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

4/11/08 001_022_GEOCRMC03_890512.indd 10:52:47 AM 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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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

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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

4/11/08 001_022_GEOCRMC03_890512.indd 10:53:26 AM 17

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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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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

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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

5/26/08023_042_GEOCRMC03_890512.indd 3:00:07 PM 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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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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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)