Download Parallel and Perpendicular Lines

Chapter
4
Parallel and Perpendicular
Lines
Chapter Content
Lessons
Standards
4.1
G-CO.1; G-CO.9
Parallel Lines and Angles
Parallel and Perpendicular Lines
Corresponding Angles
Alternate Interior and Exterior Angles
Consecutive Interior Angles
4.2
More on Parallel Lines and Angles
G-CO.9
Angle Relationships
Parallel Lines and Angles
Multi-Part Problem Practice
4.3
Perpendicular Lines
G-CO.9; G-CO.12
Pairs of Perpendicular Lines
Construction: Perpendicular and Parallel Lines
Distance and Perpendicular Lines
4.4
Parallel Lines, Perpendicular Lines, and Slope
G-GPE.5
Determining When Lines Are Parallel or Perpendicular
Rotations and Perpendicular Lines
Distance Between a Point and a Line
Multi-Part Problem Practice
4.5
Parallel Lines and Triangles
Interior and Exterior Angles
Angles in a Right Triangle
CHAPTER 4 REVIEW
CUMULATIVE REVIEW FOR CHAPTERS 1–4
148
Chapter 4:
: Parallel and Perpendicular Lines
G-CO.10
Chapter Vocabulary
alternate exterior angle
coplanar
parallel
alternate interior angle
corollary
perpendicular bisector
auxiliary line
exterior angle
skew line
consecutive interior angle
interior angle
transversal
LESSON 4.1
4.1 Parallel Lines and Angles
Parallel and Perpendicular Lines
Two lines that never intersect are either parallel lines or
skew lines. If they are also coplanar—in the same plane—
the lines are parallel.
• Lines a and b are parallel, which is written as a i b.
They are indicated by the pair of parallel lines in
the diagram.
c
b
M
a
a || b
N
If two lines are not in the same plane, and they never intersect,
then they are called skew lines.
• Lines b and c are skew lines.
Two planes that do not intersect are parallel planes.
• The planes M and N are parallel, which is written M i N.
The symbol for
parallel is a
pair of vertical
lines: i
With parallel lines defined, we can state the parallel postulate.
4.1 Parallel Lines and Angles 149
Similar to
the case for
parallel lines,
there is also
a unique
perpendicular
line through a
given point.
Corresponding Angles
A transversal is a line that intersects two other lines.
Corresponding angles are formed by a transversal and
located at the same position relative to the transversal.
In other words, they are at the same location of each
intersection, such as above a line and to the left of the
transversal. In the diagram, the corresponding angles
are the pairs A and E, B and F, C and G, and D and H.
A B
D C
E F
H G
The
corresponding
angles
postulate lets
us define the
relationship
between
four pairs of
angles.
MODEL PROBLEMS
1. A pair of corresponding angles created by a transversal and two lines have measures described by
the expressions 2x 1 3 and 3x 2 11. If x 5 14, the lines must be
A.
B.
C.
D.
Parallel.
Perpendicular.
Neither parallel nor perpendicular.
Not planar.
SOLUTION
The answer is A. Evaluating the expressions with x 5 14, the result is 31 in both cases. The angles are
congruent, and with congruent corresponding angles, the lines must be parallel.
Model Problems continue . . .
150
Chapter 4: Parallel and Perpendicular Lines
MODEL PROBLEMS continued
a
2. What is mH?
b
SOLUTION
A = 42°
a and b are parallel lines. Angle H is a corresponding
angle with D, so if we can calculate mD, we know
mH , since the angles are congruent.
A and D are supplementary
D
mA 1 mD 5 180°
E
C
F
H
d
G
A and D are linear angles, which
means they are supplementary and
sum to 180°. Substitute for mA and
solve.
42° 1 mD 5 180°
mD 5 138°
D and H are congruent
B
mH 5 mD 5 138°
The two angles are corresponding
angles and the lines are parallel, so
mH 5 mD 5 138°.
Alternate Interior and Exterior Angles
A transversal crosses two lines. Alternate interior
angles are inside the parallel lines on opposite sides of
the transversal. C and E are alternate interior angles,
as are D and F. They are interior to (inside) the
parallel lines and on alternate (different) sides of
the transversal.
D
E
The pair
of orange
arrowheads
indicate
parallel
lines.
C
F
Alternate exterior angles are outside the parallel lines, on opposite
sides of the transversal. A and G are alternate exterior angles,
as are B and H. They are exterior to (outside) the parallel lines
and on alternate (different) sides of the transversal.
A
H
B
G
4.1 Parallel Lines and Angles 151
Instead of stating that corresponding angles are congruent (when a transversal
intersects parallel lines) as a postulate, we could state it as a theorem. To do this,
assume that the alternate exterior angles theorem is a postulate. We use a paragraph
proof to prove the theorem.
Paragraph Proof
With alternate exterior angles stated as a postulate, we have one set of angles shown
to be congruent. Alternate interior angles would then be congruent too, since they
are vertical angles to the alternate exterior angles, and vertical angles are congruent.
The other pairs of corresponding angles would have to be congruent since they are
supplementary to congruent angles.
In this activity, prove the alternate exterior angles theorem converse by proving that
lines m and n are parallel.
152 Chapter 4: Parallel and Perpendicular Lines
MODEL PROBLEMS
1. MP 7 In the diagram, angles A and F have
no direct relationship. What is mF?
A = 130°
B
D
C
SOLUTION
Look for angle
that relates to F
Start the problem by looking
for an angle with a known
relationship to F. For instance,
B and F are corresponding
angles, so if B is known, then
F can be calculated.
and B are
A
supplementary
mA 1 mB 5 180°
130° 1 mB 5 180°
m
B 5 50°
and F are
B
corresponding
angles
B  F
mB 5 mF 5 50°
E F
H G
A and B are supplementary angles. Use
an equation that says the angles sum to
180° and solve for mB.
ince the two angles are congruent, they
S
have the same measure.
2. MP 1 mA is 50% greater than mD. What is mF?
A B
D C
E F
H G
SOLUTION
elationship of
R
A and D
Calculate D
Write equation
Solve
mA 5 1.5(mD)
mA 1 mD 5 180°
1.5(mD) 1 mD 5 180°
2.5(mD) 5 180°
mD 5 72°
D and F are
alternate
interior angles
F  D
mF 5 mD 5 72°
mA is 50% greater than mD. This
means mA equals 1.5 times mD.
ngles A and D are supplementary,
A
so they sum to 180°.
Replace mA with 1.5mD.
olve for mD by combining like terms,
S
and then dividing both sides by 2.5.
D and F are congruent since they are
alternate interior angles.
Consecutive Interior Angles
Consecutive interior angles are angles on the same side of the
transversal and between the lines. In the diagram, D and E are
consecutive interior angles, and C and F are consecutive interior
angles.
D C
E F
4.1 Parallel Lines and Angles153
MODEL PROBLEMS
y
x
1. What is mE?
D = 35°
A
C
B
H
E
G
F
SOLUTION
Look for a relationship between E and D. B is the connection between D and E.
Vertical angles
mB 5 mD 5 35°
Angles B and D are vertical angles,
and therefore congruent.
Consecutive
interior
angles are
supplementary
mB 1 mE 5 180°
Substitute D for B in the relationship of
consecutive interior angles.
MP 2, 4 3rd and 4th Avenues are parallel. The measure of the
angle between 4th and 32nd Street is 65°. The measure of the
angle between 3rd and Broadway is 85°. What is the measure
of the angle between Broadway and 32nd Street?
SOLUTION
Identify
consecutive
interior angles
(65° 1 x) 1 85° 5 180°
Solve equation
65° 1 x 1 85° 5 180°
150° 1 x 5 180°
x 5 30°
154
Broadway
32n x
85°
65° d Stre
et
The angles between Broadway
and 4th Avenue and Broadway
and 3rd Avenue are consecutive
interior angles.
Consecutive
interior angles
sum to 180°
Chapter 4: Parallel and Perpendicular Lines
3rd Avenue
Complete the calculation.
4th Avenue
35° 1 mE 5 180°
mE 5 145°
Solve
2.
mD 1 mE 5 180°
Consecutive interior angles
sum to 180°.
Solve for x. The measure of
the angle between Broadway
and 32nd Street is 30°.
We take
advantage of
the consecutive
interior angles
theorem to
solve this
problem.
PRACTICE
1. Non-coplanar lines that never intersect are
called
A.
B.
C.
D.
7. /3 1 /5 5 148°. Determine the measure
of /2.
Parallel.
Perpendicular.
Transversal.
Skew.
4 1
3 2
8 5
7 6
p
2. Lines j and k are skew. Which of these is
true? Choose all that apply.
A.
B.
C.
D.
E.
j and k intersect.
j and k do not intersect.
j and k are in the same plane.
j and k are in different planes.
j and k are parallel.
3. On a plane, point A doesn’t lie on the line a.
Three new lines are drawn through point A.
How many of these lines could intersect a?
A.
B.
C.
D.
Only one
Only two
All three
Zero
x
A.
B.
C.
D.
y
32°
106°
74°
148°
8. Angles A and G are
A B
D C
E F
H G
4. On a plane, point A doesn’t lie on line b.
Three new, distinct lines are drawn through
point A. How many of these lines could be
perpendicular to line b?
A. One could be
B. One or two could be
C. All three could be
A.
B.
C.
D.
Consecutive interior angles.
Alternate interior angles.
Alternate exterior angles.
Corresponding angles.
9. Choose all true statements that apply to the
diagram below.
5. Point A is not on line j. How many lines pass
through A and are perpendicular to j?
A. 0
B. 1
C. 2
D. An infinite number
6. Which of these are pairs of corresponding
angles? Choose all that apply.
B C F G
A D E H
A. A and B
B. A and C
C. A and E
J
K
I L
Lines l and m are skew lines.
Lines l and m are perpendicular.
Lines l and n are perpendicular.
Lines l, m, and n are in the same
plane, P.
E. Lines m and n are skew lines.
F. Lines m and n are perpendicular.
G. Lines l and n are parallel.
H. Lines m and n are in the same plane, P.
A.
B.
C.
D.
D. A and H
E. A and I
Practice Problems continue . . .
4.1 Parallel Lines and Angles155
Practice Problems continued . . .
10. The lines m and n are intersected by a
transversal d. List all pairs of corresponding
angles formed in the diagram below.
14. m/FPM 5 34°. Find the measure of /NMP.
m
1
4
2
3
5
6
8
M
F
N
P
n
7
d
11. In the diagram below, if m/6 5 100°, what
is the measure of /2?
m
2
1
n
5
6
8
15. Lines x and y are parallel. If m/C 5 120°,
what is m/F?
3
B
4
A
7
12. The lines IJ and KL are parallel. Use the
diagram below to answer the questions.
A
B
C
D
E
K
D
F
E
J
H
F
G
L
aList all the pairs of alternate interior
angles.
bList all the pairs of alternate exterior
angles.
cIf m/D 5 40°, what is the measure of
/E?
dIf m/B 5 145°, what is the measure of
/H?
13. MP 2 When two parallel lines are
intersected by a transversal, one of the
angles formed is equal to 150°. Make a
sketch, labeling all the angles formed.
Then find the measures of all the angles
in your diagram.
G
H
y
x
p
I
C
16. MP 2, 3 The lines a, b, c, and p are coplanar.
a is perpendicular to p, b is perpendicular to
p, and c intersects a. Do lines b and c intersect?
Make a sketch and explain your answer.
17. MP 2, 4 5th and 10th Avenues are parallel;
K Street runs parallel to J Street. Lincoln
Way is perpendicular to 10th Avenue.
K Street and Lincoln Way make an angle of
39°, and 5th Avenue and J Street make an
angle of 129°.
N
W
KS
S
reet
J St
39°
Lincoln Way
129°
5th Ave.
t
tree
E
10th Ave.
aAmid is going northeast on J Street and
turning left on Lincoln Way. By what
angle does he need to turn?
bRachel goes north on 10th Avenue and
needs to turn right onto K Street. By what
angle does she need to turn?
Practice Problems continue . . .
156 Chapter 4: Parallel and Perpendicular Lines
Practice Problems continued . . .
18. MP 3 In the diagram below, line m
intersects the sides of the angle M> at
the >points A and B. Can both MA and
MB be perpendicular to line m? Use the
perpendicular postulate to explain your
answer.
23. The parallel lines a and b are intersected by a
transversal c.
c
a
1
4
b
2
3
5
A
M
B
m
19. MP 3 m/2 1 m/8 5 46°. Find the measures
of all angles in the diagram below. Show your
work.
4
24.
A B
D C
E F
H G
2
b
3
6
5
7
8
Exercises 20–22: Lines x and y are parallel.
x
aWhat is m/C, if m/F = 110°?
b Calculate the sum of m/D and m/E.
25. MP 3 The lines a and b intersect. Selena
says she can draw a third line, c, that is
­
parallel to both a and b. Is Selena right?
Explain why or why not.
c
C
B
D
A
7
aIf /2 = 108°, what is the measure of /5?
bIf /3 = 45°, what is the measure of /8?
cIf /1 = 82°, what is the measure of /6?
dIf /7 = 29°, what is the measure of /1?
a
1
8
6
F
E
G
H
y
20. m/B is 4 times m/A. What is m/F?
21. m/B is 3 times m/A. What is m/F?
22. List all pairs of consecutive interior angles.
26. In a triangle ABC, how many lines passing
through the point C can be drawn so they
are parallel to the side AB? Make a sketch
and use the parallel postulate to explain
your answer.
2
7. Lines a, b, and c are coplanar. The
lines a and b are both perpendicular to line c.
Show that a is parallel to b. Make a sketch
to explain your answer. Hint: Suppose that
line a is not parallel to line b, and use the
perpendicular postulate to show that this
leads to a contradiction.
28. MP 3, 4 The system of three weights
is balanced as shown. All vertical strings are
parallel. Show that m/A 1 m/B 5 m/C.
A
B
C
4.1 Parallel Lines and Angles157
LESSON 4.2
4.2 More on Parallel Lines and Angles
Angle Relationships
We prove alternate exterior angles are congruent when a transversal intersects
two parallel lines.
Prove: /A > /G
Given: m i n
A B
D C
E F
H G
m
n
Statement
Reason
min
Given
We were told that m and n are
parallel.
/A > /E
Corresponding
angles postulate
Since A and E are corresponding
angles, the corresponding angles
postulate says they are congruent.
/E > /G
Vertical angles
Since E and G are vertical angles,
congruence theorem they are congruent.
/A > /G
Transitive property
of congruence
The transitive property of
congruence tells us that if /A > /E
and /E > /G, then /A > /G.
In this activity, you prove the alternate interior angles theorem.
158 Chapter 4: Parallel and Perpendicular Lines
MODEL PROBLEM
Find the measures of all the angles. Show two different ways to solve for the angles. The measure
of angle G is 65°.
a
B
A
D
C
E
H
b
F
G
SOLUTION
Method 1
Calculate
corresponding angles
mC 5 mG 5 65°
C and G are corresponding,
so they are congruent.
alculate
C
supplementary angles
mC 1 mD 5 180°
C and D are supplementary,

so calculate mD.
65° 1 mD 5 180°
mD 5 115°
Calculate
supplementary angles
mG 1 mH 5 180°
65° 1 mH 5 180°
alculate another angle using
C
a supplementary angle.
mH 5 115°
alculate vertical
C
angles
mA 5 mC 5 mE 5 mG 5 65°
Fill in the other angles.
mB 5 mD 5 mF 5 mH 5 115°
Method 2
Use alternate exterior
angles
mG 5 mA 5 65°
G and A are alternate exterior
angles, so they are congruent.
Use vertical angles
mA 5 mC 5 65°
There are two pairs of vertical
angles.
mE 5 mG 5 65°
Use supplementary
angles
mA 1 mB 5 180°
mC 1 mD 5 180°
Use four pairs of supplementary
angles.
mE 1 mF 5 180°
mG 1 mH 5 180°
Calculate
supplementary angles
65° 1 mB 5 180°
mB 5 115°
mB 5 mD 5 mF 5 mH 5 115°
We show how we calculate one
angle. The remaining angles
would be calculated the same way.
4.2 More on Parallel Lines and Angles159
Parallel Lines and Angles
Earlier, we stated a postulate: If two parallel lines are intersected by a transversal,
then the corresponding angles are congruent. The converse is true: If the
corresponding angles are congruent, then the lines are parallel. Since converses
are not always true, we write the converse as a postulate.
This postulate lets us write a series of theorem converses:
160 Chapter 4: Parallel and Perpendicular Lines
MODEL PROBLEMS
1.
MP 3, 6
a If C  F, which lines must be parallel?
Explain.
b If B  D, which lines must be parallel?
Explain.
B
C
i
D
E
o
F
a
c If mB 1 mF 5 180°, which lines must
be parallel?
w
SOLUTION
a i i o because of the alternate interior angles theorem converse. This is a pair of alternate interior
angles, with line w being the transversal. Since the angles are congruent, the lines the transversal
intersects are parallel: i i o.
b w i a because of the corresponding angles theorem converse. This is a pair of corresponding
angles, with line i being the transversal. Since the angles are congruent, the lines the transversal
intersects are parallel: w i a.
c Since mB 1 mF 5 180° then lines i and o must be parallel, because angle B is a vertical angle
with the unnamed angle that is a consecutive interior angle to F, making F and the unnamed angle
supplementary.
2. MP 6 What do the angle measures have to equal for the lines to be parallel?
3(x – 10°) (x + 10°)
SOLUTION
If the lines are
parallel, the
angles must be
supplementary
3(x 2 10°) 1 (x 1 10°) 5 180°
The two angles whose
measures are given by algebraic
expressions are consectuve
interior angles. Their measures
must sum to 180° if the lines are
parallel.
Solve equation
3x 2 30° 1 x 1 10° 5 180°
Distribute the 3. Combine like
terms, and then solve for x.
4x 2 20° 5 180°
x 5 50°
Use value of x
in expressions
3(x 2 10°) 5 3(50° 2 10°) 5 120°
x 1 10° 5 50° 1 10° 5 60°
Angle measures 5 120° and 60°
Substitute the value of x into
the expressions for the angles.
Check the calculations: The
angle measures sum to 180°.
Model Problems continue . . .
4.2 More on Parallel Lines and Angles161
MODEL PROBLEMS continued
3.
MP 2, 5, 7 A designer is trying to ensure that levels 1 and 2
are parallel in a game. If he is sketching out a diagram for the
factory, how might he do so?
SOLUTION
Construct level 1 using a
straightedge
Construct an angle using a
protractor
The designer can use
a postulate concerning
corresponding angles and
parallel lines, as well as a
protractor and straightedge.
Use the straightedge—a ruler,
in this case—to draw a line
segment that represents level 1.
Place the straightedge to the
left of level 1 and measure
the angle with a protractor.
This particular angle is 120°.
Construct a corresponding angle
using a protractor
Move the protractor down
along the straightedge, and
measure a corresponding
angle of 120°.
Construct level 2 using the
corresponding angles postulate
converse
Using a straightedge, construct
level 2 at the 120° mark on
the protractor. Since the
corresponding angles are
congruent, the levels are parallel.
162 Chapter 4: Parallel and Perpendicular Lines
PRACTICE
1. If m1 5 146°, what must the sum of the
angles 4 and 6 be so the lines a and b are
parallel?
1
2
4
b
A. 17°
B. 34°
1
3
5
a
6
8
a
A. 36°
B. 54°
A. 74°
B.
C.
75°
c
II. The sum of corresponding angles is 180°.
III. Interior consecutive angles are
congruent.
IV. Interior consecutive angles are
supplementary.
I only
I and III only
I and II and IV only
I and IV only
8
7
105°
5. In the diagram below, the lines a, b, and c are
intersected by a transversal d. Which lines
must be parallel? The diagram is not drawn
to scale.
42°
140°
138°
C. 126°
D. 144°
I. Alternate interior angles are congruent.
6
D. 106°
d
3. Two lines are intersected by a transversal.
Which of the following conditions must be
true so the lines are parallel?
A.
B.
C.
D.
3
5
F
p
4
b
2. What is the measure of angle F?
36°
2
a
7
C. 68°
D. 146°
b
4. If the lines a and b are parallel, and m3 is
40% greater than m4, what is m8?
a
b
c
d
A.
B.
C.
D.
a and b
a and c
b and c
None of the above
6. Two parallel lines are intersected by a
transversal. One of the angles formed is 70°.
Can any of the other angles be equal to 30°?
A. Yes
B. No
C. Not enough information to decide
Practice Problems continue . . .
4.2 More on Parallel Lines and Angles163
Practice Problems continued . . .
7. In the diagram below, m3 5 m4 5 128°,
m5 5 52°. Which lines must be parallel?
The diagram is not drawn to scale.
Exercises 11–12: Solve for x.
11.
x – 4°
l
7(x – 8°)
3
m
2
5
1
12.
n
4
2x + 61°
p
3(x + 7°)
A.
B.
C.
D.
m and n only
n and p only
m, n, and p are all parallel
No parallel lines on the diagram
13. MP 3 In the diagram, the pairs of sides
of the angles K and M are parallel: a is
parallel to c, and b is parallel to d. Prove that
mK 1 mM 5 180°.
8. According to the congruent angles marked
in the diagram below, which lines must be
parallel?
a
a
c
b
m
A. a and b
B. m and n
d
C. All of the above
D. None of the above
9. According to the congruent angles marked
in the diagram below, which lines must be
parallel?
c
d
p
q
A. c and d
B. p and q
b
K
n
L
M
Exercises 14–17: In the diagram, the lines x and y
are parallel.
x
y
t
1
4
2
3
5
8
6
7
C. All of the above
D. None of the above
10. Suppose mG is known. Which of the
following techniques could be used to find
mA? Choose all that apply.
A B
D C
E F
H G
a
14. If m4 5 124°, find all other angles.
15. If m6 1 m4 5 250°, find m7.
16. If m3 is 72° less than m8, find m2.
17. If m6 is 5 times m1, find m4.
b
A. Alternate exterior angles to find A.
B. Vertical angles to find E, then
corresponding angles to find A.
C. Alternate interior angles to find F, then
vertical angles to find A.
D. Consecutive interior angles to find D,
then corresponding angles to find A.
164 Chapter 4: Parallel and Perpendicular Lines
Practice Problems continue . . .
Practice Problems continued . . .
Exercises 18–21: Use the diagram to answer the
questions.
1
2
4
3
6
5
a
8
b
7
18. m2 5 45° and m7 is three times m4. Are
the lines a and b parallel? Why or why not?
19. 2 measures 68°. What must the measure of
angle 5 be so the lines a and b
are parallel?
25. MP 2, 3 ABC is equal to 60°. BCD is
equal< to> 120°.< Dilmah
made a sketch and
>
said AB and CD must be parallel, since
120° 1 60° 5 180°.
< Toshi
>
<also> sketched the
angles and said AB and CD intersect. Try to
create both the sketches Dilmah and Toshi
could have made and explain who is right.
26. MP 2, 4 The sketch shows Jen’s favorite
chair from the side. The black segments
represent the backrest, seat, and one of the
legs of the chair. The gray segments represent
one of the chair’s arms.
20. 3 measures 100°. What must the m
­ easure
of angle 8 be so the lines a and b are parallel?
110˚
110˚
21. If m1 5 x 1 1 and m6 5 2x 1 2, what
must m5 be so the lines a and b are
parallel?
22. MP 3 Using the diagram, prove the
alternate interior angles theorem converse.
A B
D C
E
m
F
n
H G
Prove: m n
Given: D F
23. Explain how you would prove that if the
alternate interior angles formed by two lines
and a transversal are not congruent, then the
two lines intersect.
aAre the chair’s arms parallel to the chair’s
seat? Why or why not?
bJen likes to sit with her arms parallel to
her thighs, so she decides to put a pillow
under her hands. Finding the angle by
which she needs to raise her arms will
help Jen to figure out how thick the
pillow should be. What must that angle
be?
27. In the diagram, the corresponding sides
of the angles K and M are parallel: a is
parallel to c, and b is parallel to d. Prove that
the angles K, L, and M are congruent.
a
24. Are lines m and n parallel? Justify your
answer.
m
A = 57°
B
D
K
n
C
F = 122°
G
E
H
c
L
M
b
d
28. Two parallel lines are intersected by a
transversal. Prove that angle bisectors (lines
dividing an angle into two adjacent congruent
angles) of the alternate interior angles are
parallel to each other. Make a sketch and label
the angles, referring to them in your proof.
Practice Problems continue . . .
4.2 More on Parallel Lines and Angles165
Practice Problems continued . . .
29. MP 5 To draw a line parallel to the line m,
through the point M, Clive uses a straightedge
and a drawing triangle, as shown in the
picture. First, he sets the triangle so one side
lies on the line m and draws a line along the
longer side of the triangle so it passes through
the point M. After that, he moves the triangle
along this new line until its vertex is at the
point M. He then draws a line along the
original side of the triangle and continues this
line using the straightedge. Explain why this
method produces parallel lines.
30. Using the diagram provided and the
knowledge that 3  10 and 1  6,
prove that 15  13. Note that no
information is given about whether lines
are parallel or not.
M
c
d
1 2
5 4 3
6 7
9 8
a
11
10
12
14 13
15 16
18 17
b
m
• Multi-Part PROBLEM Practice •
MP 2, 4 Use the diagram to answer the questions.
136th Avenue is parallel to 130th Avenue.
N
W
a Is Jefferson Boulevard parallel to 136th Avenue?
Why or why not?
S
110°
y
Wa
set
Jefferson Blvd.
Sun
50°
63°
130th Ave.
136th Ave.
b A lightrail track is planned to be built south from the
intersection of Jefferson Boulevard and Sunset Way. What should
the measure of the acute angle between the train route
and Sunset Way be so the tracks run parallel to 136th Avenue?
E
La
rk
St
re
e
t
c What would be the measure of the acute angle between the
lightrail route and Lark Street?
LESSON 4.3
4.3 Perpendicular Lines
Pairs of Perpendicular Lines
166
Chapter 4: Parallel and Perpendicular Lines
This is a
theorem about
perpendicular
lines and their
relationship to
right angles.
The symbol for
perpendicular
lines is an
upside-down
T: These two
theorems
are about
perpendicular
transversals.
m
Prove: m  y
A
B
Statement
We prove the
perpendicular
transversal
theorem.
x
y
Reason
Diagram
The diagram tells us that lines x
and y are parallel, and that line
m is perpendicular to line x.
mA 5 90°
Definition of perpendicular
lines
By the definition of
perpendicular lines, we know
that A is a right angle, and
therefore its measure must be
90°.
mA 5 mB 5 90°
Angles A and B are
corresponding angles
A  B
mA 5 mB 5 90°
Angles A and B are congruent,
because they are corresponding
angles of parallel lines. So they
have the same measure, 90°.
B is a right angle
Definition of right angle
Since angle B has a measure of
90°, angle B is a right angle by
definition.
my
Definition of perpendicular
lines
Finally, by definition of
perpendicular lines, line m is
perpendicular to line y, and we
have proved our theorem.
xiy
mx
4.3 Perpendicular Lines 167
MODEL PROBLEMS
1. a Is p parallel to q?
b Is p perpendicular to r?
q
p
r
s
SOLUTION
a Yes, because of the lines perpendicular to a transversal theorem. The theorem states that if two
lines (such as lines p and q) are perpendicular to the same line (such as line s), then they are parallel
to each other.
b Yes, because of the perpendicular transversal theorem. The theorem states that if two lines (such as
r and s) are perpendicular to one of a pair of parallel lines (line q), then they are perpendicular to the
other (line p).
2. Prove: o  m
m
n
o
p
SOLUTION
Statement
Reason
pm
Diagram
This is shown in the diagram.
oip
Lines perpendicular to a transversal
theorem
Two lines, o and p, perpendicular to the
same line, n, are parallel.
om
Perpendicular transversal theorem
Since m is perpendicular to p, one of two
parallel lines, m is also perpendicular to
o, the other parallel line.
168 Chapter 4: Parallel and Perpendicular Lines
Construction: Perpendicular and Parallel Lines
We construct a perpendicular bisector, a perpendicular line that divides a line
segment in half. To do so, we do two things at the same time.
(1) We construct a line perpendicular to our original line using a compass and a
straightedge.
(2) We bisect (divide into two equal parts) the line segment defined by the two
points on the line in the first step.
Constructing a perpendicular line with a compass:
1. Start with two points on line.
2. Draw
an arc. Keep the compass wider than distance
halfway between the points. Put one end of the compass
on a point and draw an arc with the other.
3. Draw
another arc from second point. Keep compass
open same amount.
4. Connect intersection points of arc. Use a straightedge.
We can also construct a line that is perpendicular to our original line and that
passes through a given point not on the original line:
1. Place compass at point. Draw arc through line.
4.3 Perpendicular Lines 169
2. Mark two intersection points. Create two arcs using
points on line. Follow same steps as shown above.
3. Connect intersection points of arcs. Use a straightedge.
Why does this construction
create a perpendicular line?
We add to our diagram to
answer this question.
An intersecting
line that forms
right angles is
perpendicular
to another
line, so this
proof explains
the geometry
behind the
construction.
Statement
Reason
AC  DC and
AB  DB
Each pair was drawn with
same compass setting
Both pairs of segments are congruent.
BC  BC
Reflexive property
The two triangles share this side.
nABC  nDBC
SSS congruence
The triangles have three pairs of
congruent sides. (We discuss SSS in
depth in the next chapter.)
ACB  DCB
Definition of congruence
They are two corresponding angles in
congruent triangles.
ACB, DCB are They are linear pair and
right angles
congruent.
170
Chapter 4: Parallel and Perpendicular Lines
The measures of linear pair angles
sum to 180° and the angles are
congruent. Only 90° angles match this
description.
We construct a line parallel to our original line by constructing two perpendicular
lines:
1. Start
with a perpendicular line. Follow the same
steps as shown above.
2. Construct
another perpendicular line. Line is
parallel to original line.
Finally, we construct a parallel line through a point, C:
1. Connect point C and line using straightedge. Lines
intersect at D.
2. Create two arcs. Set compass width greater than
halfway between points C and D, but smaller than
the distance between C and D. Draw arcs with
compass placed at C and D.
4.3 Perpendicular Lines 171
3. Draw
two small arcs. Set compass width to lower
previously drawn arc. Draw arcs with compass
placed at intersection points of the previously drawn
arcs and the construction line. Label intersection
point E.
< >
4. Draw parallel line, CE . Use a straightedge.
Why
does the above construction
create a parallel line?
Statement
Reason
KC > MD and
CE > DP and
KE > MP
Each pair was drawn with Each connects points drawn with same
same compass setting
compass setting.
nKCD > nMDP
SSS congruence
Three pairs of corresponding sides are
congruent. (We discuss SSS in depth in
the next chapter.)
/KCE > /MDP
Definition of congruence
They are two corresponding angles in
congruent triangles.
CD is a transversal
Diagram
Given in the diagram.
/KCE and /MDP
are corresponding
angles
Diagram
They occupy corresponding positions
relative to the lines.
Lines are parallel
Corresponding angles are
congruent
Since the corresponding angles are
congruent, the lines must be parallel.
172 Chapter 4: Parallel and Perpendicular Lines
Distance and Perpendicular Lines
The distance between a point and a line is the shortest distance between the point
and the line. That distance is the length of the perpendicular segment from the
point to the line.
MODEL PROBLEM
What is the distance between the point and the line?
y
5
4
3
2
1
–5
–4
–3
–2
–1
1
–1
2
3
4
x
5
–2
–3
–4
SOLUTION
–5
Draw perpendicular
segment
y
5
4
3
2
1
–5
–4
–3
–2
–1
–1
1
2
3
4
5
x
–2
–3
–4
–5
Use ruler postulate
distance 5 uy2 2 y1u
distance 5 u21 2 3u 5 4
PRACTICE
1. Lines a and b are parallel. Line c forms an
angle of 91° with line a. Choose the true
statement.
A. b is parallel to c.
B. c is perpendicular to b.
C. None of the above.
2. If the angle between the lines a and b is 89°,
and c is parallel to b, then
A.
B.
C.
D.
a is perpendicular to c.
c is parallel to a.
b is perpendicular to a.
None of the above.
Practice Problems continue . . .
4.3 Perpendicular Lines 173
Practice Problems continued . . .
7. How do we know that lines g and h are
parallel?
3. Which two lines must be parallel?
a
b
g
h
c
d
j
e
k
A. c and d
B. a and e
C. a and b
4. What is the distance between point M and
line c?
y
5
8. Lines m and n are perpendicular. Which
of the following statements must be true?
Choose all that apply.
4
3
2
m
1
–5
–4
–3
–2
–1
1
2
3
4
5
x
–1
–2
–3
c
A.
B.
C.
D.
–4
n
F
E
G
H
M
A. Angles E, F, G, and H are right angles.
B. /E > /F
C. Angles E and F are supplementary
angles.
D. Angles E and G are vertical angles.
–5
3
4
5
24
5. If line c is perpendicular to one of a pair
of perpendicular lines, and all lines are
coplanar, then
A. c is perpendicular to the other line of
the pair.
B. c is parallel to the other line of the pair.
C. c is parallel to both lines.
D. None of the above.
6. If point P is 8 units away from the y-axis,
which of these coordinates may represent
the point P?
A. (2, 8)
B. (26, 28)
A. Given
B. Perpendicular transversal theorem:
g and h perpendicular to j
C. Lines perpendicular to a transversal
theorem: g and h perpendicular to k
D. Perpendicular transversal theorem:
g and h perpendicular to k
C. (28, 3)
D. (4, 4)
174 Chapter 4: Parallel and Perpendicular Lines
9. What is the distance between the point and
the line?
y
6
5
4
3
2
1
–6 –5 –4 –3 –2 –1
–1
1
2
3
4
5
6
x
–2
–3
–4
–5
–6
Practice Problems continue . . .
Practice Problems continued . . .
10. MP 3 Using the diagram, prove the lines
perpendicular to a transversal theorem.
m
n
A B
D C
E F
H G
20. Find the distance between the point (27, 25)
and the x-axis.
21. Find the distance between the point
(20.57, 221) and the y-axis.
p
22. Find the distance between the point
(20.54, 27) and the y-axis.
23. Find the distance between the point A and
the line a.
y
Prove: m n
10
Given: m p, and n p
11. MP 2, 3 Prove that the angle bisectors
of adjacent supplementary angles are
­perpendicular to each other. Provide a
sketch with your answer.
12. On a plane, a is perpendicular to b, b is
parallel to c, and n is perpendicular to a.
Is line n perpendicular to c? Explain your
answer.
Exercises 13–17: In the diagram, the measure of
angle GDK is 35°, c ' a, d ' e, a i b, and e i f.
c
M
d
A
a
b
e
K
f
G
F
D
B
e
C
E
H
J
8
6
A
4
2
–10 –8
–6
–4
–2
2
–2
–4
4
6
8
10
x
a
–6
–8
–10
24. Cori says that the distance from a point
(x, y) to the y-axis is the point’s x-coordinate.
Using the point (24, 23), explain whether
Cori’s statement is correct.
25. The coordinates of point M are (x, 23). If
the distance from point M to the y-axis is
9 units, list all values of x.
26. The coordinates of point N are (5, y). If the
distance from point N to the x-axis is 3 units,
list all values of y.
15. Find the measure of /BCD.
Exercises 27–30: The line going through
point M and perpendicular to line a intersects
line a at point C. The coordinates of the points
are given. Find the distance from point M to
line a.
16. Find the measure of /EHF.
27. M(8, 23), C(1, 211)
17. What is the measure of /CEJ?
28. M(7, 23), C(23, 210)
18. When two parallel lines are intersected by
a transversal, the alternate exterior angles
add up to 180°. What could be said about
the angle between the lines and the
transversal? Why?
29. M(7, 24), C(24, 210)
13. List all pairs of perpendicular lines.
14. What is the measure of /GDM?
30. M(10, 23), C(4, 210)
Practice Problems continue . . .
19. Find the distance between the point (25, 23)
and the x-axis.
4.3 Perpendicular Lines 175
Practice Problems continued . . .
31. Using information from the diagram, find
the angle between the rays a and b.
a
b
116°
64°
154°
m
n
35. MP 4 Two mirrors are placed so they
form a right angle. A light ray is directed
into one mirror so it forms a 45° angle with
it. Using the fact that the incoming and
outgoing angles are congruent, prove that
the ray reflected from the second mirror is
parallel to the ray striking the first mirror.
32. MP 3, 5 Line m passes through point M.
Explain how to construct a line p through
point M and perpendicular to line m using
a compass and a straightedge. Perform the
constructions.
33. Describe the set of all points that are the same
distance d from a given line a, and which are
in a single plane passing through line a.
34. Describe the set of all points that are the
same distance from parallel lines a and b and
line in the plane containing the lines.
incoming
angle
outgoing
angle
45°
1
36. MP 4, 5 Using geometric construction
tools, construct a map of Commercial
Avenue, 2nd Street, and 3rd Street with the
following constraints: 2nd and 3rd Streets
are parallel to each other; Commercial
Avenue intersects both 2nd and 3rd Streets,
but not at right angles.
LESSON 4.4
4.4 Parallel Lines, Perpendicular Lines, and Slope
Determining When Lines Are Parallel or
Perpendicular
Parallel Lines
y
We discuss how to determine if two lines are parallel.
(1) If both lines are vertical, then they are parallel.
Any two horizontal lines are also parallel.
10
9
8
7
6
5
4
3
2
1
• Two vertical lines, such as x 5 4 and x 5 5, are parallel.
• Two horizontal lines, such as y 5 2 and y 5 21, are parallel.
(2) If both the lines are not conveniently vertical or horizontal,
then you have to consider their slopes to determine if they
are parallel. If their slopes are the same, the lines are parallel.
• Lines with same slope, such as y 5 3x 1 2 and y 5 3x 2 1,
are parallel.
176 Chapter 4: Parallel and Perpendicular Lines
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1
–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
y = 3x + 2
y = 3x – 1
1 2 3 4 5 6 7 8 9 10
x
Perpendicular Lines
As with parallel lines, the slopes of perpendicular lines have a specific mathematical
relationship.
(1) If one line is vertical and another horizontal, they are perpendicular.
(2) If the lines are not vertical and horizontal, you have to consider
their slopes to determine whether they are perpendicular. The
slopes of perpendicular lines are negative reciprocals.
• The two lines in the diagram are perpendicular because their
1
slopes, 22 and , are negative reciprocals. To put it another
2
way, the product of the slopes is 21.
Rotations and Perpendicular Lines
y
10
9
8
7
6
5
4
3
2
1
y=
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1
–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
Perpendicular lines have slopes that are negative reciprocals. We
show how this is true with a 90° clockwise rotation. We call the
perpendicular lines a and p. The equations for the lines will be
y 5 max, and y 5 mpx.
1
2
x+2
1 2 3 4 5 6 7 8 9 10
x
y = –2x – 1
y
(xa, ya)
(0, 0)
a
x
p
(xp , yp) = (ya, –xa)
Slope of line a
Rotate the line
90° clockwise
Slopes of line a and p are
­negative reciprocals
ma 5 Slopea 5
y2 2 y 1
x2 2 x 1
Slopea 5
y2 2 y 1
x2 2 x 1
Slopea 5
ya 2 0
ya
5
xa
xa 2 0
(xa, ya) → (xp, yp) 5 (ya, 2xa)
mp 5
yp
2xa
5
ya
xp
This is the equation for the slope
of a line.
We use (0, 0) for (x1, y1), (xa, ya) for
(x2, y2), and calculate the slope.
Now rotate the line using the
­coordinate relationship for a
90° clockwise rotation. The point
remains on the line as it rotates.
Use the equation for the slope
again, substituting the ­coordinates
of the point on the rotated
line. Comparing it to our original
slope, we see that the slopes are
negative reciprocals.
4.4 Parallel Lines, Perpendicular Lines, and Slope 177
MODEL PROBLEMS
1. A line is rotated 90°. The slope of the original line and slope of the line created by the rotation, are
A. Equal.
B. Opposite.
C. Negative reciprocals.
D. None of the above.
SOLUTION
The answer is C. Rotating a line 90° will create a perpendicular line. Perpendicular lines have negative
reciprocal slopes.
2. Write equations for b and c.
1
The equation for a is y 5 x 1 2.
2
y
10
a
b
–10
10
SOLUTION
Equation for b
–10
Equation for c
3.
c
Line b is parallel to a, because both lines are
Same slope, passes
through origin y 5
x
1
x
2
Slope is negative reciprocal
y-intercept at y 5 2
y 5 22x 1 1
perpendicular to c. Thus, a and b have the same slope,
1
. The y-intercept of b is zero since it passes through
2
the origin.
Line c is perpendicular to a. Thus, the slope of c is the
1
negative reciprocal of , the slope of a. The y-intercept
2
of c is 2, since c passes through the point (0, 2).
MP 1, 2, 3 A teacher says: “Triangle ABC has vertex A at (2, 3), vertex B at (1, 10), and vertex
C at (5, 7). ABC is a right triangle, with C its right angle.” Is he correct?
SOLUTION
Calculate slope AC
Calculate slope CB
Slopes are negative
reciprocals
We can use
If the legs are at right angles, the slopes
of two line
they are perpendicular
segments to
and will have slopes that
determine if
are negative reciprocals.
a triangle is a
Calculate the slope of AC.
right triangle.
rise
7 2 10
3
Calculate the slope of the
5
5 2
Slope CB 5
run
521
4 other leg.
3
4
The slopes are negative
? a2 b 5 21
3
4
reciprocals: their product
C is right angle
is 21. This means the lines
rise
723
4
Slope AC 5
5
5
run
522
3
are perpendicular and it is
a right triangle.
178
Chapter 4: Parallel and Perpendicular Lines
Distance Between a Point and a Line
Earlier, we calculated the distance between a point and a horizontal or vertical line, or
the shortest path between the two. Now we do the more difficult task of computing
the distance between a point and a line that is neither horizontal nor vertical.
To calculate the distance, we need to determine the perpendicular distance between
the point and the line, since that will be the shortest path. This means we must first
determine a line perpendicular to the line and passing through the point.
MODEL PROBLEM
Find the distance between the point and the line.
y
10
5
–10
–5
5
10
x
–5
SOLUTION
Calculate slope of
line
–10
slope 5
y2 2 y1
21 2 3
24
5
5
5 22 Calculate the slope of the line.
x2 2 x1
220
2
y 5 22x 1 3
Write equation
y5
Write equation for
­perpendicular line
Write in slope-intercept form. The line
crosses the y-axis at 3, so b 5 3.
1
x1b
2
The distance will be along the
perpendicular line. Its slope is the
negative reciprocal of the slope of the
given line.
Calculate x of
intersection point
1
(5) 1 b
2
1
b5
2
1
1
y5 x1
2
2
1
1
x 1 5 22x 1 3
2
2
x 5 1
Substitute
y 5 22x 1 3 5 22(1) 1 3 5 1
Identify the
intersection point
(1, 1)
Use distance
formula
distance 5 "(x2 2 x1)2 1 (y2 2 y1)2
Use the distance formula to ­calculate
the distance between the point (5, 3),
and the point of intersection (1, 1).
Evaluate
distance 5 "(2)2 1 (4)2 5 "20
distance 5 2"5 < 4.47
Square the terms and add. The d
­ istance
is about 4.47.
Substitute point
into line equation
Calculate b by substituting the
x- and y-values of the point we want to
find the distance to into the equation.
(3) 5
Set the two equations for the lines equal
to each other. Where the lines intersect,
their y- and x-values will be the same.
distance 5 "(3 2 1)2 1 (5 2 1)2
Solve for y.
4.4 Parallel Lines, Perpendicular Lines, and Slope 179
PRACTICE
1. Which of the following equations represents
a line parallel to the graph of the equation
y 5 9x 1 0.4?
A. y 5 0.4x 1 9
B. y 5 29x
C. y 5 9x 2 3.6
2. Which of the following equations
­represents a line parallel to the graph of the
equation y 5 3x 1 0.4?
A. y 5 0.4x 1 3
B. y 5 23x
C. y 5 3x 1 2.4
3. The graph of the equation y 5 1 is parallel
to the graph of which of the following
­equations? Choose all that apply.
A.
B.
C.
D.
E.
F.
G.
y 5 1x
y 5 21
y5x11
y55
x50
y50
x55
y 5 4x
y 5 24
y5x14
y55
B.
C.
x 5 5.3
y 5 25.3
E. y 5 0
x 5 0.3
F.
7. A beetle sits at the origin on a coordinate
plane. If he crawls along the shortest
­distance to the line y 5 2x 1 4, how far, in
units, does he need to crawl?
A. 2"2
B. 3
C. "2
D. 4
8. Which of the following equations r­ epresents
the line that passes through the points
(22, 24) and (3, 231)?
C. y 5 211x 1 2
D. y 5 210x 2 6
E. x 5 0
F.
y50
G. x 5 5
5. Which of the following equations represents
a line perpendicular to the graph of the
equation y 5 9x 1 4?
A.
B.
C.
A. y 5 212x 1 4
B. y 5 29x 2 3
4. The graph of the equation y 5 4 is parallel
to the graph of which of the following
­equations? Select all that apply.
A.
B.
C.
D.
6. The graph of the equation y 5 5.3 is
­perpendicular to which of the following
graphs? Select all that apply.
1
D. x 5 0
A. y 5 2
5.3
1
y 5 2 x 1 (22)
9
1
x14
9
y 5 29x
y5
Exercises 9–11: Given three points, determine if
they define a right triangle, a non-right triangle,
or do not define a triangle.
9. (26, 25), (5, 22), and (22, 2)
A. Not a right triangle
B. Right triangle
C. The points do not define a triangle
10. (24, 22), (6, 3), and (3, 9)
A. Not a right triangle
B. Right triangle
C. The points do not define a triangle
11. (24, 22), (7, 22), and (23, 22)
A. Not a right triangle
B. Right triangle
C. The points do not define a triangle
Exercises 12–14: Write the equation of the line
that passes through the points.
12. (24, 242) and (2, 18)
13. (23, 23) and (1, 21)
180 Chapter 4: Parallel and Perpendicular Lines
Practice Problems continue . . .
Practice Problems continued . . .
14. (21, 216) and (2, 20)
Exercises 26–28: Find the equation of the line
passing through a given point and having a
given relationship with another line.
15. Show that the equation of a line that goes
through the points (x1, y1) and (x2, y2) is:
y 2 y2
x 2 x2
5
y2 2 y1
x2 2 x1
26. A line passes through the point (21, 9) and
is perpendicular to the line y 5 –2x 1 1.
27. A line passes through the point (23, 8) and
is perpendicular to the line y 5 27x 1 4.
16. Are the lines described by the equations
y 5 5x 1 7 and y 5 2x 1 3 parallel? Justify
your answer.
28. A line passes through the point (0, 1) and is
parallel to y = –2x + 6.
17. What is the slope of a line perpendicular to
y 5 2x?
29. Find the shortest distance between the point
(5, 2) and the line y 5 3x 1 6.
18. What is the slope of a line perpendicular to
y 5 0.5x 1 4?
30. The line m goes through the points (23, 5)
and (1, 2). The line n goes through the points
(0, 0) and (24, 23). Are these lines parallel?
Why or why not?
19. What is the slope of a line perpendicular to
y 5 24x 1 1?
20. Find the equation of the line that passes
through (0, 25) and is perpendicular to
1
1
y52 x1 .
3
4
31. Use the diagram to answer the questions.
y
10
9
8
7
6
5
4
3
2
1
21. Write the equation of the line shown.
y
9
8
7
6
5
4
3
2
1
–9 –8 –7 –6 –5 –4 –3 –2 –1
–1
–2
–3
–4
–5
–6
–7
–8
–9
1 2 3 4 5 6 7 8 9
a
b
1 2 3 4 5 6 7 8 9 10
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1
–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
x
x
aWhat is the slope of line segment a?
bWhat is the slope of line segment b?
32. On the diagram, the triangles ABC and
DBK are similar, that is, their corresponding
angles are congruent. Prove that the lines
have the same slope.
22. What is the slope of a line that passes
through the origin and is parallel to
1
y 5 x 1 8?
5
23. What is the slope of a line that passes
through the origin and is parallel to
1
y 5 x 1 8?
6
24. Give an equation of a line that is parallel to
the line x 5 22.8 and passes through the
point (27.2, 0).
y
A
D
B
x
C
K
25. The line a goes through the points
(6, 24) and (25, 0). The line b passes
through the points (2, 27) and (29, 23). Are
these lines parallel? Why or why not?
Practice Problems continue . . .
4.4 Parallel Lines, Perpendicular Lines, and Slope 181
Practice Problems continued . . .
33. The sides of a triangle are given by the
equations y 5 0.4x 2 34.5, y 5 24x 1 1.9,
and y 5 22.5x 1 8. Determine if it is a right
triangle.
37. State and graph the equation of a line
that goes through the point (1, 2) and is
perpendicular to the line that connects the
points (4, 3) and (22, 1).
34. MP 3 Two vertices of a triangle are (0, 5)
and (0, 25). Define the third vertex in such
a way that the resulting triangle has a right
angle. (Note that there are many possible
correct answers.)
38. Two robots, Daisy and Chaise, are standing on
the adjacent vertices of a playing field shaped
as a quadrilateral. Coordinates of the four
vertices are (2, 2), (5, 1), (3, 6), and (0, 3). On a
signal, each robot sends a laser beam toward
the opposite vertex. Find the coordinates of
the point where these beams meet.
35. Graph the lines y 5 x 1 2 and y 5 x 2 4. On
your graph, mark a segment showing the
shortest distance between the lines. Find
the distance between the lines to the nearest
tenth of a unit.
36. State and graph the equation of a line that goes
through the point (4, 3) and is parallel to the
line that connects the points (1, 2) and (21,25).
39. MP 3 A teacher says: “Triangle ABC has
vertex A at (25, 22), vertex B at (2, 210),
and vertex C at (0, 0). ABC is a right triangle,
with C its right angle.” Is he correct? Justify
your answer.
• Multi-Part PROBLEM Practice •
MP 3, 7 Points A (24, 6), B (2, 10), C (11, 3), and D (2, 23) form a quadrilateral.
a Graph the quadrilateral on a coordinate plane.
b What type of quadrilateral is ABCD? Explain.
c Move point C or D to make ABCD a parallelogram but not a rectangle. Justify your action.
d Move point C or D to make ABCD a rectangle. Justify your action.
e Is your rectangle from part d a square? Explain.
LESSON 4.5
4.5 Parallel Lines and Triangles
Interior and Exterior Angles
Interior angles of a polygon are the angles inside a polygon. Exterior angles of
a polygon are angles formed by a side of a polygon and the extension of its
adjacent side.
An interior angle
Exterior
and its exterior
angles
angle form a linear
Exterior
angle
pair, so that the
Interior
Interior
angles
angles
measures of the
angles sum to 180°.
182
Chapter 4: Parallel and Perpendicular Lines
You may already be familiar with the following theorem—that the measures of
the angles in a triangle sum to 180°.
We will prove that the measures of a triangle’s angles sum to 180°. To prove this
theorem, we do a little planning first, adding a line not present in the diagram.
An auxiliary line is a line added to help in a proof.
Prove: m4 1 m2 1 m5 5 180°
We can add a
point or segment
as an auxiliary in
a proof.
A
2
4
5
B
C
Draw a line parallel to segment BC:
We want to prove the measures
of the angles in a triangle sum
to 180°. To do this, we start by
drawing the line AD. We know
that the measures of angles 1,
2, and 3 sum to 180°, since the
angles form a line.
Statement
< >
AD i BC
A
1 2
4
B
D
3
5
C
If we can show
that 1 and 3
are congruent
to 4 and 5, the
triangle’s interior
angles, we will
have proven that
the sum of interior
angles of a
triangle is 180°.
Reason
Diagram
The auxiliary line is parallel to BC.
m1 1 m2 1 m3 5 180°
Definition of a straight
angle
Since angles 1, 2, and 3 form a line, we
know that their measures sum to 180°.
1  4
Alternate interior angles
theorem
AB is a transversal of the parallel line
and line segment, so these alternate
interior angles are congruent.
3  5
Alternate interior angles
theorem
AC is also a transversal of the parallel
line and line segment, so these alternate
interior angles are congruent too.
Substitution
Replace angles 1 and 3 from the previous
steps with 4 and 5 to prove that the
interior angle measures of a triangle
sum to 180°.
m4 1 m2 1 m5 5 180°
4.5 Parallel Lines and Triangles 183
We state the exterior angle theorem.
MODEL PROBLEMS
A
1. What is m/Z?
60°
Z
SOLUTION
40°
B
C
Exterior angle
theorem
m/Z 5 m/A 1 m/C
The exterior angle theorem states that the measure
of the angle equals the sum of the measures of the
two remote interior angles.
Evaluate
m/Z 5 60° 1 40° 5 100°
Substitute and add.
2. What is m/N, given m/N 5 (2x 1 10°)? M
20°
2x + 10°
N
SOLUTION
Exterior angle
theorem
K
m/KPM 5 m/N 1 m/M The exterior angle theorem states that the
3x – 20° 5 (2x 1 10°) 1 20° measure of the exterior angle equals the sum of
the measures of the two remote interior angles.
Substitute the expressions and values shown in
the diagram.
Solve equation 3x 2 20° 5 2x 1 30°
3x 5 2x 1 50°
x 5 50°
Substitute
3x – 20°
P
Combine the constants, then isolate the variable.
m/N 5 2x 1 10°
Substitute the value of x into the expression for
m/N 5 2(50°) 1 10° 5 110° m/N.
184 Chapter 4: Parallel and Perpendicular Lines
Angles in a Right Triangle
The corollary below follows from the theorem that a triangle’s angles sum to 180°.
A corollary is a theorem easily proven from another theorem.
To put it another
way, the other
two angles in a
right triangle are
complementary.
MODEL PROBLEM
What is z?
SOLUTION
J
z + 15°
K
2 (z – 12°)
Use triangle sum corollary
Distribute, combine
L
2(z 2 12°) 1 z 1 15° 5 90°
2z 2 24° 1 z 1 15° 5 90°
3z 2 9 5 90°
Solve equation
Substitute
3z 5 99°
z 5 33°
We substitute our value for z into the expressions 2(z 2 12°)
and z 1 15°. The angles equal 42° and 48°. They sum to 90°,
as expected.
PRACTICE
1. In a triangle ABC, A is a right angle. If the
sum of angles A and B is less than 150°,
which of the following best describes the
measure of angle C?
A.
B.
C.
D.
Greater than 150°
Equal to 30°
Greater than 30°
Greater than 45°
2. In a right triangle XYZ, Y is a right angle
and YW is perpendicular to XZ where W is
a point on XZ. If X = 34°, then mZYW is:
A.
B.
C.
D.
34°
56°
90°
146°
Practice Problems continue . . .
4.5 Parallel Lines and Triangles 185
Practice Problems continued . . .
3. Which angles are interior angles? Choose all
that apply.
e f
d c
h
e
f
g
h
E.
F.
G.
H.
4. Which angles are exterior angles? Choose all
that apply.
e f
d c
h
g
a b
A.
B.
C.
D.
a
b
c
d
E.
F.
G.
H.
e
f
g
h
5. In a triangle, the measure of the first angle
is 12° less than the measure of the second
one, and the measure of the third angle is
twice as great as the measure of the first one.
What are the ­measures of the angles of the
triangle?
A.
B.
C.
D.
12. Angles’ ratio: 1 : 2 : 3
13. In a triangle, the measure of the second
angle is twice as big as the measure of the
first angle. The measure of the third angle is
6 times as big as the measure of the second
one. Find the angles of the triangle.
a b
a
b
c
d
Exercises 11–12: Find the measures of all interior
angles of a triangle, using the given ratio
between them.
11. Angles’ ratio: 5 : 6 : 7
g
A.
B.
C.
D.
10. /A 5 60°, /B 5 60°
48°, 96°, 36°
54°, 108°, 42°
54°, 108°, 94°
54°, 84°, 42°
Exercises 6–10: Given the measures of two
interior angles A and B of a triangle ABC, find
the measure of the third interior angle, C.
6. /A 5 50°, /B 5 30°
7. /A 5 75°, /B 5 40°
8. /A 5 55°, /B 5 80°
9. /A 5 35°, /B 5 120°
14. Dorothy says two angles in a triangle she
drew have the measures 94° and 98°. Can
Dorothy be right about her triangle? Why or
why not?
15. MP 2 The measure of one of the exterior
angles of a ­triangle is 40°, and the measure
of one of the interior angles is 30°. Make
a sketch, marking these angles, and find
the measures of all interior angles of the
triangle.
16. MP 3, 6 Can there be both an obtuse and
a right angle in a triangle? If so, give an
­example. If not, explain why.
17. Using the diagram, find the measures of the
angles of the triangle.
(2x + 1)°
(6x + 12)°
(x – 4)°
18. Find the measure of exterior angle adjacent
to interior angle C of the triangle ABC, if
/A 5 40° and /B 5 84°.
19. One acute angle of a right triangle is
20° smaller than the other. Find the measure
of the larger acute angle of the triangle.
20. One acute angle of a right triangle is 5 times
as the other. Find the measures of the acute
angles of the triangle.
21. One acute angle of a right triangle is 2 times
as great as the other. Find the measures of
the acute angles of the triangle.
22. Two exterior angles of a triangle are equal to
116° and 144°. Find the measures of all three
interior angles of the triangle.
Practice Problems continue . . .
186 Chapter 4: Parallel and Perpendicular Lines
Practice Problems continued . . .
23. Two exterior angles of a triangle are equal to
101° and 141°. Find all three interior angles
of the triangle.
30. In the diagram, ABC is a right triangle and
CD ' AB. Prove that m/A 5 m/BCD.
A
D
24. The sum of the exterior angles adjacent to
angles A and B of the triangle ABC is equal
to 240°. Find the interior angle C of the
triangle.
25. One of the angles of a triangle is 50°, and
the difference of the other two angles is 20°.
Find the angles of the triangle.
26. Two exterior angles of a triangle are equal to
120° and 135°. Find the third exterior angle
of the triangle.
27. Two exterior angles of a triangle are equal to
100° and 145°. Find the third exterior angle
of the triangle.
28. An exterior angle of a right triangle is equal
to 135°. Find the acute angles of the triangle.
29. One acute angle of a right triangle is 10°
smaller than the other. Find the larger acute
angle of the triangle.
C
B
31. Find the angles of a right triangle with two
congruent acute angles.
32. The sum of the exterior angles adjacent to
angles A and B of the triangle ABC is equal to
230°. Find the interior angle C of the triangle.
33. MP 3 What is the sum of the three exterior
angles of a triangle, where one angle is
­measured at each vertex? Justify your answer.
34. MP 3 The sum of any two interior angles
of a particular triangle is greater than 90°.
Determine the type of triangle: obtuse, right,
or acute. Explain your answer.
35. MP 3, 6 Prove the sum of the measures of
the exterior angles of a triangle is 360°.
Chapter 4 Review
1. The two lines shown are parallel. Which
expression must be true?
K
B
F
A.
B.
C.
D.
K5F
K 1 B 5 180°
K 1 F 5 180°
K1F5B
2. Angle Z forms a vertical pair with angle B,
and angle Z is also a corresponding angle
with Y. Z and Y are angles formed by a
transversal and two parallel lines. Angles B
and Y must be
A.
B.
C.
D.
Supplementary.
Complementary.
Congruent.
Have no definite relationship.
3. Point A is not on line j. How many lines pass
through A and are parallel to j?
A.
B.
C.
D.
0
1
2
An infinite number
4. The angles A and E are
C B
D A
G F
H E
A.
B.
C.
D.
Consecutive interior.
Alternate interior.
Alternate exterior.
Corresponding.
Chapter Review continues . . .
Chapter 4 Review 187
Chapter Review continued . . .
5. Point b is not on line z. You can correctly say:
A. No line can pass through b and be
parallel to z.
B. Only one line passes through b and is
perpendicular to z.
C. b and z cannot be coplanar.
D. None of the above.
10. When two parallel lines are intersected
by a transversal, the difference of two
consecutive interior angles formed is 50°.
Find these angles.
11. In the diagram, assume the measure of angle
A is twice the measure of angle B. What is
the measure of angle G?
A B
D C
E F
H G
6. Two lines never intersect. This means:
A.
B.
C.
D.
They must be coplanar.
They must be parallel lines.
They must be skew lines.
None of the above.
7. If m i n, how do we know that p ' n?
12. Given m/1 5 (2y 2 1), m/2 5 (3x 2 52),
m/5 5 (2x 1 34), and m/7 5 (3y 2 10),
what is m/8?
p
m
2
1 3
4 5
q
n
m
p
6 7
8 9
A. p is perpendicular to m, and m is
parallel to n.
B. p is perpendicular to m, and m is
perpendicular to n.
C. q is perpendicular to n.
8. Do the points (25, 4), (8, 3), and (22, 22)
define a right triangle?
n
13. Prove that the opposite angles of the
quadrilateral ABCD are congruent.
A
1
4
D
3
a
B
2
C
b
A. No.
B. Yes.
C. The points do not define a triangle at
all.
D. It is not possible to tell.
9. Using the diagram, prove the consecutive
interior angles theorem converse.
A B
D C
E F
H G
Prove: m n
m
n
Given: mD + mE = 180°
Chapter Review continues . . .
188 Chapter 4: Parallel and Perpendicular Lines
Chapter Review continued . . .
14. Two parallel lines are intersected by a
transversal. Two consecutive interior angles
have measures of 2(x 2 1) and 3x 2 3. What
is the measure of the smaller angle?
15. When two parallel lines p and q are
­intersected by a transversal n, the interior
consecutive angles A and B are congruent.
Prove that the transversal is perpendicular
to both parallel lines.
19. In the diagram, AB || DE and m/CDE 5 21°.
Find the measure of /ABC.
D
E
C
B
16. Find the distance between the point M and
the line t.
y
5
4
3
2
t
1
–5
–4
–3
–2
–1
1
2
3
4
5
x
–1
–2
–4
–5
17. Find the distance between the
point P and the line l.
22. Using the diagram, prove the perpendicular
lines and right angles theorem.
Prove: Angles A, B, C, and D are right
angles.
Given: m ' n
y
5
4
3
m
2
1
P
–5
–4
–3
20. Lines b and d are perpendicular to each
other. Line a is parallel to line b. Line c
intersects line b, forming an angle of 142°.
All lines are coplanar. Find the smaller of
the angles that line c forms with line a. Then
find the smaller of the angles that line c
forms with line d. Provide a sketch with
your answer.
21. MP 5 Point M does not lie on line m.
Explain how to construct a line p through
point M and perpendicular to line m using
a compass and a straightedge. Perform the
constructions.
–3
M
A
–2
–1
1
2
3
4
5
x
–1
A B
D C
–2
–3
n
–4
–5
l
18. A snail moves along a straight line with a
constant speed, changing direction by a right
angle every 3 minutes. It follows this pattern
for 2 hours and 45 minutes. During the last
minute of its trip, will the snail be parallel to
the line it started along or perpendicular to it?
23. What is the slope of a line perpendicular to
1
y 5 x 1 9?
4
24. What is the slope of a line perpendicular to
1
y 5 x 1 7?
6
25. Explain how to determine if three given
points could be the vertices of a right
triangle.
Chapter Review continues . . .
Chapter 4 Review 189
Chapter Review continued . . .
26. Kathy found the slopes of two lines that she
thought might be perpendicular. Both slopes
were negative. What should Kathy conclude
about the angle between these lines?
27. Explain why a triangle cannot have two
right angles.
28. What is the measure of angle B?
20°
A B
40°
29. Find the measure of the exterior angle
adjacent to interior angle C of the
triangle ABC, if A 5 45° and B 5 110°.
190
Chapter 4: Parallel and Perpendicular Lines
30. The measure of the exterior angle adjacent
to the interior angle M of the triangle KLM is
110°. The two non-adjacent interior angles of
the triangle are congruent. Find all interior
angles of the triangle.
31. Using the diagram, prove the triangle sum
corollary.
Prove: mA + mB 5 90°
Given: mC 5 90°
A
C
B
Cumulative Review
for
Chapters
1–4
1. Two lines are described by the equations
y 5 2zx 1 5 and y 5 zx 2 3. The lines are
A.
B.
C.
D.
Parallel.
Perpendicular.
Overlapping.
None of the above.
E
A
F
2. How could the angle be written? Choose all
that apply.
B
A
C
A. ABC
B. CBA
C. ACB
D. A
E. C
< >
< >
3. Choose all that apply. AB and AC are
C
A
B
F
A.
B.
C.
D.
5. How would the congruent triangles be
described?
Coplanar in plane F.
Parallel.
Perpendicular.
None of the above.
4. Two of the lines shown are parallel. This
means that angles V and W
C
B
D
A. nABC  nDEF
B. nABC  nFDE
C. nABC  nFED
6. Consider the true statement: “If it is an even
number, then it is divisible by 2.” Which
of the following is also a true statement?
Choose all that apply.
A. Converse
B. Inverse
C. Contrapositive
7. Dr. Watson said: “If the patient has the
boola-boola disease, he sneezes 4 times a
minute. If a patient mumbles ‘Eli, Eli’ twice
an hour, the patient has the boola-boola
disease.” Dr. Watson should conclude
A. If a patient sneezes 4 times a minute, he
mumbles ‘Eli, Eli’ twice an hour.
B. If a patient sneezes 4 times a minute,
the patient has the boola-boola disease.
C. A person who does not have the boolaboola disease never sneezes.
D. If a patient mumbles ‘Eli, Eli’ twice an
hour, he sneezes 4 times a minute.
V
W
A.
B.
C.
D.
Are congruent.
Are supplementary.
Are complementary.
Have no defined mathematical
relationship.
Chapters 1–4 191
8. Check the statement, or statements, that are
true.
A
B
A.
B.
C.
D.
30°
120°
C
60° 150°
D
Angles A and B are complementary.
Angles A and B are supplementary.
Angles B and C are complementary.
Angles B and D are supplementary.
9. What is m/C?
A = 127°
D
B
C
10. Point P lies on YZ. YZ is 28 and YP is 21.
What is PZ?
HJ
8
5 , how far is
1. J is on HI, and HI 5 33. If
1
JI
3
J from H?
12. You want to translate a figure 8 units
­ orizontally and 6 units vertically. What
h
constants do you add to the x- and
y-coordinates for this translation?
13. MP 3 Prove the transitive property for
parallel lines: If the lines a and b are p
­ arallel,
and the lines b and c are parallel, then a is
also parallel to c. Hint: Suppose that
line a is not parallel to line c, and use the
parallel postulate to show that this leads to a
contradiction.
­ arallel to
14. In the diagram below, line a is p
line b, /1  /3, and the measure of
/9 5 120°. Prove the shaded triangle is
equilateral.
1 2 3
6 5 4
a
b
7 8
10 9
192 Cumulative Review
11 12
14 13
15. The measure of angle D is twice the measure
of angle E. What is the measure of angle B?
A B
D C
E F
H G
16. Two parallel lines are intersected by a
­transversal. Prove that the angle bisectors
of two consecutive interior angles are
­perpendicular. Make a sketch, label the
angles, and refer to them in your proof.
17. A teacher says: “Triangle ABC has vertex A
at (24, 21), vertex B at (2, 28), and vertex C
at (0, 0). ABC is a right triangle, with C its
right angle.” Is she correct? Justify your
answer.
18. A graphic artist is rescaling a picture from
72 pixels to represent a square to 90 ­pixels.
If the original picture had a width of
300 pixels, how many pixels wide is the
new picture?
19. What is the measure of angle 2x?
3x
2x
x
20. nABC is translated as shown. What is the
translation?
y
9
8
7
6
5
4
3
2
1
–9 –8 –7 –6 –5 –4 –3 –2 –1
–1
–2
–3
–4
–5
–6
–7
–8
–9
B
A
A’
B’
C
1 2 3 4 5 6 7 8 9
x
C’
21. Figure A is dilated by a scale factor of 5 to
form figure B, which is then dilated by a
7
scale factor of to form figure C. What
4
is the scale factor that dilates figure A to
figure C?
22. Two triangles are similar. Triangle ABC has
36 times the area of triangle DEF. How
many times longer would a side of ­triangle
ABC be than the corresponding side in
triangle DEF?
23. Using the point (a, b), show algebraically
that four 90° counterclockwise rotations
about the origin return you to the original
point (a, b).
24. What are the coordinates of the upper right
vertex of a figure if it is dilated by a factor
of 7? The original figure’s coordinates are
(1, 3), (1, 5), (27, 3), and (27, 5), and the
center of dilation is the origin.
Exercises 25–28: Write the biconditional for each
definition.
25. The real numbers are composed of the
rational and irrational numbers.
26. An integer prime is a number with exactly
32. Kim’s office is 14 feet by 20 feet. She wishes
to lengthen the walls of her office by a scale
factor of 1.5. What will be the dimensions of
the expanded office?
Exercises 33–34: Determine whether the
following statement is always true, sometimes
true, or never true. Justify your answer.
33. Inductive reasoning leads to a correct
conclusion.
34. Deductive reasoning based on true
statements leads to a correct conclusion.
35. MP 2, 3 ,4 There were 437 homes in the
county of Piranha Lake in 1995. Ten years
later, the number of homes had increased
to 761. Therefore, the number of homes
in Piranha Lake in 2010 was exactly 923.
Is this an example of inductive or deductive
reasoning using a graph? Justify your
answer.
two positive distinct factors.
27. Two angles that sum to 180° are called
supplementary angles.
28. Parallel lines are lines on a plane that do not
intersect.
29. MP 5 Draw and label the sides and angles
of two similar quadrilaterals.
30. Two rectangles are similar. Their widths are
corresponding sides: the width of rectangle
A is 5, and the width of rectangle B is 3. If
the larger rectangle has a height of 15, what
is the height of the smaller rectangle?
31. The corresponding sides of two similar
octagons (8-sided figures) have a scale
factor of 2.4. If the smaller octagon has a
perimeter of 20, what is the perimeter of
the larger octagon?
Chapters 1–4 193