
2
1
m
2
ℓ
m
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Theorems 3-5, 3-6, and
3-7
HSM11_GEMC_0303_T93311
HSM11_GEMC_0303_T93310
Theorem 3-5 Converse of the Alternate Interior Angles Theorem
If two lines and a transversal form alternate interior angles that are congruent, then
the two lines are parallel.
Theorem 3-6 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 3-7 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 3-5
l8 ,
If /4 > then b6c.
Theorem 3-6
Theorem 3-7
l8 If /3 and are supplementary,
then b6c.
l7 ,
If /1 > then b6c.
6 7
5 8
4 3
1 2
b
c
HSM11_GEMC_0303_T93308
67
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
2
7
/6 and /7 are alternate / same-side angles.
/6 and /7 are corresponding / exterior / interior angles.
I can use Theorem 3-2 / Theorem 3-4 to prove the lines parallel.
Using /6 > /7, lines a and b /  and m are parallel and the transversal is a/ b / / m .
hsm11gmse_0303_t00246.ai
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 ≅ ∠7

1
m
5 6
3
7
∠7 ≅ ∠5
∠3 ≅ ∠7
∠3 ≅ ∠5
∠1 ≅ ∠3
Transitive Prop.
Vertical angles
Transitive
Prop.
hsm11gmse_0303_t00252
Vertical angles
of ≅.
are ≅.
of ≅.
are ≅.
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 rHSM11_GEMC_0303_T93314
and s are parallel. What is another way to
explain why r ns? Justify your answer.
16. Justify each step.
3
t
2
1
/1 > /2
Given
/2 > /3
Vertical angles are congruent.
/1 > /3
Transitive Property of Congruence
17. Angles 1 and 3 are alternate / corresponding .
HSM11_GEMC_0303_T93315
18. What postulate or theorem can you now use to explain why r6s?
Converse of the Corresponding Angles Theorem
________________________________________________________________________
Chapter 3
s
r
68 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Given
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 Theorem, if c 6 d then corresponding angles are
complementary / congruent / supplementary .
21. Use the theorem to solve for w.
hsm11gmse_0303_t00255
3w 2 2 5 55
3w 5 57
w 5 19
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. Explanations may vary. Sample:
My classmate identified the wrong pair of parallel sides. lADC and lBCD
________________________________________________________________________
hsm11gmse_0303_t003
are supplementary and same-side interior angles. The transversal is DC . ________________________________________________________________________
AD n BC by the Converse of the Same-Side Interior Angles Theorem.
________________________________________________________________________
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.
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69
Lesson 3-3
3-4
Parallel and
Perpendicular Lines
Vocabulary
Review
Complete each statement with always, sometimes or never.
1. A transversal 9 intersects at least two lines.
always
2. A transversal 9 intersects two lines at more than two points.
never
3. A transversal 9 intersects two parallel lines.
sometimes
4. A transversal 9 forms angles with two other lines.
always
Vocabulary Builder
Transitive
transitive (adjective)
TRAN
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.
5. If a . b
6. If Joe is younger than Ann
7. If you travel from
and b . c,
and Ann is younger than
Station 2 to Station 3
then a S c .
Sam, then
and you travel from
Joe is younger
Station 3 to
than Sam
.
Station 4
then you travel from
Station 2 to Station 4.
Chapter 3
70
,
Theorem 3-8 Transitive Property of Parallel Lines and Theorem 3-9
8. Complete the table below.
Theorem 3-8
Theorem 3-9
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⊥t
and
b ∙ c
n⊥t
then
a∙c
m
∙
n
HSM11_GEMC_0304_T13316
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 60 8 and 30 8 .
hsm11gmse_0304_t00260
90 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
parallel
The top and bottom
to the right piece.
Opposite sides are
pieces will be
parallel
perpendicular
to the side pieces.
.
The top piece will be
parallel
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.
HSM11_GEMC_0304_T93317
71
Lesson 3-4
Theorem 3-10 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 3-10.
ℓ
m
Use the information in each diagram to complete each statement.
15. a
g
n
16.
t
p
HSM11_GEMC_0303_T93318
c
g and a ' t , so g
a 6 t .
' HSM11_GEMC_0304_T14343
c ' n and n 6 p , so c ' p .
HSM11_GEMC_0304_T14344
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
b
18. Lines that are perpendicular to the same line are parallel / perpendicular .
19. Can you conclude that a6b ? Explain.
hsm11gmse_0304_t00263
Yes. Explanations may vary. Sample: Lines a and b are both
________________________________________________________________________
perpendicular to line d, so anb by Theorem 3-8.
________________________________________________________________________
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.
M
A
B
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
HSM11_GEMC_0304_T93918
also parallel / perpendicular to the other.
Avenue B and Main Street are parallel / perpendicular streets.
Chapter 3
72 Lesson Check • Do you UNDERSTAND?
Which theorem or postulate from earlier in the chapter supports the conclusion
in Theorem 3-9? In the Perpendicular Transversal Theorem? Explain.
n
Use the diagram at the right for Exercises 22 and 23.
22. Complete the conclusion to Theorem 3-9.
ℓ
In a plane, if two lines are perpendicular to the same line, then 9.
m
they are parallel to each other
_______________________________________________________________
23. Complete the statement of Theorem 3-4.
If two lines and a transversal form 9 angles that are
congruent, then the lines are parallel. corresponding
HSM11_GEMC_0304_T93319
c
Use the diagram at the right for Exercises 24 and 25.
24. Complete the conclusion to the Perpendicular Transversal Theorem.
a
In a plane, if a line is perpendicular to one of two parallel lines,
then it is also 9.
b
perpendicular to the other.
________________________________________________________________________
25. Explain how any congruent angle pairs formed by parallel lines support the
conclusion to the Perpendicular Transversal Theorem.
HSM11_GEMC_0304_T93320
Answers may vary. Sample: If both alternate interior angles are ________________________________________________________________________
right angles, then the line perpendicular to one parallel line is ________________________________________________________________________
perpendicular to the other parallel line.
________________________________________________________________________
________________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
parallel
perpendicular
Rate how well you can understand parallel and perpendicular lines.
Need to
review
0
2
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6
8
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get it!
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73
Lesson 3-4
Parallel Lines and
Triangles
3-5
Vocabulary
Review
Identify the part of speech for the word alternate in each sentence below.
1. You vote for one winner and one alternate.
noun
2. Your two friends alternate serves during tennis.
verb
3. You and your sister babysit on alternate nights.
adjective
4. Write the converse of the statement.
Statement: If it is raining,
raining then I need an umbrella.
Converse:
If I need an umbrella, then it is raining.
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.
T
5. A tripod is a stand that has three legs.
F
6. A triangle is a polygon with three or more sides.
F
7. A triatholon is a race with two events — swimming and bicycling.
T
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
Postulate 3-2 Parallel Postulate
P
Through a point not on a line, there is one and only one line parallel to the
given line.

1 line(s) through P parallel to line /.
9. You can draw Theorem 3-11 Triangle Angle-Sum Theorem
hsm11gmse_0305_t00264
The sum of the measures of the angles of a triangle is 180.
Find each angle measure.
10. C
11. M
45°
100°
30°
B
A
L
N
50 m/C 5 45
m/L 5 HSM11_GEMC_0305_T93322
HSM11_GEMC_0305_T93323
Problem 1 Using the Triangle Angle-Sum Theorem
Got It? Use the diagram at the right. What is the value of z?
Complete each statement.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
59
12. m/A 5 43
A
B
49
x y
59 D
z
C
43 1 49 5 92
13. m/ABC 5 180
14. m/A 1 m/ABC 1 m/C 5 hsm11gmse_0305_t00267
180
59 1 92 1 z 5 59 2 180 2 92 5 29
z 5 Check your result by solving for z another way.
15. Find m/BDA.
16. Then find m/BDC.
mlA 1 mlABD 1
mlBDA 5 180
59 1 43 1 x 5 180
x 5 180 2 102
5 78
mlBDA 1 mlBDC 5 180
x 1 y 5 180
y 5 180 2 78
5 102
17. Use your answers to Exercises 15 and 16 to find the value of z.
z 1 mlCBD 1 mlBDC 5 180
z 1 49 1 102 5 180
z 5 180 2 (49 1 102) 5 29
75
Lesson 3-5
Theorem 3-12 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.
1
18. ml1
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
HSM11_GEMC_0305_T93324
3
6
4
1
2
HSM11_GEMC_0305_T93321
Exterior Angle Theorem
Problem 2 Using the Triangle
HSM11_GEMC_0305_T93330
Got It? Two angles of a triangle measure 53. What is the
measure of an exterior angle at each vertex of the triangle?
a
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.
x
53°
53°
y
22. Complete the flow chart.
Triangle Angle-Sum
53 + 53 + a = 180
HSM11_GEMC_0305_T93325
a = 180 – 106
= 74
Exterior Angle
x = a + 53
= 74 + 53
= 127
Chapter 3
Exterior Angle
y = a + 53
= 74 + 53
= 127
76 HSM11_GEMC_0305_T93326
Exterior Angle
z = 53 + 53
= 106
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
z
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
180 2 80 5 100
25. Can you find m/A if you know two of the angle measures? Explain. Explanations may vary. Sample:
Yes. Use the Triangle Angle-Sum Theorem. mlA 5 180 2 100 2 30 5 50
__________________________________________________________________________________
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
mƋ1 à mƋ3 à mƋ2 â180
Triangle Angle-Sum Theorem
mƋ1 à mƋ3 â mƋ4
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Linear Pair Postulate
à mƋ2 â180
mƋ4
27. Explain how the Triangle Exterior Angle Theorem makes sense based on the
Triangle Angle-Sum Theorem. Answers may vary. Sample:
Using the Triangle Angle-Sum Theorem,180 2 ml2 5 ml1 1 ml3. Since a linear pair is
______________________________________________________________________________________
supplementary, 180 2 ml2 5 ml4. Then by the Transitive Property, ml4 5 ml1 1 ml3.
______________________________________________________________________________________
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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77
Lesson 3-5
Constructing Parallel
and Perpendicular Lines
3-6
Vocabulary
Review
Write T for true or F for false.
T
F
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
5.
5
3
2
4
alternate interior
corresponding
6
alternate exterior
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.
construct
NOUN
The 9 on the highway caused quite a traffic jam.
construction
ADJECTIVE
The time you spent working on your homework was 9.
constructive
Chapter 3
78
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3.
Problem 1 Constructing Parallel Lines
Got It? Reasoning The diagram at the right shows the construction
of line m through point N with line m parallel to line <. Why must
lines < and m be parallel?
m
N 1
7. The diagram shows the construction of congruent

H
NHJ .
angles 1 and J
8. Circle the description(s) of the angle pairs that were constructed.
alternate interior
congruent
corresponding
same-side interior vary. Sample:
9. Now explain why lines / and m must be parallel. Explanations mayhsm11gmse_0306_t00300.ai
You constructed the transversal to form congruent
________________________________________________________________________
corresponding angles with lines < and m, so < n m by the Converse
________________________________________________________________________
of the Corresponding Angles Theorem.
________________________________________________________________________
Problem 2 Constructing a Special Quadrilateral
*Got) It?
* ) Draw a segment. Label its length m. Construct quadrilateral ABCD with
AB n CD , so that AB 5 m and CD 5 2m.
Underline the correct word or symbol to complete each sentence.
10. Construct parallel / perpendicular lines.
* )
* )
* )
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
11. Draw AB . Draw point D not on AB . Draw AD . The length of
AB / AD is m.
12. At D, construct /TDZ perpendicular / congruent to /DAB so that /TDZ and
* ) * )
/DAB are corresponding angles. Then DZ 6 AB .
* )
13. Now, you need a side of length 2m. Construct C on DZ so that DC 5 2m.
Draw BC / BA .
14. Do the construction below.
T
D
A
Z
m
C
B
HSM11_GEMC_0306_T93919
79
Lesson 3-6
Problem 3 Perpendicular at a Point on a Line
* )
* )
* )
* )
Got It? Use a straightedge to draw EF . Construct FG so that FG ' EF at point F.
15. Use the diagram at the right. Write each construction step.
G
Step 1 Draw a line. Label points E and F.
__________________________________________
E
F
H
Step 2 Make the compass opening EF. With the
compass tip on point F, draw an arc that
__________________________________________
intersects the line twice. Mark point H so
__________________________________________
EF 5 FH.
__________________________________________
HSM11_GEMC_0306_T93334
1
Step 3 Make the compass opening greater than 2 EH. With the compass
tip on E, draw an arc above point F.
__________________________________________________________________
__________________________________________________________________
Step 4 Without changing the compass setting, place the compass tip on
point of intersection G.
__________________________________________________________________
__________________________________________________________________
* )
Step 5 Draw FG .
Postulate 3-3 Perpendicular Postulate
Complete the statement of Postulate 3-3 below.
16. Through a point not on a line, there is one and only one line parallel / perpendicular
to the given line.
17. Circle the diagram that models Postulate 3-3.
P
P
P
HSM11_GEMC_0306_T93920
HSM11_GEMC_0306_T93922
HSM11_GEMC_0306_T93921
Chapter 3
80 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
point H. Draw an arc that intersects the arc from Step 3. Label the
__________________________________________________________________
Problem 4 Perpendicular From a Point to a Line
* )
* )
* )
* )
* )
Got It? Draw a line CX and a point Z not on CX . Construct ZB so that ZB ' CX .
Underline the correct word(s) to complete each sentence.
18. Open your compass to a size equal to / greater than the distance from Z to line /.
19. With the compass tip on point Z, draw an arc that intersects line / at one / two point(s).
20. Label the point(s) C and X / Z .
21. Place the compass point on C / Z
and make an arc below line /.
Z
22. With the same opening and the
compass tip on C / X , draw an
arc that intersects the arc you
made in Exercise 21. Label the
point of intersection B.
C
E
X
* ) * )
B
23. Draw ZB / CX .
24. Use line / and point Z at the right.
Construct a line through point Z
perpendicular to line /.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Lesson Check • Do you UNDERSTAND?
Suppose you use a wider compass setting in Exercise 18. Will you construct a different
perpendicular line? Explain.
25. Explain why you will NOT construct a different perpendicular line.
Explanations
is one and only one line
___________________________________________________________________________
subtraction may vary. Sample: There
multiplication
perpendicular to a given line.
___________________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
construction
parallel
perpendicular
Rate how well you can construct parallel and perpendicular lines.
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Lesson 3-6
3-7
Equations of Lines in the
Coordinate Plane
Vocabulary
Review
Write T for true or F for false.
T
1. An ordered pair describes the location of a point in a coordinate grid.
F
2. An ordered pair can be written as (x-coordinate, y-coordinate) or (y-coordinate,
x-coordinate).
T
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.
slope
5. The driveway 9 down to the garage.
sloped
6. The 9 lawn led to the river.
sloping
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
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
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
(1, 7)
(5, 7)
4 (2, 3)
(1, 2)
2
x
(4, 0)
6
O
2
8
(4, 2)
7 Ľ 3
Two points on line a are (2, 3) and (5, 7).
a
d
6
Write
I know the slope is the ratio
y
c
â
5 Ľ 2
Now I can simplify.
4
3
â
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Key Concept 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
3
10. In what form is the given equation written?
2
slope-intercept form
_________________________________________
11. Written as a fraction, the slope is
3
1
1
Ľ5 Ľ4 Ľ3 Ľ2 Ľ1 O
Ľ1
.
x
1
2
3
4
5
Ľ2
12. One point on the graph is ( 0 ,24).
Ľ3
13. From that point, move 3 unit(s) up and
1 unit (s) to the right.
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
1
slope m 5 22
y-intercept 5 2
Need
Plan
Write an equation of
a line.
Use
y 5 mx 1 b
,
the slope-intercept form of
a linear equation.
16. Now write the equation.
y 5 212x 1 2
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
17. The equation is found below. Write a justification for each step.
Write in point-slope form.
y 2 y1 5 m(x 2 x1)
6
Substitute.
6
Simplify.
y 2 (21) 5 5(x 2 (22))
y 1 1 5 5(x 1 2)
(ⴚ2, ⴚ1)
⫺3
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)
y 2 5 5 65x 2 18
5
25
y 5 65x 2 18
5 1 5
y 5 65x 1 75
y 1 1 5 65x 1 12
5
5
y 5 65x 1 12
5 25
y 5 65x 1 75
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
(3, 5)
4
O
x
2
4
⫺2
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
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?
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
T 20. Every point on a horizontal line through (4,23)
has y-coordinate of 23.
3
2
1
F
21. The equation of a vertical line through (4,23)
is y 5 23.
T
22. The equation of a vertical line through (4,23)
is x 5 4.
Ľ5 Ľ4 Ľ3 Ľ2 Ľ1 O
Ľ1
x
1
2
3
4
5
Ľ2
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?
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Answers may vary. Sample: Slope is rise over run, not run over rise.
_________________________________________________________________
88
10 (2)
0
m
12
m
m0
25. Find the slope, m.
2 10
212
m 5 22
8 2 8 5 0
26. The run is 8 2 8 5
0 , so the slope is undefined
.
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.
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85
Lesson 3-7
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.
b
perpendicular
1. Line b is 9 to line a.
x
O
2. Line b is 9 to the x-axis.
parallel
3. Line a is 9 to the y-axis.
parallel
4. The x-axis is 9 to the y-axis.
perpendicular
a
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.
the polygon is a triangle
_______________________________________________________________________
6. INVERSE If a polygon is not a triangle, then 9.
the sum of the measures of its angles is not 180
_______________________________________________________________________
7. CONTRAPOSITIVE If the sum of the measures of the angles of a polygon
is not 180, then 9.
the polygon is not a triangle
_______________________________________________________________________
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
9. NOUN
After your friend helps you with your homework,
you 9 by helping your friend with his chores.
reciprocate
3
The 9 of 23 is 2 .
reciprocal
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
slope
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
24
12
5
726
623
2 13
14. Are the slopes equal?
5 13
Yes / No
15. Are lines /3 and /4 parallel? Explain.
No. Explanations may vary. Sample: Since the slopes are
_______________________________________________________________________
not equal, the lines are not parallel.
_______________________________________________________________________
87
Lesson 3-8
Writing Equations of Parallel Lines
Problem 2
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 21 .
17. The equation of the line parallel to y 5 2x 2 7 will have slope m 5 21 .
18. Find the equation of the line using point-slope form. Complete the steps below.
m(x 2 x1)
y 2 y1 5
y 2 3 5 21fx 2 (25)g
Write in point-slope form.
Substitute point and slope into equation.
y235
2x 2 5
Simplify.
y5
2x 2 2
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.
F
19. The second bullet in the Take Note is the contrapositive of the first bullet.
T
20. The product of the slopes of any horizontal line and any vertical line is 21.
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.
28
27
m3 5 21
32 2 5 1 5 28
m3 3 m4 5
6
1
5 10
m4 5 8 722(22)
28
1
(28) Q 10
R 5 10
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
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Write T for true or F for false.
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
m2 is
the perpendicular line is
1
because 3 3
I can use m2 and (23, 7)
5 21.
3
3
the negative reciprocal of m1.
1
3 .
y 2 y1 5 m(x 2 x1)
to write the equation of
1
[x 2 (23)]
3
1
y 2 7 5 (x 1 3)
3
y275
the perpendicular line in
point-slope form.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
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
slope of given line = 3; slope of parallel line = 3
y 2 2 5 3(x 1 4)
Math Success
Check off the vocabulary words that you understand.
slope
reciprocal
parallel
perpendicular
Rate how well you understand perpendicular lines.
Need to
review
0
2
4
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8
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get it!
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Lesson 3-8