Download Lesson 3-1 and Lesson 3-2

Enrichment
PERIOD
A4
Glencoe Geometry
001_022_GEOCRMC03_890512.indd 10
Chapter 3
10
Glencoe Geometry
No, because the Parallel Postulates states that the line will never
intersect and that is not possible in spherical geometry because
two lines (great circles) always intersect in two points.
4. Does the fifth axiom, or Parallel Postulate, hold for spherical geometry? Explain.
Two lines (great circles) will always intersect in two points in
spherical geometry.
3. Make a conjecture about the number of points of intersection of any two lines (great
circles) in spherical geometry.
2. Try to draw two lines (great circles) or wrap two rubber bands around a ball that do not
intersect. Is it possible? no
1. Get a ball. Wrap two rubber bands around the ball to represent two lines (great circles)
on the sphere. How many points of intersection are there? 2
Exercises
The fifth axiom of Euclidean Geometry states that given any straight line and a point not
on it, there exists one and only one straight line that passes through that point and never
intersects the first line. The fifth axiom is also known as the Parallel Postulate.
Latitude and longitude meet at right angles on a sphere.
4. Right angles can be found on the sphere.
So, in spherical geometry, a great circle is both a line and a circle.
3. A circle can be drawn with any center or radius.
A line of infinite length in spherical geometry will go around itself an infinite number
of times.
2. A finite line segment can be extended infinitely in both directions.
However, a straight line in spherical geometry is a great circle. A great circle is a
circle that goes around the sphere and contains the diameter of the sphere.
1. A straight line can be drawn between any two points.
The first four axioms in spherical geometry are the same as those in the Euclidean
Geometry you have studied. However, in spherical geometry, the meanings of lines and
angles are different.
On a map, longitude and latitude appear to be lines. However, longitude and latitude exist
on a sphere rather than on a flat surface. In order to accurately apply geometry to longitude
and latitude, we must consider spherical geometry.
DATE
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
DATE
PERIOD
=
=
=
=
=
=
=
105
105
75
105
75
105
75
∠1
∠3
∠4
∠5
∠6
∠7
∠8
and
and
and
and
and
and
and
∠2
∠2
∠2
∠3
∠2
∠3
∠6
form a linear pair.
form a linear pair.
are vertical angles.
are alternate interior angles.
are corresponding angles.
are corresponding angles.
are vertical angles.
102; Alt. Int. Angles Th.
Chapter 3
11. ∠7 68; Vertical Angles Th.
9. ∠4 100; Cons Int. Angles Th.
7. ∠12 100; Supp. Angles
5 6
8 7
5 6
8 7
1 2
4 3
p
p
5 6
87
9 10
12 11
n
m
p
n
m
11
Lesson 3-2
4/11/08 10:52:50 AM
Glencoe Geometry
Th.
13 14
16 15
q
10. ∠3 80; Att. Int.
Angles Th.
w
v
12. ∠16 112; Vertical Angles Th; Cons.
Interior Angles Th.
8. ∠1 80;Corr. Angles
1 2
4 3
6. ∠14 78; Cons. Int. Angles Th;
Corre. Angles Th.
q
9 10
12 11
13 14
16 15
1 2
4 3
Angles Th.
4. ∠7 102; Corre. Angles Th.
2. ∠6 78; Cons. Int.
In the figure, m∠9 = 80 and m∠5 = 68. Find the measure
of each angle. Tell which postulate(s) or theorem(s) you used.
5. ∠15 102; Corre. Angles Th.
3. ∠11 102; Corre. Angles Th.
1. ∠5
In the figure, m∠3 = 102. Find the measure of each angle.
Tell which postulate(s) or theorem(s) you used.
Exercises
m∠1
m∠3
m∠4
m∠5
m∠6
m∠7
m∠8
Example
In the figure, m∠2 = 75. Find the measures
of the remaining angles.
Also, consecutive interior angles are supplementary.
• alternate exterior angles
• alternate interior angles
• corresponding angles
When two parallel lines are cut by a transversal,
the following pairs of angles are congruent.
Angles and Parallel Lines
Study Guide and Intervention
Parallel Lines and Angle Pairs
3-2
NAME
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11
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 3
Spherical Geometry
3-1
NAME
Answers (Lesson 3-1 and Lesson 3-2)
Chapter 3
DATE
Angles and Parallel Lines
Study Guide and Intervention
(continued)
PERIOD
15 = y
5y
75
−
=−
5
5
75 = 5y
m∠2 = m∠3
r s, so m∠2 = m∠3
because they are
corresponding angles.
p
1
4
q
A5
5x°
(13y - 5)°
(5y + 5)°
x = 11; y = 10; use
consecutive interior angles
(11x + 4)°
x = 15; y = 19; use corresponding
and supplementary angles
(4x + 10)°
(5x - 5)°
(6y - 4)°
4.
2.
(15x + 30)°
10x°
(4z + 6)°
3x°
(5x - 20)°
x = 10; y = 25; Use consecutive
interior and alternate interior angles
4y°
2y°
r
s
2x° 90° x°
2y°
z°
Glencoe Geometry
Answers
12
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Angles and Parallel Lines
Skills Practice
DATE
4. ∠1
6. ∠6
110
110
3. ∠8
5. ∠4
70
110
110
12. ∠11 100
100
11. ∠5
105
15. ∠7
105
18. ∠9
75
16. ∠15 105
14. ∠5
w
2
5
x
1
m
6
s
3
40°
(5x)°
(8x - 10)°
(6y + 20)° (7x)°
u
12
11 16
15
y
8
t
22.
s
r
z
Lesson 3-2
5/26/08 10:53:38 AM
Glencoe Geometry
x = 21, y = 29; Use alternate
interior angles to find x. Then use
supplementary angles to find y.
(4y + 4)° 60°
(3x - 3)°
x = 10, y = 15; Use alternate
interior angles to find x. Then use
supplementary angles to find y.
20.
13
x = 11, y = 13; Use corresponding
angles to find x. Then use
supplementary angles to find y.
(5y - 5)°
(11x - 1)°
(9x + 21)°
x = 28, y = 47; Use the
supplementary angles to find
x. Then use alternate exterior
angles to find y.
(3y - 1)°
Chapter 3
21.
19.
4
7
9 10 14
13
2
1
3
4
q
10
9
6 12 11
5
7
8
1 2
3 4
5 6
7 8
PERIOD
Find the value of the variable(s) in each figure. Explain your reasoning.
17. ∠14 75
105
13. ∠2
In the figure, m∠3 = 75 and m∠10 = 105. Find the measure
of each angle.
80
10. ∠2
80
9. ∠8
80
8. ∠6
100
7. ∠9
In the figure, m∠7 = 100. Find the measure of each angle.
2. ∠5
70
1. ∠3
In the figure, m∠2 = 70. Find the measure of each angle.
3-2
NAME
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13
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
001_022_GEOCRMC03_890512.indd 12
106°
2y°
x°
6.
3
x = 30; y = 15 ; z = 150 use
supplementary, alternate interior,
and consecutive interior angles
x = 74; y = 37; z = 25;
use consecutive interior, corresponding,
and supplementary angles
Chapter 3
5.
2
x = 6; y = 24; Use consecutive
interior angles
(3y + 18)°
90°
Find the value of the variable(s) in each figure. Explain your reasoning.
3.
1.
Find the value of the variable(s) in each figure. Explain your reasoning.
Exercises
20 = x
15 + 5 = x - 5 + 5
15 = x - 5
3x + 15 - 3x = 4x - 5 - 3x
p q, so m∠1 = m∠2
because they are
corresponding angles.
m∠1 = m∠2
3x + 15 = 4x - 5
If m∠1 = 3x + 15, m∠2 = 4x - 5, and m∠3 = 5y,
find the value of x and y.
Example
Algebra can be used to find unknown values in
angles formed by a transversal and parallel lines.
Algebra and Angle Measures
3-2
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Answers (Lesson 3-2)
2. ∠8
6. ∠13 106; Supp. 106; Cons. 92; Vert. m
1
8
2
n
7
9
16
10
x = 14, y = 37; Use
Supplementary and alternate
exterior angles
3x°
(4y - 10)°
(9x + 12)°
130
50°
100°
1
10.
8.
Given: || m, m || n
3y°
(2x + 13)°
98
62°
1
144°
Glencoe Geometry
001_022_GEOCRMC03_890512.indd 14
Chapter 3
14
12. FENCING A diagonal brace strengthens the wire fence and prevents
it from sagging. The brace makes a 50° angle with the wire as shown.
Find the value of the variable. 130
It is given that m, so ∠1 ∠8 by the Alternate
Exterior Angles Theorem. Since it is given that m n,
∠8 ∠12 by the Corresponding Angles Postulate.
Therefore, ∠1 ∠12, since congruence of angles is
transitive.
Sample proof:
Prove: ∠1 ∠12
(5y - 4)°
s
50°
n
m
Glencoe Geometry
y°
9 10
11 12
5 6
7 8
1 2
3 4
k
x = 28, y = 23; Use corresponding
and supplementary angles
11. PROOF Write a paragraph proof of Theorem 3.3.
9.
Find x. (Hint: Draw an auxiliary line.)
7.
r
12
11 13
14
15
4
3 5
6
Find the value of the variable(s) in each figure. Explain your reasoning.
5. ∠11 106; Supp. 3. ∠9 88; Corr. Th, Supp ∠s 4. ∠5
1. ∠10 92; Corr. Th.
In the figure, m∠2 = 92 and m∠12 = 74. Find the measure
of each angle. Tell which postulate(s) or theorem(s) you used.
Angles and Parallel Lines
Practice
10˚
1
Level 1
Level 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4/11/08 001_022_GEOCRMC03_890512.indd
10:53:11 AM
15
Chapter 3
65
1
115˚
3. CITY ENGINEERING Seventh Avenue
runs perpendicular to both 1st and 2nd
Streets, which are parallel. However,
Maple Avenue makes a 115° angle with
2nd Street. What is the measure of
angle 1?
52
52˚
1
2. BRIDGES A double decker bridge has
two parallel levels connected by a
network of diagonal girders. One of the
girders makes a 52° angle with the
lower level as shown in the figure. What
is the measure of angle 1?
170
Ramp
1st St.
2nd St.
15
Angles and Parallel Lines
DATE
PERIOD
116˚
Lesson 3-2
4/11/08 10:53:17 AM
Glencoe Geometry
x = 20; upper bank = 100 and
lower bank = 80
b. How wide is the scanning angle for
each robot? What are the angles that
the bridge makes with the upper and
lower banks?
They are consecutive interior
angles and are supplementary.
4x + 5x = 180
a. How are the angles that are covered
by the robots at the lower and upper
banks related? Derive an equation
that x satisfies based on this
relationship.
lower bank
upper bank
5. SECURITY An important bridge crosses
a river at a key location. Because it is so
important, robotic security cameras are
placed at the locations of the dots in the
figure. Each robot can scan x degrees.
On the lower bank, it takes 4 robots to
cover the full angle from the edge of the
river to the bridge. On the upper bank,
it takes 5 robots to cover the full angle
from the edge of the river to the bridge.
The rectangle must be sawed along the
dashed line in the figure. What is the
measure of angle 1? 64
1
4. PODIUMS A carpenter is building a
podium. The side panel of the podium is
cut from a rectangular piece of wood.
Word Problem Practice
1. RAMPS A parking garage ramp rises
to connect two horizontal levels of a
parking lot. The ramp makes a 10° angle
with the horizontal. What is the
measure of angle 1 in the figure?
3-2
le A
ve.
3-2
NAME
Map
PERIOD
7th Ave.
A6
ge
DATE
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 3
Brid
NAME
Answers (Lesson 3-2)
Chapter 3
Enrichment
DATE
PERIOD
A7
Glencoe Geometry
Answers
16
2.
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
DATE
y -y
1
3
5
0
7
5
-−
-4
Chapter 3
−
11. EH
2
5
undefined
9. EM
3
7. AB
Find the slope of each line.
5. T(1, -2), U(6, -2)
3. L(1, -2), N(-6, 3)
1. J(0, 0), K(-2, 8)
12. BM
10. AE
8. CD
2
17
1
-−
0
-2
6. V(-2, 10), W(-4, -3)
4. P(-1, 2), Q(-9, 6)
2. R(-2, -3), S(3, -5)
2
13
−
2
1
-−
5
2
-−
H
(–1, –4)
A(–2, –2)
y
M(4, 2)
E(4, –2)
x
x
Lesson 3-3
4/11/08 10:53:42 AM
Glencoe Geometry
D(0, –2)
O
(1, 2)
p
O (2, 0)
y
B(0, 4)
(–3, 2)
(–2, –2)
q
C(–2, 2)
Determine the slope of the line that contains the given points.
Exercises
-3 - 2
2-0
2
=−
or - −
2
1
m= −
x2 - x1
y -y
For line q, substitute (2, 0) for (x1, y1) and (-3, 2) for (x2, y2).
-2 - 1
-2 - 2
4
=−
or −
y2 - y1
m= −
x2 - x1
For line p, substitute (1, 2) for (x1, y1) and (-2, -2) for (x2, y2).
Find the slope of each line.
2
2
1
and (x2, y2) is given by the formula m = −
x - x , where x1 ≠ x2.
Example
PERIOD
The slope m of a line containing two points with coordinates (x1, y1)
Slopes of Lines
Study Guide and Intervention
Slope of a Line
3-3
NAME
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10:53:26 AM
17
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
001_022_GEOCRMC03_890512.indd 16
Chapter 3
1.
In the following pictures, draw lines to find the vanishing point or points.
NEXT
REST STOP
64 miles
The picture below shows a straight road going into the distance. The parallel lines of the
left and right sides of the road have been traced to show the vanishing point.
If you look down a road that does not have any curves or bends in it, the sides of the road
that are parallel appear to meet at a single point. This is called the vanishing point and has
been used in artwork since the 1400s.
Vanishing Point
3-2
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Answers (Lesson 3-2 and Lesson 3-3)
Slopes of Lines
Study Guide and Intervention
DATE
4-2
2
1
= −
slope of CD
=−
=−
5-1
4
2
2
A8
"
0
$
#
%
parallel
Glencoe Geometry
001_022_GEOCRMC03_890512.indd 18
Chapter 3
18
with L(2, 1) and B(7, 4)
7. passes through C(−2, 5), parallel to LB
with A(−5, 6)
6. passes through H(8, 5), perpendicular to AG
and G(−1, −2)
5. slope = 4, passes through (6, 2)
0
y
x
x
Glencoe Geometry
(5, −2)
(6, 2)
(8, 5)
4. M(0, -3), N(-2, -7), R(2, 1), S(0, -3)
perpendicular
2. M(-1, 3), N(0, 5), R(2, 1), S(6, -1)
Graph the line that satisfies each condition.
neither
3. M(-1, 3), N(4, 4), R(3, 1), S(-2, 2)
parallel
1. M(0, 3), N(2, 4), R(2, 1), S(8, 4)
See students’ work
⎯ are parallel, perpendicular, or neither. Graph
⎯⎯ and RS
Determine whether MN
each line to verify your answer.
Exercises
When graphed, the two lines intersect but not at a right angle.
y
and CD
Product of slope for AB
Since the product of their slopes is not –1, the two lines are
not perpendicular.
and CD
.
Therefore, there is no relationship between AB
3
1
=−
or 1.5
3 −
(2)
The two lines do not have the same slope, so they are not parallel.
To determine if the lines are perpendicular, find the product of their slopes
5 - (-1)
6
= − = −
slope of AB
or 3
2
1 - (-1)
Find the slope of each line.
⎯⎯ and CD
⎯⎯ are parallel, perpendicular, or
Determine whether AB
neither for A(-1, -1), B(1, 5), C(1, 2), D(5, 4). Graph each line to verify your
answer.
Example
Two lines are perpendicular if and only if the product of their slopes is -1.
Two lines have the same slope if and only if they are parallel.
If you examine the slopes of pairs of parallel lines
and the slopes of pairs of perpendicular lines, where neither line in each pair is vertical, you
will discover the following properties.
(continued)
PERIOD
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Slopes of Lines
Skills Practice
DATE
/
0
y
x
1
4
3
−
5
0
y
8
x
-2
5
1
-−
-4
PERIOD
x
Chapter 3
O
J(3, 3)
D(–3, 1) Y(3, 0)
y
x
13. passes through Y(3, 0), parallel to DJ
with D(-3, 1) and J(3, 3)
A(0, 1)
O
y
11. slope = 3, passes through A(0, 1)
O
y
x
19
T(0, –2)
C(0, 3)
O
y
x
X(2, –1)
Lesson 3-3
5/26/08 2:54:46 PM
Glencoe Geometry
14. passes through T(0, -2), perpendicular
with C(0, 3) and X(2, -1)
to CX
R(–4, 5)
3
, passes through R(-4, 5)
12. slope = - −
2
perpendicular
10. A(-4, -8), B(4, -6), M(-3, 5), N(-1, -3)
neither
8. A(-1, 4), B(2, -5), M(-3, 2), N(3, 0)
Graph the line that satisfies each condition.
parallel
9. A(-2, -7), B(4, 2), M(-2, 0), N(2, 6)
parallel
7. A(0, 3), B(5, -7), M(-6, 7), N(-2, -1)
See students’ graphs.
⎯⎯ and MN
⎯⎯ are parallel, perpendicular, or neither.
Determine whether AB
Graph each line to verify your answer.
5.
6.
4. J(-5, -2), K(5, -4)
2
3. C(0, 1), D(3, 3) −
3
Find the slope of each line.
2. G(-2, 5), H(1, -7)
1. S(-1, 2), W(0, 4) 2
Determine the slope of the line that contains the given points.
3-3
NAME
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19
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 3
Parallel and Perpendicular Lines
3-3
NAME
Answers (Lesson 3-3)
Chapter 3
Slopes of Lines
Practice
DATE
2
1
-−
6. a line perpendicular to PS
2
- −
4. GR
5
13
2. I(-2, -9), P(2, 4) −
4
L
PERIOD
G
M
A9
x
U(2, –2)
O
G(4, –2)
x
x
Z(–3, 0)
E(–2, 4)
O
y
x
K(2, –2)
14. contains Z(-3, 0), perpendicular to EK
with E(-2, 4) and K(2, -2)
P(–3, –3)
O
y
4
12. slope = −
, contains P(-3, -3)
R
Glencoe Geometry
Answers
20
Glencoe Geometry
x
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 3
275.5 in.
4. WATER LEVEL Before the rain began,
the water in a lake was 268 inches deep.
The rain began and after four hours of
rain, the lake was 274 inches deep. The
rain continued for one more hour at the
same intensity. What was the depth of
the lake when the rain stopped?
6 hours
3. ROAD TRIP Jenna is driving 400 miles
to visit her grandmother. She manages
to travel the first 100 miles of her trip in
two hours. If she continues at this rate,
how long will it take her to drive the
remaining distance?
50
3
-−
2. DESCENT An airplane descends at a
rate of 300 feet for every 5000 horizontal
feet that the plane travels. What is the
slope of the path of descent?
4/11/08 001_022_GEOCRMC03_890512.indd
10:54:02 AM
21
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
001_022_GEOCRMC03_890512.indd 20
Chapter 3
15. PROFITS After Take Two began renting DVDs at their video store, business soared.
Between 2005 and 2010, profits increased at an average rate of $9000 per year. Total
profits in 2010 were $45,000. If profits continue to increase at the same rate, what will
the total profit be in 2014? $81,000
F(0, –3)
B(–4, 2)
y
13. contains B(-4, 2), parallel to FG
with F(0, -3) and G(4, -2)
O
y
2
1
, contains U(2, -2)
11. slope = - −
3
perpendicular
10. K(-3, -7), M(3, -3), S(0, 4), T(6, -5)
perpendicular
Graph the line that satisfies each condition.
parallel
S
20
3
−
21
PERIOD
B St.
Clover St.
200 yd
Lesson 3-3
4/11/08 10:54:09 AM
Glencoe Geometry
d. The intersection of B Street and
6th Street is 600 yards east of the
intersection of B Street and Ford
Street. How many yards north is it?
1
Both have a slope of −
because
3
both are perpendicular to Ford
and 6th, and the slope of a
perpendicular is given by the
negative reciprocal.
c. What are the slopes of Clover and
B Streets? Explain.
-3; Ford Street and 6th
Street are parallel so they
have the same slope.
b. What is the slope of 6th Street?
Explain.
450 yd
a. The intersection of B Street and
Ford Street is 150 yards east of the
intersection of Ford Street and Clover
Street. How many yards south is it?
N
5. CITY BLOCKS The figure shows a map
of part of a city consisting of two pairs of
parallel roads. If a coordinate grid is
applied to this map, Ford Street would
have a slope of -3.
DATE
Ford St.
9. K(-4, 10), M(2, -8), S(1, 2), T(4, -7)
P
8. K(-5, -2), M(5, 4), S(-3, 6), T(3, -4)
O
y
Slopes of Lines
Word Problem Practice
1. HIGHWAYS A highway on-ramp rises
15 feet for every 100 horizontal feet
traveled. What is the slope of the ramp?
3-3
NAME
6th St.
neither
7. K(-1, -8), M(1, 6), S(-2, -6), T(2, 10)
See students’ work
⎯ are parallel, perpendicular, or neither.
⎯⎯ and ST
Determine whether KM
Graph each line to verify your answer.
5
2
-−
5. a line parallel to GR
3. LM
2
−
3
Find the slope of each line.
1
1. B(-4, 4), R(0, 2) - −
2
Determine the slope of the line that contains the given points.
3-3
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Answers (Lesson 3-3)
Enrichment
PERIOD
7
2
2
5
5
A
L O
J
y
C
K
A10
7
3
−
hypotenuse: PQ
3
2
-−
7
4
−
1), R(2, 3)
−− 2
6. P(-2, -3), Q(5,
−−−
slope of PQ =
−−−
slope of QR =
−−
slope of PR =
x
B
3
O
y
R
9
Q
x
Glencoe Geometry
001_022_GEOCRMC03_890512.indd 22
Chapter 3
−−
10 −−
2
PR: - −
; SQ: - −
; no
S
P
7. P(-2, 6), Q(4, 0), R(1, -4), S(-5, 2)
22
R
O
2
Q x
−−
−−
1
PR: 2; SQ: - −
; yes
S
P
y
Glencoe Geometry
8. P(0, 6), Q(3, 0), R(-4, -2), S(-5, 4)
The coordinates of quadrilateral PQRS are given. Graph quadrilateral PQRS and
find the slopes of the diagonals. State whether the diagonals are perpendicular.
hypotenuse: PR
−−
5. P(5, 1), Q(1, -1), R(-2, 5)
−−−
1
slope of PQ = −
2
−−−
slope of QR = - 2
−−
4
slope of PR = - −
The coordinates of the vertices of right PQR are given. Find the slope of each
side of the triangle. Then name the hypotenuse.
7
−− 1 −− 1 −− 5 −− 5 −−
4 −−
4
AB: −
; LK: −; BC: −; JL: −; AC: - −
; JK: - −
4. Show that the segments named in Exercise 3 are
parallel by finding the slopes of all six segments.
−−
−− −−
−− −−
−−
AB and LK, BC and JL, AC and JK
3. Which segments appear to be parallel?
J(1, 5), K(6, 1), L(-1, 0)
−− −−−
−−
2. J, K, and L are midpoints of AB, BC, and AC,
respectively. Find the coordinates of J, K, and L.
Draw JKL.
1. The coordinates of the vertices of a triangle are
A(-6, 4), B(8, 6), and C(4, -4). Graph ABC.
In coordinate geometry, the slopes of two lines determine if the lines are
parallel or perpendicular. This knowledge can be useful when working with polygons.
DATE
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Slopes of Lines
Spreadsheet Activity
DATE
3
4
1
Sheet 1
2
Sheet 2
2
Sheet 3
1
2
5. (3, -1), (9, -6)
6. (-2, 5), (-7, -2)
Chapter 3
23
Lesson 3-3
5/13/08 11:20:21 PM
Glencoe Geometry
⎯⎯ and UV
⎯⎯ are parallel,
Use a spreadsheet to determine whether PQ
perpendicular, or neither.
7. P(22, 3), Q(3, 1), U(0, 3), and V(5, 5)
8. P(3, 5), Q(1, 22), U(23, 24), and V(21, 3)
4. (3, 5), (-1, 9)
Use a spreadsheet to find the slopes of the lines that contain the given points.
1. (2, 4), (1, 7)
2. (-2, 8), (3, -5)
3. (0, 4), (7, 0)
Exercises
2
1
. Since the product of the slopes is -1, the lines are perpendicular.
the slope is −
-2
4
, the numerator of the slope is 1 and the denominator is 2. So,
or -2. For UV
is −
in row 2 and the ordered pairs for UV
in row 3 as
Step 1 Enter the ordered pairs for PQ
above.
Step 2 With cell E1 selected, click on the bottom right corner of cell E1 and drag it to E3.
This returns the numerators of the slopes. With cell F1 selected, click on the
bottom right corner of cell F1 and drag it to F3. This returns the denominators of
the slopes.
, the numerator of the slope is 4 and the denominator is -2. So, the slope
For PQ
Example 2
⎯⎯ and UV
⎯⎯ are parallel,
Use a spreadsheet to determine whether PQ
perpendicular, or neither for P(-1, 2), Q(-3, 6), U(0, 1), and V(2, 2).
6
The numerator of the slope is -2 and the
2
1
denominator is 6. So,the slope is - −
or - −
.
0
PERIOD
Example 1
Use a spreadsheet to find the slope of a line that contains the
points (-2, 3) and (4, 1).
Step 1 Use the first cell of the spreadsheet for the x value of the first point. Use cell B1
for the y value of the first point. Use cell C1 for the x value of the second point and
use cell D1 for the y value of the second point.
Step 2 In cell E1, enter an equals sign followed by the expression for the numerator of the
slope, which is D1 - B1. Then press ENTER to return the numerator of the slope
of the line.
Step 3 In cell F1, enter an equals sign
followed by the expression for the
A
B
C
D
E
F
denominator of the slope, which is
⫺2
3
4
1
⫺2
6
1
C1 - A1. Then press ENTER to
2
⫺1
2
⫺3
6
4
⫺2
return the denominator of the slope.
3
You can use a spreadsheet to investigate the slope of a line.
3-3
NAME
4/11/08 023_042_GEOCRMC03_890512.indd
10:54:15 AM
23
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 3
Slopes and Polygons
3-3
NAME
Answers (Lesson 3-3)
Chapter 3
DATE
Equations of Lines
Study Guide and Intervention
PERIOD
4
A11
2
y = -3x - 8
6. m: -3, (1,-11)
y = -2
4. m: 0, b: -2
2
1
y = -−
x+4
1
2. m: - −
, b: 4
Glencoe Geometry
Answers
24
y-5=0
12. m = 0, (-2, 5)
4
1
y + 2 = -−
(x + 3)
4
1
10. m = −
, (-3, -2)
y + 2 = -2(x - 4)
8. m = -2, (4, -2)
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
DATE
PERIOD
Equations of Lines
1
hours of service Donna would earn
For 5 −
2
( 2)
2
C = 25h + 125 = 25(5.5) + 125
= 137.5 + 125 or $262.50
Donna would earn more with the first plan.
1
hours of service Donna would earn
For 5 −
= 247.5 + 55 or $302.50
Second Plan
1
+ 55
C = 45h + 55 = 45 5 −
Chapter 3
the third company
3. A third satellite company charges a flat
rate of $69 for all channels, including the
premium channels. If Jerri wants to add
a fourth premium channel, which service
would be least expensive?
Current service: C = 10p + 34.95
Competing service: C = 8p + 39.99
25
1. Write an equation in slope-intercept form
that models the total monthly cost for
each satellite service, where p is the
number of premium channels.
Lesson 3-4
4/11/08 11:14:30 PM
Glencoe Geometry
number of premium channels
represents the rate of change, or
slope, of the equation.
4. Write a description of how the fee for the
number of premium channels is reflected
in the equation. The fee for the
competing service
2. If Jerri wants to include three premium
channels in her package, which service
would be less, her current service or the
competing service?
Jerri’s current satellite television service charges a flat rate of $34.95 per month for the basic
channels and an additional $10 per month for each premium channel. A competing satellite
television service charges a flat rate of $39.99 per month for the basic channels and an
additional $8 per month for each premium channel.
For Exercises 1–4, use the following information.
Exercises
of change, or slope, is 45.
The y-intercept is located
where there are 0 hours,
or $55.
C = mh + b
= 45h + 55
Example
Donna offers computer services to small companies in her city. She
charges $55 per month for maintaining a web site and $45 per hour for each
service call.
b. Donna may change her costs to represent them
a. Write an equation to
by the equation C = 25h + 125, where $125 is the
represent the total
fixed monthly fee for a web site and the cost per
monthly cost, C, for
hour is $25. Compare her new plan to the old one
maintaining a web site
1
and for h hours of
if a company has 5 −
hours of service calls. Under
2
service calls.
which plan would Donna earn more?
For each hour, the cost
First plan
increases $45. So the rate
using linear equations.
Many real-world situations can be modeled
Study Guide and Intervention (continued)
Write Equations to Solve Problems
3-4
NAME
4/11/08 023_042_GEOCRMC03_890512.indd
10:54:39 AM
25
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
023_042_GEOCRMC03_890512.indd 24
Chapter 3
2
5
y + 3 = -−
x
5
11. m = - −
, (0, -3)
2
y - 3 = -(x + 1)
9. m = -1, (-1, 3)
1
y+1=−
(x - 3)
2
2
1
7. m = −
, (3, -1)
See students’ work
Write an equation in point-slope form of the line having the given slope that
contains the given point. Then graph the line.
5
1
y = -−
x+−
3
3
5
1
5. m: - −
, (0 , −
)
3
3
4
1
y=−
x+5
4
1
3. m: −
, b: 5
y = 2x - 3
1. m: 2, b: -3
See students’ work
Write an equation in slope-intercept form of the line having the given slope and
y-intercept or given points. Then graph the line.
Exercises
4
Point-slope form
3
m = -−
, (x1, y1) = (8, 1)
The point-slope form of the equation of the
3
line is y - 1 = - −
(x - 8).
4
3
y - 1 = -−
(x - 8)
y - y1 = m(x - x1)
y = mx + b
Slope-intercept form
y = -2x + 4
m = -2, b = 4
The slope-intercept form of the equation of
the line is y = -2x + 4.
4
Example 2
Write an equation in
point-slope form of the line with slope
3
- − that contains (8, 1).
Example 1
Write an equation in
slope-intercept form of the line with
slope -2 and y-intercept 4.
You can write an equation of a line if you are given any of
the following:
• the slope and the y-intercept,
• the slope and the coordinates of a point on the line, or
• the coordinates of two points on the line.
If m is the slope of a line, b is its y-intercept, and (x1, y1) is a point on the line, then:
• the slope-intercept form of the equation is y = mx + b,
• the point-slope form of the equation is y - y1 = m(x - x1).
Write Equations of Lines
3-4
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Answers (Lesson 3-4)
Equations of Lines
Skills Practice
DATE
PERIOD
5
5
2
y = -−
x-6
2
4. m: - −
, (0, -6)
y = 3x - 8
2. m: 3, b: -8
A12
3
1
y+8=−
(x + 3)
3
1
8. m = −
, (-3, -8)
y + 4 = -3(x - 2)
6. m = -3, (2, -4)
12. u y = −x - 5
1
3
10. s y = -2x + 2
y = 6x - 2
Glencoe Geometry
023_042_GEOCRMC03_890512.indd 26
Chapter 3
y = -5x + 10
19. contains (2, 0) and (0, 10)
y = -x - 6
17. m = -1, contains (0, -6)
5
y = -−
x
3
r
O
26
2
1
y = -−
x-1
t
s
u
x
Glencoe Geometry
20. x-intercept is -2, y-intercept is -1
y = 4x - 3
18. m = 4, contains (2, 5)
3
5
16. m = - −
,b=0
15. m = 6, b = -2
14. the line perpendicular to line s that contains (0, 0) y = −x
1
2
13. the line parallel to line r that contains (1, -1) y = x - 2
11. t y = 3x - 3
9. r y = x + 3
y
Write an equation in slope-intercept form for each line shown or described.
2
1
y - 5 = -−
(x + 2)
2
1
7. m = - −
, (-2, 5)
y - 2 = 2(x - 5)
5. m = 2, (5, 2)
See students’ graphs.
Write equations in point-slope form of the line having the given slope that
contains the given point. Then graph the line.
7
3
y=−
x+1
7
3
3. m: −
, (0, 1)
y = -4x + 3
1. m: -4, b: 3
See students’ graphs.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Equations of Lines
Practice
DATE
PERIOD
(
2
)
7
1
y = -−
x-−
9
2
9
7
1
2. m: - −
, 0, - −
y = 4.5x + 0.25
3. m: 4.5, (0, 0.25)
5
2
5
9. c y = - − x + 4
9
O
y
2
4
1
y = -−
x+1
17. contains (-4, 2) and (8, -1)
5
y=−
x-5
15. x-intercept is 2, y-intercept is -5
y = 3x - 9
13. m = 3, contains (2, -3)
c
Chapter 3
27
x
Lesson 3-4
4/11/08 10:54:55 AM
Glencoe Geometry
18. COMMUNITY EDUCATION A local community center offers self-defense classes for
teens. A $25 enrollment fee covers supplies and materials and open classes cost $10
each. Write an equation to represent the total cost of x self-defense classes at the
community center. C = 10x + 25
y = 4x - 12
16. passes through (2, -4) and (5, 8)
3
1
y=−
x+2
14. x-intercept is -6, y-intercept is 2
9
4
y = -−
x+2
4
12. m = - −
,b=2
5
2
11. perpendicular to line c, contains (-2, -4) y = − x + 1
b
y - 4 = -1.3(x + 4)
10. parallel to line b, contains (3, -2) y = -x + 1
8. b y = -x - 5
5
7. m: -1.3, (-4, 4)
6
y + 2 = -−
(x + 5)
6
5. m: - −
, (-5, -2)
Write an equation in slope-intercept form for each line
shown or described.
y + 3 = 0.5(x - 7),
6. m: 0.5, (7, -3)
3
y-6=−
(x - 4)
2
2
3
4. m: −
, (4, 6)
See students’ work
Write equations in point-slope form of the line having the given slope that
contains the given point. Then graph the line.
2
y=−
x - 10
3
3
2
1. m: −
, b: -10
See students’ work
Write an equation in slope-intercept form of the line having the given slope and
y-intercept or given points. Then graph the line.
3-4
NAME
5/13/08 023_042_GEOCRMC03_890512.indd
11:20:32 PM
27
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 3
Write an equation in slope-intercept form of the line having the given slope and
y-intercept. Then graph the line.
3-4
NAME
Answers (Lesson 3-4)
Chapter 3
Equations of Lines
A13
5
t
Glencoe Geometry
Glencoe Geometry
anything under 5 yd
c. For how many yards would it be less
expensive for Gail to buy the primed
linen?
U = 15Y + 30
b. Let U be the cost of buying Y yards
of unprimed linen and a jar of primer.
Write an equation that relates U
and Y.
P = 21Y
a. Let P be the cost of Y yards of primed
Belgian linen. Write an equation that
relates P and Y.
5. ARTISTRY Gail is an oil painter. She
paints on canvases made from Belgian
linen. Before she paints on the linen, she
has to prime the surface so that it does
not absorb the oil from the paint she
uses. She can buy linen that has already
been primed for $21 per yard, or she can
buy unprimed linen for $15 per yard,
but then she would also have to buy a
jar of primer for $30.
R = -0.7t + 89.9; a fresh coat is
89.9% reflective.
Answers
28
PERIOD
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Enrichment
DATE
PERIOD
O
x
Chapter 3
y
O
x
2
29
rectangle, because consecutive
sides are perpendicular
2
3
x+3
y = -−
3
x
y = -−
3
2
x-2
y=−
3
2
y=−
x+1
Lesson 3-4
4/11/08 10:55:10 AM
Glencoe Geometry
3. Find the equations of the lines that form the sides to the polygon shown below. What
type of polygon is it? Explain your reasoning.
y
2. Graph the lines from Exercise 1 to determine whether your prediction was
correct.
Since there are two pairs of parallel lines, the lines will form a
parallelogram. The lines are not perpendicular, so they will not
form a rectangle.
y = 2x - 3
y = 2x + 1
2
1
y=−
x-2
2
1. The following equations when graphed will contain the sides of a polygon. Without
graphing the lines, make a prediction about what kind of figure the lines will create.
1
y=−
x+3
When equations are graphed on a coordinate grid, their lines can intersect in a
way that the segments determined by their intersection points form the sides of
a polygon.
Polygons on a Coordinate Grid
3-4
NAME
4/11/08 023_042_GEOCRMC03_890512.indd
10:55:05 AM
29
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
023_042_GEOCRMC03_890512.indd 28
Chapter 3
P = 50n - 750; to make $750,
n = 30
3. COST Carla has a business that tests
the air quality in artist’s studios. She
had to purchase $750 worth of testing
equipment to start her business. She
charges $50 to perform the test. Let n be
the number of jobs she gets and let P be
her net profit. Write an equation that
relates P and n. How many jobs does she
need to make $750?
m = 0.8t
Write an equation that relates m and t.
O
5
m
2. DRIVING Ellen is driving to a friend’s
house. The graph shows the distance
(in miles) that Ellen was from home t
minutes after she left her house.
The slope is 1.7 and is the
average number of inches
the tree grew each month.
The y-intercept is 28 and is
the height of the tree when
he began.
DATE
4. PAINT TESTING A paint company
decided to test the durability of its white
paint. They painted a square all white
with their paint and measured the
reflectivity of the square each year.
Seven years after being painted, the
reflectivity was 85%. Ten years after
being painted, the reflectivity dropped
to 82.9%. Assuming that the reflectivity
decreases steadily with time, write an
equation that gives the reflectivity R (as
a percentage) as a function of time t in
years. What is the reflectivity of a fresh
coat of their white paint?
Word Problem Practice
1. GROWTH At the same time each month
over a one year period, John recorded
the height of a tree he had planted. He
then calculated the average growth rate
of the tree. The height h in inches of the
tree during this period was given by the
formula h = 1.7t + 28, where t is the
number of months. What are the slope
and y-intercept of this line and what do
they signify?
3-4
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Answers (Lesson 3-4)
Proving Lines Parallel
Study Guide and Intervention
DATE
1
s
2
n
m
A14
B
C
(6x - 20)°
(3x + 10)°
D
A
Find m∠ABC so that
(6x - 20)°
m
m
7; Alt. Int. Th.
(8x + 8)°
(9x + 1)°
15; Alt. Ext. Th.
(5x - 5)°
Glencoe Geometry
023_042_GEOCRMC03_890512.indd 30
Chapter 3
4.
1.
5.
2.
m
6x°
(4x + 20)°
m
(3x - 20)°
30
20; Alt. Ext. Th.
2x°
10; Alt. Int. Th.
6.
3.
m
m
Glencoe Geometry
10; Corr. Th.
70°
(5x + 20)°
25; Alt. Int. Th.
(3x + 15)°
We can conclude that m n if alternate
interior angles are congruent.
m∠BAD = m∠ABC
3x + 10 = 6x - 20
10 = 3x - 20
30 = 3x
10 = x
m∠ABC = 6x - 20
= 6(10) - 20 or 40
n
m
m n.
Find x so that l m. Identify the postulate or theorem you used.
Exercises
∠1 and ∠2 are corresponding angles of lines
r and s. Since ∠1 ∠2, r s by the
Converse of the Corresponding Angles
Postulate.
r
If m∠1 = m∠2, determine
which lines, if any, are parallel. State
the postulate or theorem that justifies
your answer.
Example 2
the lines are parallel.
Example 1
then
•
•
•
•
•
corresponding angles are congruent,
alternate exterior angles are congruent,
consecutive interior angles are supplementary,
alternate interior angles are congruent, or
two lines are perpendicular to the same line,
If
If two lines in a plane are cut by a transversal and certain
conditions are met, then the lines must be parallel.
PERIOD
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
DATE
PERIOD
Given: ∠1 ∠5, ∠15 ∠5
Prove: m, r s
Proof:
1
3
2
C
B
Chapter 3
6. m
5. ∠1 ∠5
4. r s
31
6. If corr are , then lines .
5. Given
4. If corr. are , then lines .
3. Transitive Property of 2. Vertical are .
2. ∠13 ∠15
3. ∠5 ∠13
1. Given
Reasons
1. ∠15 ∠5
Statements
D
A
2. Transitive Property of 3. If alt. int. angles are , then
the lines are .
1. Given
Reasons
1. Complete the proof.
Exercises
1. ∠1 ∠2
∠1 ∠3
2. ∠2 ∠3
−− −−−
3. AB DC
Proof:
Statements
Given: ∠1 ∠2, ∠1 ∠3
−− −−
Prove: AB DC
Example
r
s
13 14
16 15
9 10
12 11
m
Lesson 3-5
4/11/08 10:55:22 AM
Glencoe Geometry
5 6
8 7
1 2
4 3
You can prove that lines are parallel by using postulates and
theorems about pairs of angles.
Proving Lines Parallel
Study Guide and Intervention (continued)
Prove Lines Parallel
3-5
NAME
4/11/08 023_042_GEOCRMC03_890512.indd
10:55:15 AM
31
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 3
Identify Parallel Lines
3-5
NAME
Answers (Lesson 3-5)
Chapter 3
Proving Lines Parallel
Skills Practice
DATE
A15
(4x)°
(x+6)°
k
130°
(2x + 6)°
m
m
22
19
9.
6.
k
k
(3x + 10)°
m. Show your work.
(5x+19)°
(7x-5)°
m
m
(4x - 10)°
2
20
10.
7.
ℓ m; Consec. Int. Th.
4. m∠5 + m∠12 = 180
a b; Corr. Post.
2. ∠9 ∠11
k
(3x+10)°
2. Angle Addition Postulate
2. m∠ABC = m∠1 + m∠2
Glencoe Geometry
Answers
32
14
Glencoe Geometry
If 2 lines are ⊥ to the same line, then
7. lines are .
6
m
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Proving Lines Parallel
Practice
DATE
2. ∠CBF ∠GFH
⎯ BH
⎯;
AJ
Alt. Ext. Th.
4. ∠ACD ∠KBF
t
m
6.
Proof:
Statements
1. ∠2 and ∠3 are supplementary.
m
B
3
2
D 1
9; alt. int. t
(5x - 15)°
(2x + 12)°
m
G
J
D
6
4 C
5
C
A
Reasons
1. Given
2. If consec. int are suppl., then
lines are .
3. Segments contained in lines
are .
7.
F
H
B
Chapter 3
33
A
Lesson 3-5
4/11/08 10:55:37 AM
Glencoe Geometry
Sample answer: If the gardener digs each row at a 90 angle to the
footpath, each row will be perpendicular to the footpath. If each of the
rows is perpendicular to the footpath, then the rows are parallel.
9. LANDSCAPING The head gardener at a botanical garden wants to plant rosebushes in
parallel rows on either side of an existing footpath. How can the gardener ensure that
the rows are parallel?
−− −−
3. AB CD
⎯
⎯ CD
2. AB
t
(5x + 18)°
21; alt. ext. (7x - 24)°
Write a two-column proof.
∠2 and ∠3 are supplementary.
−− −−−
AB CD
12; corr. (3x + 6)°
(4x - 6)°
8. PROOF
Given:
Prove:
5.
E
K
PERIOD
Find x so that l m. Identify the postulate or theorem you used.
⎯;
⎯ EG
BD
Converse Alt. Int. Th.
3. ∠EFB ∠FBC
⎯;
⎯;
⎯ EG
⎯ EG
BD
BD
Converse Cons. Int. Th. Converse Corr. Th.
1. m∠BCG + m∠FGC = 180
Given the following information, determine which lines,
if any, are parallel. State the postulate or theorem that
justifies your answer.
3-5
NAME
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33
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
023_042_GEOCRMC03_890512.indd 32
Chapter 3
−− −−−
7. BA CD
6. Definition of perpendicular
5. Transitive Property of Equality
5. m∠ABC = 90
(5x+18)°
m
m
11 12
14 13
4. Definition of complementary angles
−− −−−
6. BA ⊥ BC
b
3 4
6 5
4. m∠1 + m∠2 = 90
3. ∠1 and ∠2 are complementary. 3. Given
1. Given
k
(6x + 4)°
(8x - 8)°
Statements
−−− −−−
1. BC ⊥ CD
Reasons
a
1 2
8 7
9 10
16 15
PERIOD
11. PROOF Provide a reason for each statement in the proof of Theorem 3.7.
B 2
C
Given: ∠1 and ∠2 are complementary.
−−− −−−
1
BC ⊥ CD
−− −−−
Prove: BA CD
A
D
Proof:
8.
5. k
Find x so that ℓ m; Alt. Ext. Th.
3. ∠2 ∠16
a b; Alt. Int. Th.
1. ∠3 ∠7
Given the following information, determine which lines,
if any, are parallel. State the postulate or theorem that
justifies your answer.
3-5
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Answers (Lesson 3-5)
A16
Glencoe Geometry
023_042_GEOCRMC03_890512.indd 34
Chapter 3
Parallelogram; The top edges
are perpendicular to the vertical
line so they are a single line. The
bottom edge is also a single line
and perpendicular to the same
line as the top, so it is parallel
to the top. The top edge is
transversal to the left and right
slanted edges and the angles
are supplementary. So, the left
and right edges are parallel.
3. PATTERNS A rectangle is cut along the
slanted, dashed line shown in the figure.
The two pieces are rearranged to form
another figure. Describe as precisely as
you can the shape of the new figure.
Explain.
What more can you say about these two
gray books? They are parallel.
2. BOOKS The two gray books on the
bookshelf each make a 70° angle with
the base of the shelf.
34
Proving Lines Parallel
PERIOD
70˚
2
108˚
Glencoe Geometry
Sample answer: One side
of the “A” is longer than
the other.
b. When building the “A,” Harold makes
sure that angle 1 is correct, but when
he measures angle 2, it is not correct.
What does this imply about the “A”?
a. What should the measures of angles
1 and 2 be so that the horizontal part
of the “A” is truly horizontal? 108
1
5. SIGNS Harold is making a giant letter
“A” to put on the rooftop of the “A is for
Apple” Orchard Store. The figure shows
a sketch of the design.
To pull off this display, what should the
measure of angle 1 be? 80
30˚
1
4. FIREWORKS A fireworks display is
being readied for a celebration. The
designers want to have four fireworks
shoot out along parallel trajectories.
They decide to place two launchers on a
dock and the other two on the roof of a
building.
Word Problem Practice
DATE
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Enrichment
DATE
Chapter 3
−−− −−
7. AD CF
4. ∠1 ∠2
5. ∠1 and ∠2 are corresponding angles for
and CF
lines AD
CF
6. AD
3. ∠5 ∠6
2. ∠2 and ∠6 are complementary
1. ∠1 and ∠5 are complementary
Statements
−−− −−
Prove: AD CF
∠5 ∠6
∠2 and ∠6 are complementary
Given: ∠1 and ∠5 are complementary
35
3
5
2
7
1
4
6
Reasons
1
%
PERIOD
&
6
$
2
'
Lesson 3-5
4/11/08 10:55:48 AM
Glencoe Geometry
converse of Corresponding
Angles Theorem
Given
angles complementary to
congruent angles are
congruent
Given
segments contained in parallel
lines are parallel
definition of corresponding
angles
Given
#
5
"
The reasons necessary to complete the following proof are
scrambled up below. To complete the proof, number the
reasons to match the corresponding statements.
Scrambled-Up Proof
3-5
NAME
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35
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 3
1. RECTANGLES Jim made a frame for
a painting. He wants to check to make
sure that opposite sides are parallel by
measuring the angles at the corners and
seeing if they are right angles. How
many corners must he check in order to
be sure that the opposite sides are
parallel? 3
3-5
NAME
Answers (Lesson 3-5)
Chapter 3
DATE
Perpendiculars and Distance
Study Guide and Intervention
A17
X
S
R
X
Q
SX
T
B
S
Glencoe Geometry
Answers
36
R
6. S to RT
T
P X Q
4. S to PQ
X
D
2. D to AB
T
R
A
X
S
B
C
A
A
Q
F
F
G
E
E
Glencoe Geometry
B
B
distance between
M and
PQ
M
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
DATE
PERIOD
m
x
O
2
= √5
The distance between and m is √
5 units.
(2 - 0)2 + (0 - 1)2
= √
The point of intersection of p and m is (2, 0).
Use the Distance Formula to find the
distance between (0, 1) and (2, 0).
d = √
(x2 - x1)2 + (y2 -y1)2
2
1
= -−
(2) + 1 = -1 + 1 = 0
2
Substitute 2 for x to find the y-coordinate.
1
y=-−
x+1
Use substitution.
1
x+1
2x - 4 = - −
2
4x - 8 = -x + 2
5x = 10
x=2
To find the point of intersection of p and m,
solve a system of equations.
Line m: y = 2x - 4
1
Line p: y = - −
x+1
2 √
2
11
Chapter 3
2. y = x + 3
y=x-1
1. y = 8
y = -3
37
√
5
Lesson 3-6
4/11/08 10:56:03 AM
Glencoe Geometry
3. y = -2x
y = -2x - 5
Find the distance between each pair of parallel lines with the given equations.
Exercises
intersection for p and is (0, 1).
2
1
x + 1. The point of
equation of p is y = - −
2
x
1
Line p has slope - −
and y-intercept 1. An
(0, 1)
Draw a line p through (0, 1) that is
perpendicular to and m.
y p
m
O
y
Example
Find the distance between the parallel lines l and m with the
equations y = 2x + 1 and y = 2x - 4, respectively.
The distance between parallel lines is the length
of a segment that has an endpoint on each line and is perpendicular to them. Parallel lines
are everywhere equidistant, which means that all such perpendicular segments have the
same length.
Perpendiculars and Distance
Study Guide and Intervention (continued)
Distance Between Parallel Lines
3-6
NAME
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10:55:55 AM
37
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
023_042_GEOCRMC03_890512.indd 36
Chapter 3
T
P
5. S to QR
U
R
3. T to RS
A
C
1. C to AB
Construct the segment that represents the distance indicated.
Exercises
P
PERIOD
Construct the segment that represents the distance
⊥ AF
.
. Draw EG
Extend AF
−−−
.
EG represents the distance from E to AF
⎯.
from E to AF
Example
When a point is
not on a line, the distance from the point to the line is the
length of the segment that contains the point and
is perpendicular to the line.
Distance From a Point to a Line
3-6
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Answers (Lesson 3-6)
Perpendiculars and Distance
Skills Practice
DATE
C
D
E
2. G to EF
G
F
R
A18
Q
Glencoe Geometry
023_042_GEOCRMC03_890512.indd 38
Chapter 3
√
26
11. y = -5x
y = -5x + 26
8
8. y = 7
y = -1
3 √
2
12. y = x + 9
y=x+3
11
9. x = -6
x=5
38
2 √
5
Glencoe Geometry
13. y = -2x + 5
y = -2x - 5
√
10
10. y = 3x
y = 3x + 10
Find the distance between each pair of parallel lines with the given equations.
3 √
58
7. Line ℓ contains points (−7, 8) and (0, 5). Point P has coordinates (−5, 32).
2 √
10
6. Line ℓ contains points (−4, −2) and (2, 0). Point P has coordinates (3, 7).
2 √
2
5. Line ℓ contains points (2, 4) and (5, 1). Point P has coordinates (1, 1).
5
4. Line ℓ contains points (0, −2) and (6, 6). Point P has coordinates (−1, 5).
S
P
3. Q to SR
COORDINATE GEOMETRY Find the distance from P to ℓ.
A
B
1. B to AC
PERIOD
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Perpendiculars and Distance
Practice
DATE
N
O
D
A
2. A to DC
B
C
W
T
V
2 √
5
9. y = 2x + 7
y = 2x - 3
(–2, –3)
y = -x + 1
3 √
10
10. y = 3x + 12
y = 3x - 18
O
y
Chapter 3
39
x
Lesson 3-6
4/11/08 10:56:22 AM
Glencoe Geometry
Sample answer: The shortest path would be a perpendicular segment
from where they are to the bank of the canal.
12. CANOEING Bronson and a friend are going to carry a canoe across a flat field to the
bank of a straight canal. Describe the shortest path they can use.
11. Graph the line y = -x + 1. Construct a perpendicular
segment through the point at (-2, -3). Then find the
distance from the point to the line. 3 √
2
2 √
2
8. y = -x
y = -x - 4
Find the distance between each pair of parallel lines with the given equation.
√
178
7. Line l contains points (−2, 4) and (1, −9). Point P has coordinates (14, −6).
2 √
5
6. Line l contains points (5, 18) and (9, 10). Point P has coordinates (−4, 26).
3 √
2
5. Line l contains points (3, 5) and (7, 9). Point P has coordinates (2, 10).
5
4. Line l contains points (−2, 0) and (4, 8). Point P has coordinates (5, 1).
S
U
PERIOD
3. T to VU
COORDINATE GEOMETRY Find the distance from P to l.
M
1. O to MN
Construct the segment that represents the distance indicated.
3-6
NAME
5/26/08 023_042_GEOCRMC03_890512.indd
11:18:07 AM
39
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 3
Construct the segment that represents the distance indicated.
3-6
NAME
Answers (Lesson 3-6)