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Chapter 3Resource Masters

Geometry

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

33

© Glencoe/McGraw-Hill vii Glencoe Geometry

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 3.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to yourGeometry Study Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

alternate exterior angles

alternate interior angles

consecutive interior angles

corresponding angles

equidistant

ee·kwuh·DIS·tuhnt

non-Euclidean geometry

yoo·KLID·ee·yuhn

parallel lines

parallel planes

(continued on the next page)

⎧ ⎪ ⎨ ⎪ ⎩

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

© Glencoe/McGraw-Hill viii Glencoe Geometry

Vocabulary Term Found on Page Definition/Description/Example

plane Euclidean geometry

point-slope form

rate of change

skew lines

slope

slope-intercept form

spherical geometry

SFIR·uh·kuhl

transversal

⎧ ⎪ ⎨ ⎪ ⎩

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

33

Learning to Read MathematicsProof Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

33

© Glencoe/McGraw-Hill ix Glencoe Geometry

Proo

f Bu

ilderThis is a list of key theorems and postulates you will learn in Chapter 3. As you

study the chapter, write each theorem or postulate in your own words. Includeillustrations as appropriate. Remember to include the page number where youfound the theorem or postulate. Add this page to your Geometry Study Notebookso you can review the theorems and postulates at the end of the chapter.

Theorem or Postulate Found on Page Description/Illustration/Abbreviation

Theorem 3.1Alternate Interior Angles Theorem

Theorem 3.2Consecutive Interior Angles Theorem

Theorem 3.3Alternate Exterior Angles Theorem

Theorem 3.4Perpendicular Transversal Theorem

Theorem 3.5

Theorem 3.6

Theorem 3.7

(continued on the next page)

© Glencoe/McGraw-Hill x Glencoe Geometry

Theorem or Postulate Found on Page Description/Illustration/Abbreviation

Theorem 3.8

Theorem 3.9

Postulate 3.1Corresponding Angles Postulate

Postulate 3.2

Postulate 3.3

Postulate 3.4

Postulate 3.5Parallel Postulate

Learning to Read MathematicsProof Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

33

Study Guide and InterventionParallel Lines and Transversals

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1

© Glencoe/McGraw-Hill 125 Glencoe Geometry

Less

on

3-1

Relationships Between Lines and Planes When two lines lie in the same plane and do not intersect, they are parallel.Lines that do not intersect and are not coplanar are skew lines.In the figure, ! is parallel to m, or ! || m. You can also write PQ!"# || RS!"#. Similarly, if two planes do not intersect, they are parallel planes.

a. Name all planes that are parallel to plane ABD.plane EFH

b. Name all segments that are parallel to C!G!.B!F!, D!H!, and A!E!

c. Name all segments that are skew to E!H!.B!F!, C!G!, B!D!, C!D!, and A!B!

For Exercises 1–3, refer to the figure at the right.

1. Name all planes that intersect plane OPT.

2. Name all segments that are parallel to N!U!.

3. Name all segments that intersect M!P!.


4. Name all segments parallel to Q!X!.

5. Name all planes that intersect plane MHE.

6. Name all segments parallel to Q!R!.

7. Name all segments skew to A!G!.

T

SG

A

X

EN

MH

O

Q

R

M

N O

P

R

U T

S

A

B C

D

E

F G

H

P Q

R S

n

m

!

ExampleExample

ExercisesExercises


Angle Relationships A line that intersects two or more other lines in a plane is calleda transversal. In the figure below, t is a transversal. Two lines and a transversal form eightangles. Some pairs of the angles have special names. The following chart lists the pairs ofangles and their names.

Angle Pairs Name

!3, !4, !5, and !6 interior angles

!3 and !5; !4 and !6 alternate interior angles

!3 and !6; !4 and !5 consecutive interior angles

!1, !2, !7, and !8 exterior angles

!1 and !7; !2 and !8 alternate exterior angles

!1 and !5; !2 and !6; !3 and !7; !4 and !8

corresponding angles

Identify each pair of angles as alternate interior, alternate exterior, corresponding, or consecutiveinterior angles.

a. !10 and !16 b. !4 and !12alternate exterior angles corresponding angles

c. !12 and !13 d. !3 and !9consecutive interior angles alternate interior angles

Use the figure in the Example for Exercises 1–12.

Name the transversal that forms each pair of angles.

1. !9 and !13 2. !5 and !14 3. !4 and !6

Identify each pair of angles as alternate interior, alternate exterior,corresponding, or consecutive interior angles.

4. !1 and !5 5. !6 and !14 6. !2 and !8

7. !3 and !11 8. !12 and !3 9. !4 and !6

10. !6 and !16 11. !11 and !14 12. !10 and !16

!

p q

n1 2 9 10111234

141365151678

t

n

m1 2

34

5 678

Study Guide and Intervention (continued)

Parallel Lines and Transversals

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1

ExampleExample

ExercisesExercises

Skills PracticeParallel Lines and Transversals

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1


Less

on

3-1


1. Name all planes that are parallel to plane DEH.

2. Name all segments that are parallel to A!B!.

3. Name all segments that intersect G!H!.

4. Name all segments that are skew to C!D!.

Identify the sets of lines to which each given line is a transversal.

5. r

6. s

7. w

Identify each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles.

8. !2 and !8 9. !3 and !6

10. !1 and !9 11. !3 and !9

12. !6 and !12 13. !7 and !11

Name the transversal that forms each pair of angles. Then identify the special name for the angle pair.

14. !4 and !10 15. !2 and !12

16. !7 and !3 17. !13 and !10

18. !8 and !14 19. !6 and !14

h

g

j

k

12

910

1112

13 1514

1634

65 7

8

a

b

cd

1 2

9 101112

3465

78

rs

t

w

z

A

E F

B

D

H G

C



1. Name all planes that intersect plane STX.

2. Name all segments that intersect Q!U!.

3. Name all segments that are parallel to X!Y!.

4. Name all segments that are skew to V!W!.

Identify the sets of lines to which each given line is a transversal.

5. e

6. h

Identify each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles.

7. !9 and !13 8. !6 and !16

9. !3 and !10 10. !8 and !14

Name the transversal that forms each pair of angles. Then identify the special name for the angle pair.

11. !2 and !12 12. !6 and !18

13. !13 and !19 14. !11 and !7

FURNITURE For Exercises 15–16, refer to the drawing of the end table.

15. Find an example of parallel planes.

16. Find an example of parallel lines.

cdb

a1

9 101112

13 141516

1719

1820

56

78

234

m

!

pn

1

9 1011

1213

1415

16

5 678

234

g

he

i

f

Q R

ST

U

V W

X

Y

Z

Practice Parallel Lines and Transversals

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1

Reading to Learn MathematicsParallel Lines and Transversals

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1


Less

on

3-1

Pre-Activity How are parallel lines and planes used in architecture?

Read the introduction to Lesson 3-1 at the top of page 126 in your textbook.

• Give an example of parallel lines that can be found in your classroom.

• Give an example of parallel planes that can be found in your classroom.

Reading the Lesson

1. Write a geometrical term that matches each definition.

a. two planes that do not intersect

b. lines that are not coplanar and do not intersect

c. two coplanar lines that do not intersect

d. a line that intersects two or more lines in a plane at different points

e. a pair of angles determined by two lines and a transversal consisting of an interiorangle and an exterior angle that have different vertices and that lie on the same sideof the transversal

2. Refer to the figure at the right. Give the special name for each angle pair.

a. !3 and !5

b. !6 and !12

c. !4 and !8

d. !2 and !3

e. !8 and !12

f. !5 and !9

g. !4 and !10

h. !6 and !7

Helping You Remember

3. A good way to remember new mathematical terms is to relate them to words that youuse in everyday life. Many words start with the prefix trans-, which is a Latin rootmeaning across. List four English words that start with trans-. How can the meaning ofthis prefix help you remember the meaning of transversal?

!m

np

12

910

1112

34

65

78


Perspective DrawingsTo draw three-dimensional objects, artists makeperspective drawingssuch as the ones shown.To indicate depth in aperspective drawing, someparallel lines are drawn as converging lines. Thedotted lines in the figuresbelow each extend to avanishing point, or spotwhere parallel lines appear to meet.

Draw lines to locate the vanishing point in each drawing of a box.

1. 2. 3.

4. The fronts of two cubes are shown below. Using point P as the vanishing point for both cubes, complete the perspective drawings of the cubes.

5. Find an example of a perspective drawing in a newspaper or magazine.Trace the drawing and locate a vanishing point.

P

Vanishing points

Railroad tracks Cube Cabinet

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1

Study Guide and InterventionAngles and Parallel Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2


Less

on

3-2

Parallel Lines and Angle Pairs When two parallel lines are cut by a transversal,the following pairs of angles are congruent.

• corresponding angles• alternate interior angles• alternate exterior angles

Also, consecutive interior angles are supplementary.

In the figure, m!2 ! 75. Find the measures of the remaining angles.m!1 ! 105 !1 and !2 form a linear pair.m!3 ! 105 !3 and !2 form a linear pair.m!4 ! 75 !4 and !2 are vertical angles.m!5 ! 105 !5 and !3 are alternate interior angles.m!6 ! 75 !6 and !2 are corresponding angles.m!7 ! 105 !7 and !3 are corresponding angles.m!8 ! 75 !8 and !6 are vertical angles.

In the figure, m!3 ! 102. Find the measure of each angle.

1. !5 2. !6

3. !11 4. !7

5. !15 6. !14

In the figure, m!9 ! 80 and m!5 ! 68. Find the measure of each angle.

7. !12 8. !1

9. !4 10. !3

11. !7 12. !16

w v

p

q

1 234

6578

9 101112

1413

1516

p q

m

n

1 234

6578

9 101112

14131516

p

m

n

1 234

6578

ExampleExample

ExercisesExercises


Algebra and Angle Measures Algebra can be used to find unknown values in angles formed by a transversal and parallel lines.

If m!1 ! 3x " 15, m!2 ! 4x # 5, m!3 ! 5y,and m!4 ! 6z " 3, find x and y.

p || q, so m!1 ! m!2 because they are corresponding angles.

3x " 15 ! 4x # 53x " 15 # 3x ! 4x # 5 # 3x

15 ! x # 515 " 5 ! x # 5 " 5

20 ! x

p q

r

s

1 2

34


Angles and Parallel Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2

ExampleExample

r || s, so m!2 ! m!3 because they are corresponding angles.

m!2 ! m!375 ! 5y

$755$ ! $

55y$

15 ! y

ExercisesExercises

Find x and y in each figure.

1. 2.

3. 4.

Find x, y, and z in each figure.

5. 6.

2x$

2y$

90$ x$

z$

2y$106$

x$

(4z " 6)$

(5x # 20)$

3x$2y $

4y $

(11x " 4)$

(13y # 5)$

(5y " 5)$

5x$

(15x " 30)$

(3y " 18)$ 10x$

90$

(5x # 5)$(6y # 4)$

(4x " 10)$

Skills PracticeAngles and Parallel Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2


Less

on

3-2


1. !3 2. !5

3. !8 4. !1

5. !4 6. !6


7. !9 8. !6

9. !8 10. !2

11. !5 12. !11


13. !2 14. !5

15. !7 16. !15

17. !14 18. !9


19. 20.

Find m!1 in each figure.

21. 22.140$

32$

160$

1

20$

7x$

(8x # 10)$

(6y " 20)$

5x$

40$(3y # 1)$

x

w

zy

1 2

43

65

8

9 1011 12

13 1415 16

7

sm

ut

1 2

4 3

658

91011

127

q

r

s

1 243

6587



1. !10 2. !8

3. !9 4. !5

5. !11 6. !13


7. 8.

Find m!1 in each figure.

9. 10.

11. PROOF Write a paragraph proof of Theorem 3.3.Given: ! || m, m || nProve: !1 " !12

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 y. 50$

y$

m

n

1 23 4

657 8

!

k

9 1011 12

144$

62$

1

100$

50$

1

3y$ (2x " 13)$(5y # 4)$3x$

(9x " 12)$

(4y # 10)$

n

m

rs

12

43

65

8

910

1112

1314

1516

7

Practice Angles and Parallel Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2

Reading to Learn MathematicsAngles and Parallel Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2


Less

on

3-2

Pre-Activity How can angles and lines be used in art?

Read the introduction to Lesson 3-2 at the top of page 133 in your textbook.• Your textbook shows a painting that contains two parallel lines and a

transversal. What is the name for !1 and !2?• What is the relationship between these two angles?

Reading the Lesson1. Choose the correct word to complete each sentence.

a. If two parallel lines are cut by a transversal, then alternate exterior angles are

(congruent/complementary/supplementary).

b. If two parallel lines are cut by a transversal, then corresponding angles are


c. If parallel lines are cut by a transversal, then consecutive interior angles are


d. In a plane, if a line is perpendicular to one of two parallel lines, then it is

(parallel/perpendicular/skew) to the other.

Use the figure for Exercises 2 and 3.

2. a. Name four pairs of vertical angles.

b. Name all angles that form a linear pair with !7.

c. Name all angles that are congruent to !1.

d. Name all angles that are congruent to !4.

e. Name all angles that are supplementary to !3.

f. Name all angles that are supplementary to !2.

3. Which conclusion(s) could you make about lines u and v if m!4 ! m!1?

A. t || u B. t ⊥ u C. v ⊥ u D. v ⊥ t E. v || t

Helping You Remember4. How can you use an everyday meaning of the adjective alternate to help you remember

the types of angle pairs for two lines and a transversal?

t

u

v

12

34

65

78


More Optical IllusionsIn drawings, diagonal lines may create the illusion of depth.For example, the figure at the right can be thought of as picturing a flat figure or a cube. The optical illusions on this page involve depth perception.

Answer each question.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2

1. How many cubes do you see in thedrawing?

3. Does the drawing show a view from the top or the bottom of the stairs?

2. Can this figure show an actual object?

4. Which line segment is longer, A!B! orC!D!? Measure to check your answer.

A

B C

D

5. Which person in the drawing at the right appears to be tallest? Measure to check your answer.

6. Draw two more objects the same size on the figure at the right. Does one appear larger than the other?

Study Guide and InterventionSlopes of Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-33-3


Less

on

3-3

Slope of a Line The slope m of a line containing two points with coordinates (x1, y1)

and (x2, y2) is given by the formula m ! $yx

2

2

#

#

yx

1

1$, where x1 % x2.

Find the slope of each line.For line p, let (x1, y1) be (1, 2) and (x2, y2) be (#2, #2).

m ! $yx

2

2

#

#

yx

1

1$

! $##

22

##

21$ or $

43$

For line q, let (x1, y1) be (2, 0) and (x2, y2) be (#3, 2).

m ! $yx

2

2

#

#

yx

1

1$

! $#23##

02$ or #$

25$

Determine the slope of the line that contains the given points.

1. J(0, 0), K(#2, 8) 2. R(#2, #3), S(3, #5)

3. L(1, #2), N(#6, 3) 4. P(#1, 2), Q(#9, 6)

5. T(1, #2), U(6, #2) 6. V(#2, 10), W(#4, #3)

Find the slope of each line.

7. AB!"# 8. CD!"#

9. EM!"# 10. AE!"#

11. EH!"# 12. BM!"#

x

y

O

C(–2, 2)

A(–2, –2)

H(–1, –4)

B(0, 4)

M(4, 2)

E(4, –2)D(0, –2)

x

y

O

(1, 2)

(–2, –2)

(–3, 2)

(2, 0)

ExampleExample

ExercisesExercises


Parallel and Perpendicular Lines If you examine the slopes of pairs of parallel linesand the slopes of pairs of perpendicular lines, where neither line in each pair is vertical, youwill discover the following properties.

Two lines have the same slope if and only if they are parallel.Two lines are perpendicular if and only if the product of their slopes is #1.


Slopes of Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-33-3

Find the slope of a lineparallel to the line containing A(#3, 4)and B(2, 5).Find the slope of AB!"#. Use (#3, 4) for (x1, y1)and use (2, 5) for (x2, y2).

m ! $yx

2

2

#

#

yx

1

1$

! $2

5#

#(#

43)

$ or $15$

The slope of any line parallel to AB!"# must be $

15$.

Find the slope of a lineperpendicular to PQ!"# for P(#2, #4) andQ(4, 3).Find the slope of PQ!"#. Use (#2, #4) for (x1, y1) and use (4, 3) for (x2, y2).

m ! $yx

2

2

#

#

yx

1

1$

! $34

#

#

((#

#

42))

$ or $76$

Since $76$ & ##$

67$$ ! #1, the slope of any line

perpendicular to PQ!"# must be #$67$.

Example 1Example 1 Example 2Example 2

ExercisesExercises

Determine whether MN!"# and RS!"# are parallel, perpendicular, or neither.

1. M(0, 3), N(2, 4), R(2, 1), S(8, 4) 2. M(#1, 3), N(0, 5), R(2, 1), S(6, #1)

3. M(#1, 3), N(4, 4), R(3, 1), S(#2, 2) 4. M(0, #3), N(#2, #7), R(2, 1), S(0, #3)

5. M(#2, 2), N(1, #3), R(#2, 1), S(3, 4) 6. M(0, 0), N(2, 4), R(2, 1), S(8, 4)

Find the slope of MN!"# and the slope of any line perpendicular to MN!"#.

7. M(2, #4), N(#2, #1) 8. M(1, 3), N(#1, 5)

9. M(4, #2), N(5, 3) 10. M(2, #3), N(#4, 1)

Skills PracticeSlopes of Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-33-3


Less

on

3-3


1. S(#1, 2), W(0, 4) 2. G(#2, 5), H(1, #7)

3. C(0, 1), D(3, 3) 4. J(#5, #2), K(5, #4)


5. NP!"# 6. TW!"#

7. a line parallel to TW!"# 8. a line perpendicular to NP!"#

Determine whether AB!"# and MN!"# are parallel, perpendicular, or neither.

9. A(0, 3), B(5, #7), M(#6, 7), N(#2, #1) 10. A(#1, 4), B(2, #5), M(#3, 2), N(3, 0)

11. A(#2, #7), B(4, 2), M(#2, 0), N(2, 6) 12. A(#4, #8), B(4, #6), M(#3, 5), N(#1, #3)

Graph the line that satisfies each condition.

13. slope ! 3, contains A(0, 1) 14. slope ! #$32$, contains R(#4, 5)

15. contains Y(3, 0), parallel to DJ!"# 16. contains T(0, #2), perpendicular to CX!"#

with D(#3, 1) and J(3, 3) with C(0, 3) and X(2, #1)

T(0, –2)

C(0, 3)

X(2, –1)x

y

OD(–3, 1)

J(3, 3)

Y(3, 0)x

y

O

R(–4, 5)

x

y

O

A(0, 1)x

y

O

x

y

ON T

W

P



1. B(#4, 4), R(0, 2) 2. I(#2, #9), P(2, 4)


3. LM!"# 4. GR!"#

5. a line parallel to GR!"# 6. a line perpendicular to PS!"#

Determine whether KM!"# and ST!"# are parallel, perpendicular, or neither.

7. K(#1, #8), M(1, 6), S(#2, #6), T(2, 10) 8. K(#5, #2), M(5, 4), S(#3, 6), T(3, #4)

9. K(#4, 10), M(2, #8), S(1, 2), T(4, #7) 10. K(#3, #7), M(3, #3), S(0, 4), T(6, #5)

Graph the line that satisfies each condition.

11. slope ! #$12$, contains U(2, #2) 12. slope ! $

43$, contains P(#3, #3)

13. contains B(#4, 2), parallel to FG!"# 14. contains Z(#3, 0), perpendicular to EK!"#

with F(0, #3) and G(4, #2) with E(#2, 4) and K(2, #2)

15. PROFITS After Take Two began renting DVDs at their video store, business soared.Between 2000 and 2003, profits increased at an average rate of $12,000 per year. Totalprofits in 2003 were $46,000. If profits continue to increase at the same rate, what willthe total profit be in 2009?

K(2, –2)

E(–2, 4)

Z(–3, 0)

x

y

O

F(0, –3)

B(–4, 2)

G(4, –2)x

y

O

P(–3, –3)

x

y

OU(2, –2)

x

y

O

x

y

O

R

G

S

P

M

L

Practice Slopes of Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-33-3

Reading to Learn MathematicsSlopes of Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-33-3


Less

on

3-3

Pre-Activity How is slope used in transportation?

Read the introduction to Lesson 3-3 at the top of page 139 in your textbook.• If you are driving uphill on a road with a 4% grade, how many feet will

the road rise for every 1000 horizontal feet traveled? • If you are driving downhill on a road with a 7% grade, how many meters

will the road fall for every 500 meters traveled?

Reading the Lesson1. Which expressions can be used to represent the slope of the line containing points (x1, y1)

and (x2, y2)? Assume that no denominator is zero.

A. $''

yx$ B. $hv

oreirztoicnatlarlirsuen

$ C. $yx

2

2

#

#

yx

1

1$ D. $

cchh

aann

ggee

iinn

xy$

E. $yx

2

1

#

#

yx

1

2$ F. $

yx

1

1

#

#

yx

2

2$ G. $

xy

2

2

#

#

xy

1

1$ H. $

yy

2

1

#

#

xx

2

1$

2. Match the description of a line from the first column with the description of its slopefrom the second column.

Type of Line Slope

a. a horizontal line i. a negative number

b. a line that rises from left to right ii. 0

c. a vertical line iii. undefined

d. a line that falls from left to right iv. a positive number

3. Find the slope of each line.

a. a line parallel to a line with slope $34$

b. a line perpendicular to the x-axis

c. a line perpendicular to a line with slope 5

d. a line parallel to the x-axis

e. y-axis


4. A good way to remember something is to explain it to someone else. Suppose your friendthinks that perpendicular lines (if neither line is vertical) have slopes that arereciprocals of each other. How could you explain to your friend that this is incorrect andgive her a good way to remember the correct relationship?


Slopes and PolygonsIn coordinate geometry, the slopes of two lines determine if the lines areparallel or perpendicular. This knowledge can be useful when working withpolygons.

1. The coordinates of the vertices of a triangle are A(#6, 4), B(8, 6), and C(4, #4). Graph "ABC.

2. J, K, and L are midpoints of A!B!, B!C!, and A!C!,respectively. Find the coordinates of J, K, and L.Draw "JKL.

3. Which segments appear to be parallel?

4. Show that the segments named in Exercise 3 are parallel by finding the slopes of all six segments.

The coordinates of the vertices of right "PQR are given. Find the slope of eachside of the triangle. Then name the hypotenuse.

5. P(5, 1) Q(1, #1) R(#2, 5) 6. P(#2, #3) Q(5, 1) R(2, 3)

slope of P!Q! ! slope of P!Q! !

slope of Q!R! ! slope of Q!R! !

slope of P!R! ! slope of P!R! !

hypotenuse: hypotenuse:

The coordinates of quadrilateral PQRS are given. Graph quadrilateral PQRS andfind the slopes of the diagonals. State whether the diagonals are perpendicular.

7. P(#2, 6), Q(4, 0), R(1, #4), S(#5, 2) 8. P(0, 6), Q(3, 0), R(#4, #2), S(#5, 4)

S

P

Q

R

x

y

O

S

P

Q

R

x

y

O

A

B

C

K

J

L x

y

O

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

3-33-3

Study Guide and InterventionEquations of Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-43-4


Less

on

3-4

Write Equations of Lines You can write an equation of a line if you are given any ofthe 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 an equation inslope-intercept form of the line withslope #2 and y-intercept 4.y ! mx " b Slope-intercept form

y ! #2x " 4 m ! #2, b ! 4

The slope-intercept form of the equation ofthe line is y ! #2x " 4.

Write an equation inpoint-slope form of the line with slope #%

34% that contains (8, 1).

y # y1 ! m(x # x1) Point-slope form

y # 1 ! #$34$(x # 8) m ! #$

34$, (x1, y1) ! (8, 1)

The point-slope form of the equation of theline is y # 1 ! #$

34$(x # 8).


ExercisesExercises

Write an equation in slope-intercept form of the line having the given slope and y-intercept.

1. m: 2, y-intercept: #3 2. m: #$12$, y-intercept: 4

3. m: $14$, y-intercept: 5 4. m: 0, y-intercept: #2

5. m: #$53$, y-intercept: $

13$ 6. m: #3, y-intercept: #8

Write an equation in point-slope form of the line having the given slope thatcontains the given point.

7. m ! $12$, (3, #1) 8. m ! #2, (4, #2)

9. m ! #1, (#1, 3) 10. m ! $14$, (#3, #2)

11. m ! #$52$, (0, #3) 12. m ! 0, (#2, 5)


Write Equations to Solve Problems Many real-world situations can be modeledusing linear equations.

Donna offers computer services to small companies in her city. Shecharges $55 per month for maintaining a web site and $45 per hour for eachservice call.


Equations of Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-43-4

ExampleExample

a. Write an equation torepresent the totalmonthly cost C formaintaining a web siteand for h hours ofservice calls.

For each hour, the costincreases $45. So the rate ofchange, or slope, is 45. They-intercept is located wherethere are 0 hours, or $55.C ! mh " b

! 45h " 55

b. Donna may change her costs to represent themby the equation C ! 25h " 125, where $125 is thefixed monthly fee for a web site and the cost perhour is $25. Compare her new plan to the old one if a company has 5 %

12% hours of service calls. Under

which plan would Donna earn more?

First plan

For 5$12$ hours of service Donna would earn

C ! 45h " 55 ! 45#5$12$$ " 55

! 247.5 " 55 or $302.50

Second Plan

For 5$12$ hours of service Donna would earn

C ! 25h " 125 ! 25(5.5) " 125

! 137.5 " 125 or $262.50Donna would earn more with the first plan.

ExercisesExercises

For Exercises 1–4, use the following information.Jerri’s current satellite television service charges a flat rate of $34.95 per month for the basicchannels and an additional $10 per month for each premium channel. A competing satellitetelevision service charges a flat rate of $39.99 per month for the basic channels and anadditional $8 per month for each premium channel.

1. Write an equation in slope-intercept formthat models the total monthly cost foreach satellite service, where p is thenumber of premium channels.

3. A third satellite company charges a flatrate of $69 for all channels, including thepremium channels. If Jerri wants to adda fourth premium channel, which servicewould be least expensive?

2. If Jerri wants to include three premiumchannels in her package, which servicewould be less, her current service or thecompeting service?

4. Write a description of how the fee for thenumber of premium channels is reflectedin the equation.

Skills PracticeEquations of Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-43-4


Less

on

3-4


1. m: #4, y-intercept: 3 2. m: 3, y-intercept: #8

3. m: $37$, (0, 1) 4. m: #$

25$, (0, #6)

Write equations in point-slope form and slope-intercept form of the line havingthe given slope and containing the given point.

5. m: 2, (5, 2) 6. m: #3, (2, #4)

7. m: #$12$, (#2, 5) 8. m: $

13$, (#3, #8)

Write an equation in slope-intercept form for each line.

9. r 10. s

11. t 12. u

13. the line parallel to line r that contains (1, #1)

14. the line perpendicular to line s that contains (0, 0)

Write an equation in slope-intercept form for the line that satisfies the givenconditions.

15. m ! 6, y-intercept ! #2 16. m ! #$53$, y-intercept ! 0

17. m ! #1, contains (0, #6) 18. m ! 4, contains (2, 5)

19. contains (2, 0 ) and (0, 10) 20. x-intercept is #2, y-intercept is #1

x

y

O

r

s

t

u



1. m: $23$, y-intercept: #10 2. m: #$

79$, #0, #$

12$$ 3. m: 4.5, (0, 0.25)

Write equations in point-slope form and slope-intercept form of the line havingthe given slope and containing the given point.

4. m: $32$, (4, 6) 5. m: #$

65$, (#5, #2)

6. m: 0.5, (7, #3) 7. m: #1.3, (#4, 4)

Write an equation in slope-intercept form for each line.

8. b 9. c

10. parallel to line b, contains (3, #2)

11. perpendicular to line c, contains (#2, #4)

Write an equation in slope-intercept form for the line that satisfies the givenconditions.

12. m ! #$49$, y-intercept ! 2 13. m ! 3, contains (2, #3)

14. x-intercept is #6, y-intercept is 2 15. x-intercept is 2, y-intercept is #5

16. passes through (2, #4) and (5, 8) 17. contains (#4, 2) and (8, #1)

18. COMMUNITY EDUCATION A local community center offers self-defense classes forteens. A $25 enrollment fee covers supplies and materials and open classes cost $10each. Write an equation to represent the total cost of x self-defense classes at thecommunity center.

x

y

O

c

b

Practice Equations of Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-43-4

Reading to Learn MathematicsEquations of Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-43-4


Less

on

3-4

Pre-Activity How can the equation of a line describe the cost of cellulartelephone service?

Read the introduction to Lesson 3-4 at the top of page 145 in your textbook.If the rates for your cellular phone plan are described by the equation inyour textbook, what will be the total charge (excluding taxes and fees) for amonth in which you use 50 minutes of air time?

Reading the Lesson1. Identify what each formula represents.

a. y # y1 ! m(x # x1)

b. m ! $yx

2

2

#

#

yx

1

1$

c. y ! mx " b

2. Write the point-slope form of the equation for each line.

a. line with slope #$12$ containing (#2, 5)

b. line containing (#4.5, #6.5) and parallel to a line with slope 0.5

3. Which one of the following correctly describes the y-intercept of a line?

A. the y-coordinate of the point where the line intersects the x-axis

B. the x-coordinate of the point where the line intersects the y-axis

C. the y-coordinate of the point where the line crosses the y-axis

D. the x-coordinate of the point where the line crosses the x-axis

E. the ratio of the change in y-coordinates to the change in x-coordinates

4. Find the slope and y-intercept of each line.

a. y ! 2x # 7

b. x " y ! 8.5

c. 2.4x # y ! 4.8

d. y # 7 ! x " 12

e. y " 5 ! #2(x " 6)

Helping You Remember5. A good way to remember something new is to relate it to something you already know.

How can the slope formula help you to remember the equation for the point-slope form of a line?


Absolute ZeroAll matter is made up of atoms and molecules that are in constantmotion. Temperature is one measure of this motion. Absolute zerois the theoretical temperature limit at which the motion of themolecules and atoms of a substance is the least possible.

Experiments with gaseous substances yield data that allow you toestimate just how cold absolute zero is. For any gas of a constantvolume, the pressure, expressed in a unit called atmospheres,varies linearly as the temperature. That is, the pressure P and thetemperature t are related by an equation of the form P ! mt " b,where m and b are real numbers.

1. Sketch a graph for the data in the table.

2. Use the data and your graph to find values for m and b in the equationP ! mt " b, which relates temperature to pressure.

3. Estimate absolute zero in degrees Celsius by setting P equal to 0 in the equation above and using the values m and b that you obtained in Exercise 2.

25–25 50 75 100 125

0.5

1.5

1.0

t

P

O

t P(in $C) (in atmospheres)

#25 0.91

0 1.00

25 1.09

100 1.36

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

3-43-4

Study Guide and InterventionProving Lines Parallel

NAME ______________________________________________ DATE ____________ PERIOD _____

3-53-5


Less

on

3-5

Identify Parallel Lines If two lines in a plane are cut by a transversal and certainconditions are met, then the lines must be parallel.

If then

• corresponding angles are congruent, the lines are parallel.• 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 m!1 ! m!2,determine which lines, if any, areparallel.

Since m!1 ! m!2, then !1 " !2. !1 and!2 are congruent corresponding angles, so r || s.

n

m

r s

1 2

Find x and m!ABCso that m || n .

We can conclude that m || n if alternateinterior angles are congruent.

m!DAB ! m!CDA3x " 10 ! 6x # 20

10 ! 3x # 2030 ! 3x10 ! x

m!ABC ! 6x # 20! 6(10) # 20 or 40

n

mA

B

C

D

(3x " 10)$

(6x # 20)$


ExercisesExercises

Find x so that ! || m .

1. 2. 3.

4. 5. 6.mn

!(5x " 20)$

70$

m!

(3x # 20)$2x$

m!

(9x " 1)$(8x " 8)$

m!

(3x " 15)$

m! (4x " 20)$

6x $m!

(5x # 5)$

(6x # 20)$


Prove Lines Parallel You can prove that lines are parallel by using postulates andtheorems about pairs of angles. You also can use slopes of lines to prove that two lines areparallel or perpendicular.


Proving Lines Parallel

NAME ______________________________________________ DATE ____________ PERIOD _____

3-53-5

ExampleExample

For Exercises 1–6, fill in the blanks.Given: !1 " !5, !15 " !5Prove: ! || m, r || s

Statements Reasons

1. !15 " !5 1.

2. !13 " !15 2.

3. !5 " !13 3.

4. r || s 4.

5. 5. Given

6. 6. If corr # are ", then lines ||.

7. Determine whether PQ!"# ⊥ TQ!"#. Explain why or why not.

x

y

O

P Q

T

!

m

r s

1 24 3

5 68 7

9 101112

14131516

a Given: !1 " !2, !1 " !3

Prove: A!B! || D!C!

Statements Reasons1. !1 " !2 1. Given

!1 " !32. !2 " !3 2. Transitive Property of "3. A!B! || D!C! 3. If alt. int. angles are ", then

the lines are ||.

1 2

3A B

CD

b. Which lines are parallel?Which lines are perpendicular?

slope of P!Q! ! 0 slope of S!R! ! 0

slope of P!S! ! $43$ slope of Q!R! ! $

43$

slope of P!R! ! #2 slope of S!Q! ! $12$

So P!Q! || S!R!, P!S! || Q!R!, and P!R! ⊥ S!Q!.

x

y

O 4 8–4–8

8

4

–4

–8

P(–2, 4) Q(8, 4)

S(–8, –4) R(2, –4)

ExercisesExercises

Skills PracticeProving Lines Parallel

NAME ______________________________________________ DATE ____________ PERIOD _____

3-53-5


Less

on

3-5

Given the following information, determine which lines,if any, are parallel. State the postulate or theorem that justifies your answer.

1. !3 " !7 2. !9 " !11

3. !2 " !16 4. m!5 " m!12 ! 180


5. 6. 7.

8. PROOF Provide a reason for each statement in the proof of Theorem 3.7.Given:!1 and !2 are complementary.

B!C! ⊥ C!D!Prove: B!A! || C!D!Proof:Statements Reasons1. B!C! ⊥ C!D! 1.2. m!ABC ! m!1 " m!2 2.3. !1 and !2 are complementary. 3.4. m!1 " m!2 ! 90 4.5. m!ABC ! 90 5.6. B!A! ⊥ B!C! 6.7. B!A! || C!D! 7.

Determine whether each pair of lines is parallel. Explain why or why not.

9. 10.

x

y

O

U(4, 2)

T(–4, 0)

S(5, –3)R(0, –4)

x

y

O

A(1, 3)

B(–2, –3)E(0, –3)

F(2, 1)

1

A D

CB 2

m

k!(6x " 4)$

(8x # 8)$mk!

(4x # 10)$

(3x " 10)$m

k !

(2x " 6)$

130$

m

!

a b1 2 3 4

8 7 6 5

9 10 11 1216 15 14 13


Given the following information, determine which lines,if any, are parallel. State the postulate or theorem that justifies your answer.

1. m!BCG " m!FGC ! 180 2. !CBF " !GFH

3. !EFB " !FBC 4. !ACD " !KBF


5. 6. 7.

8. PROOF Write a two-column proof.Given:!2 and !3 are supplementary.Prove: A!B! || C!D!

9. LANDSCAPING The head gardener at a botanical garden wants to plant rosebushes inparallel rows on either side of an existing footpath. How can the gardener ensure thatthe rows are parallel?

1 42 5

3 6 AB

CD

(2x " 12)$(5x # 15)$

t

m

!

(5x " 18)$(7x # 24)$

t

m

!

(3x " 6)$

(4x # 6)$tm

!

A

B CD

E FH

K

GJ

Practice Proving Lines Parallel

NAME ______________________________________________ DATE ____________ PERIOD _____

3-53-5

Reading to Learn MathematicsProving Lines Parallel

NAME ______________________________________________ DATE ____________ PERIOD _____

3-53-5


Less

on

3-5

Pre-Activity How do you know that the sides of a parking space are parallel?


How can the workers who are striping the parking spaces in a parking lotcheck to see if the sides of the spaces are parallel?

Reading the Lesson

1. Choose the word or phrase that best completes each sentence.

a. If two coplanar lines are cut by a transversal so that corresponding angles are

congruent, then the lines are (parallel/perpendicular/skew).

b. In a plane, if two lines are perpendicular to the same line, then they are

(perpendicular/parallel/skew).

c. For a line and a point not on the line, there exists

(at least one/exactly one/at most one) line through the point that is parallel to thegiven line.

d. If two coplanar lines are cut by a transversal so that consecutive interior angles are

(complementary/supplementary/congruent), then the lines are

parallel.

e. If two coplanar lines are cut by a transversal so that alternate interior angles are

congruent, then the lines are (perpendicular/parallel/skew).

2. Which of the following conditions verify that p || q ?

A. !6 " !12 B. !2 " !4

C. !8 " !16 D. !11 " !13

E. !6 and !7 are supplementary. F. !1 " !15

G. !7 and !10 are supplementary. H. !4 " !16


3. A good way to remember something new is to draw a picture. How can a sketch help youto remember the Parallel Postulate?

p

t

q

r

1 2

9 10

121314

1516

11

3 46 5

78


Scrambled-Up ProofThe reasons necessary to complete the following proof arescrambled up below. To complete the proof, number thereasons to match the corresponding statements.

Given: C!D! ⊥ B!E!A!B! ⊥ B!E!A!D! " C!E!B!D! " D!E!

Prove: A!D! || C!E!

Proof:

Statements Reasons

1. C!D! ⊥ B!E! Definition of Right Triangle

2. AB ⊥ B!E! Given

3. !3 and !4 are right angles. Given

4. "ABD and "CDE are right triangles. Definition of Perpendicular Lines

5. A!D! " C!E! Given

6. B!D! " D!E! CPCTC

7. "ABD " "CDE In a plane, if two lines are cut by a transversalso that a pair of corresponding angles iscongruent, then the lines are parallel.(Theorem 7-5)

8. !1 " !2 Given

9. A!D! || C!E! HL

A C

B ED

5

3 1 4 2

6

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

3-53-5

Study Guide and InterventionPerpendiculars and Distance

NAME ______________________________________________ DATE ____________ PERIOD _____

3-63-6


Less

on

3-6

Distance From a Point to a Line 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 isperpendicular to the line.

Draw the segment that represents the distance

from E to AF!"#.Extend AF!"#. Draw E!G! ⊥ AF!"#.E!G! represents the distance from E to AF!"#.

Draw the segment that represents the distance indicated.

1. C to AB!"# 2. D to AB!"#

3. T to RS!"# 4. S to PQ!"#

5. S to QR!"# 6. S to RT!"#

XR T

S

R

T

S

P

Q

X

RT

S

P QX

U

RS

T

X

A

CD

BXA

C

BX

A F G

B E

A F

B E

Pdistance betweenM and PQ!"#

Q

M

ExampleExample

ExercisesExercises


Distance Between Parallel Lines The distance between parallel lines is the lengthof a segment that has an endpoint on each line and is perpendicular to them. Parallel linesare everywhere equidistant, which means that all such perpendicular segments have thesame length.

Find the distance between the parallel lines ! and m whoseequations are y ! 2x " 1 and y ! 2x # 4, respectively.


Perpendiculars and Distance

NAME ______________________________________________ DATE ____________ PERIOD _____

3-63-6

ExampleExample

Draw a line p through (0, 1) that isperpendicular to ! and m .

Line p has slope #$12$ and y-intercept 1. An

equation of p is y ! #$12$x " 1. The point of

intersection for p and ! is (0, 1).

x

y

O

mp !

(0, 1)

x

y

O

m! To find the point of intersection of p and m ,solve a system of equations.

Line m : y ! 2x # 4Line p: y ! #$

12$x " 1

Use substitution.

2x # 4 ! #$12$x " 1

4x # 8 ! #x " 25x ! 10x ! 2

Substitute 2 for x to find the y-coordinate.

y ! #$12$x " 1

! #$12$(2) " 1 ! #1 " 1 ! 0

The point of intersection of p and m is (2, 0).

Use the Distance Formula to find thedistance between (0, 1) and (2, 0).

d ! %(x2 #!x1)2 "! ( y2 #! y1)2!

! %(2 # 0!)2 " (!0 # 1!)2!! %5!

The distance between ! and m is %5! units.

ExercisesExercisesFind the distance between each pair of parallel lines.

1. y ! 8 2. y ! x " 3 3. y ! #2xy ! #3 y ! x # 1 y ! #2x # 5

Skills PracticePerpendiculars and Distance

NAME ______________________________________________ DATE ____________ PERIOD _____

3-63-6


Less

on

3-6


1. B to AC!"# 2. G to EF!"# 3. Q to SR!"#

Construct a line perpendicular to ! through K. Then find the distance from K to !.

4. 5.

Find the distance between each pair of parallel lines.

6. y ! 7 7. x ! #6 8. y ! 3xy ! #1 x ! 5 y ! 3x " 10

9. y ! #5x 10. y ! x " 9 11. y ! #2x " 5y ! #5x " 26 y ! x " 3 y ! #2x # 5

x

y

O

!

K

x

y

O

!

K

S

P Q

RD

E F

GA

B

C



1. O to MN!"# 2. A to DC!"# 3. T to VU!"#

Construct a line perpendicular to ! through B. Then find the distance from B to !.

4. 5.

Find the distance between each pair of parallel lines.

6. y ! #x 7. y ! 2x " 7 8. y ! 3x " 12y ! #x # 4 y ! 2x # 3 y ! 3x # 18

9. 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.

10. CANOEING Bronson and a friend are going to carry a canoe across a flat field to thebank of a straight canal. Describe the shortest path they can use.

(–2, –3)

y ! #x " 1

x

y

O

x

y

O

!

B

x

y

O

!

B

T

VW

S U

A B

CD

M N

O

Practice Perpendiculars and Distance

NAME ______________________________________________ DATE ____________ PERIOD _____

3-63-6

Reading to Learn MathematicsPerpendiculars and Distance

NAME ______________________________________________ DATE ____________ PERIOD _____

3-63-6


Less

on

3-6

Pre-Activity How does the distance between parallel lines relate to hanging new shelves?


Name three examples of situations in home construction where it would beimportant to construct parallel lines.

Reading the Lesson1. Fill in the blank with a word or phrase to complete each sentence.

a. The distance from a line to a point not on the line is the length of the segment to the line from the point.

b. Two coplanar lines are parallel if they are everywhere .c. In a plane, if two lines are both equidistant from a third line, then the two lines are

to each other.d. The distance between two parallel lines measured along a perpendicular to the two

lines is always .e. To measure the distance between two parallel lines, measure the distance between

one of the lines and any point on the .

2. On each figure, draw the segment that represents the distance indicated.

a. P to ! b. D to AB!"#

c. E to FG!"# d. U to RV!"#

Helping You Remember3. A good way to remember a new word is to relate it to words that use the same root. Use

your dictionary to find the meaning of the Latin root aequus. List three words other thanequal and equidistant that are derived from this root and give the meaning of each.

TV

R S

U

E

F G

A

D C

B

P

!


Parallelism in SpaceIn space geometry, the concept of parallelism must be extended to include two planes and a line and a plane.

Definition: Two planes are parallel if and only if they do not intersect.

Definition: A line and a plane are parallel if and only if they do not intersect.

Thus, in space, two lines can be intersecting, parallel, or skew while two planes or a line and a plane can only beintersecting or parallel. In the figure at the right, t $ M,t $ P, P || H, and " and n are skew.

The following five statements are theorems about parallel planes.

Theorem: Two planes perpendicular to the same line are parallel.Theorem: Two planes parallel to the same plane are parallel.Theorem: A line perpendicular to one of two parallel planes is

perpendicular to the other.Theorem: A plane perpendicular to one of two parallel planes is

perpendicular to the other.Theorem: If two parallel planes each intersect a third plane, then

the two lines of intersection are parallel.

Use the figure given above for Exercises 1–10. State yes or no to tell whether thestatement is true.

1. M & P 2. " & n 3. M & H 4. " & P 5. " $ t

6. n & H 7. " $ P 8. t & H 9. M $ t 10. t $ H

Make a small sketch to show that each statement is false.

11. If two lines are parallel to the same 12. If two planes are parallel, then any line plane, then the lines are parallel. in one plane is parallel to any line in the

other plane.

13. If two lines are parallel, then any plane 14. If two lines are parallel, then any plane containing one of the lines is parallel to containing one of the lines is parallel to any plane containing the other line. the other line.

t

! M

n P

H

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

3-63-6

© Glencoe/McGraw-Hill A2 Glencoe Geometry

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fig

ure,

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par

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l to

m,o

r !

|| m.Y

ou c

an a

lso

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te

PQ

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tw

o pl

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ey a

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b.N

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segm

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th

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es 1

–3,r

efer

to

the

figu

re a

t th

e ri

ght.

1.N

ame

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lane

s th

at in

ters

ect

plan

e O

PT

.

MN

O,M

PS

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2.N

ame

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tha

t ar

e pa

ralle

l to

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.

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ame

all s

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t in

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ect

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rcis

es 4

–7,r

efer

to

the

figu

re a

t th

e ri

ght.

4.N

ame

all s

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ents

par

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l to

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.

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at in

ters

ect

plan

e M

HE

.

MH

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w t

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An

gle

Rel

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A li

ne t

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inte

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ts t

wo

or m

ore

othe

r lin

es in

a p

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da

tran

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he f

igur

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low

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a t

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wo

lines

and

a t

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orm

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htan

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.Som

e pa

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of t

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ve s

peci

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ames

.The

fol

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char

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ts t

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airs

of

angl

es a

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heir

nam

es.

Ang

le P

airs

Nam

e

!3,

!4,

!5,

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inte

rior

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rior

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3 an

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te

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th

e fi

gure

in

th

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xam

ple

for

Exe

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es 1

–12.

Nam

e th

e tr

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es.

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tify

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es a

s a

lter

na

te i

nte

rior

,alt

ern

ate

ext

erio

r,co

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din

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tive

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teri

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.

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118.

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corr

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utiv

e in

teri

oral

t.in

teri

or

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d !

1611

.!11

and

!14

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10 a

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16

alt.

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rior

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rior

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3-1

3-1

Exam

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Exam

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Exercis

esExercis

es

Answers (Lesson 3-1)


An

swer

s

Skil

ls P

ract

ice

Para

llel L

ines

and

Tra

nsve

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s

NA

ME

____

____

____

____

____

____

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3-1

3-1

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Lesson 3-1

For

Exe

rcis

es 1

–4,r

efer

to

the

figu

re a

t th

e ri

ght.

1.N

ame

all p

lane

s th

at a

re p

aral

lel t

o pl

ane

DE

H.

plan

e FB

G

2.N

ame

all s

egm

ents

tha

t ar

e pa

ralle

l to

A!B!

.C!

D!,E!

F!,G!

H!

3.N

ame

all s

egm

ents

tha

t in

ters

ect

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.C!

G!,D!

H!,E!

H!,F!

G!

4.N

ame

all s

egm

ents

tha

t ar

e sk

ew t

o C!

D!.

A!E!,

B!F!,

E!H!,F!

G!

Iden

tify

th

e se

ts o

f li

nes

to

wh

ich

eac

h g

iven

lin

e is

a

tran

sver

sal.

5.r

sand

t,s

and

w,s

and

z,ta

nd w

,tan

d z,

wan

d z

6.s

rand

t,r

and

w,r

and

z,ta

nd w

,tan

d z,

wan

d z

7.w

rand

s,r

and

t,ra

nd z

,san

d t,

sand

z,t

and

z

Iden

tify

eac

h p

air

of a

ngl

es a

s a

lter

na

te i

nte

rior

,alt

ern

ate

ex

teri

or,c

orre

spon

din

g,or

con

secu

tive

in

teri

oran

gles

.

8.!

2 an

d !

89.

!3

and

!6

alte

rnat

e ex

teri

orco

nsec

utiv

e in

teri

or

10.!

1 an

d !

911

.!3

and

!9

corr

espo

ndin

gal

tern

ate

inte

rior

12.!

6 an

d !

1213

.!7

and

!11

alte

rnat

e ex

teri

orco

rres

pond

ing

Nam

e th

e tr

ansv

ersa

l th

at f

orm

s ea

ch p

air

of a

ngl

es.T

hen

id

enti

fy t

he

spec

ial

nam

e fo

r th

e an

gle

pai

r.

14.!

4 an

d !

1015

.!2

and

!12

g;al

tern

ate

inte

rior

g;al

tern

ate

exte

rior

16.!

7 an

d !

317

.!13

and

!10

j;co

rres

pond

ing

k;c

onse

cutiv

e in

teri

or

18.!

8 an

d !

1419

.!6

and

!14

h;al

tern

ate

inte

rior

h;co

rres

pond

ing

h

g

j

k

12

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For

Exe

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–4,r

efer

to

the

figu

re a

t th

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ght.

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ame

all p

lane

s th

at in

ters

ect

plan

e S

TX

.TU

Y,R

SW

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UV,

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e al

l seg

men

ts t

hat

inte

rsec

t Q!

U!.

Q!R!

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Z!

3.N

ame

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egm

ents

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t ar

e pa

ralle

l to

X!Y!

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ame

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egm

ents

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t ar

e sk

ew t

o V!

W!.

Q!U!

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X!,T!U!

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U!Z!

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tify

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e se

ts o

f li

nes

to

wh

ich

eac

h g

iven

lin

e is

a

tran

sver

sal.

5.e fa

nd g

,fan

d h,

fand

i,g

and

h,g

and

i,h

and

i

6.h ea

nd f,

eand

g,e

and

i,fa

nd g

,fan

d i,

gan

d i

Iden

tify

eac

h p

air

of a

ngl

es a

s a

lter

na

te i

nte

rior

,alt

ern

ate

ex

teri

or,c

orre

spon

din

g,or

con

secu

tive

in

teri

oran

gles

.

7.!

9 an

d !

138.

!6

and

!16

corr

espo

ndin

gal

tern

ate

exte

rior

9.!

3 an

d !

1010

.!8

and

!14

cons

ecut

ive

inte

rior

alte

rnat

e in

teri

or

Nam

e th

e tr

ansv

ersa

l th

at f

orm

s ea

ch p

air

of a

ngl

es.T

hen

id

enti

fy t

he

spec

ial

nam

e fo

r th

e an

gle

pai

r.

11.!

2 an

d !

1212

.!6

and

!18

a;al

tern

ate

inte

rior

d;co

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pond

ing

13.!

13 a

nd !

1914

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and

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b;al

tern

ate

exte

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e in

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FUR

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or E

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the

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.

15.F

ind

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s.S

ampl

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:the

top

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16.F

ind

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ampl

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tab

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3-1

3-1



Rea

din

g t

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earn

Math

emati

csPa

ralle

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es a

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rans

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NA

ME

____

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3-1

3-1

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Lesson 3-1

Pre-

Act

ivit

yH

ow a

re p

aral

lel

lin

es a

nd

pla

nes

use

d i

n a

rch

itec

ture

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 3-

1 at

the

top

of

page

126

in y

our

text

book

.

•G

ive

an e

xam

ple

of p

aral

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ines

tha

t ca

n be

fou

nd in

you

r cl

assr

oom

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ampl

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s:ed

ges

of f

loor

alo

ng o

ppos

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alls

;ve

rtic

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dges

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a do

or•

Giv

e an

exa

mpl

e of

par

alle

l pla

nes

that

can

be

foun

d in

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r cl

assr

oom

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ampl

e an

swer

s:ce

iling

and

flo

or;o

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Rea

din

g t

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Less

on

1.W

rite

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eom

etri

cal t

erm

tha

t m

atch

es e

ach

defi

niti

on.

a.tw

o pl

anes

tha

t do

not

inte

rsec

tpa

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nes

b.lin

es t

hat

are

not

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and

do

not

inte

rsec

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nes

c.tw

o co

plan

ar li

nes

that

do

not

inte

rsec

tpa

ralle

l lin

esd.

a lin

e th

at in

ters

ects

tw

o or

mor

e lin

es in

a p

lane

at

diff

eren

t po

ints

tran

sver

sal

e.a

pair

of

angl

es d

eter

min

ed b

y tw

o lin

es a

nd a

tra

nsve

rsal

con

sist

ing

of a

n in

teri

oran

gle

and

an e

xter

ior

angl

e th

at h

ave

diff

eren

t ve

rtic

es a

nd t

hat

lie o

n th

e sa

me

side

of t

he t

rans

vers

alco

rres

pond

ing

angl

es

2.R

efer

to

the

figu

re a

t th

e ri

ght.

Giv

e th

e sp

ecia

l nam

e fo

r ea

ch

angl

e pa

ir.

a.!

3 an

d !

5co

rres

pond

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angl

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and

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alte

rnat

e ex

teri

or a

ngle

sc.

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and

!8

alte

rnat

e in

teri

or a

ngle

sd.

!2

and

!3

cons

ecut

ive

inte

rior

ang

les

e.!

8 an

d !

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rres

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angl

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alte

rnat

e in

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sg.

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vert

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ang

les

h.

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and

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linea

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Hel

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Rem

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mat

hem

atic

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Man

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Pers

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poin

t fo

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plet

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e pe

rspe

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f th

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bes.

5.F

ind

an e

xam

ple

of a

per

spec

tive

dra

win

g in

a n

ewsp

aper

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azin

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race

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dra

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____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

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___

3-1

3-1



An

swer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Ang

les

and

Para

llel L

ines

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

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___

3-2

3-2

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1G

lenc

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etry

Lesson 3-2

Para

llel L

ines

an

d A

ng

le P

airs

Whe

n tw

o pa

ralle

l lin

es a

re c

ut b

y a

tran

sver

sal,

the

follo

win

g pa

irs

of a

ngle

s ar

e co

ngru

ent.

•co

rres

pond

ing

angl

es•

alte

rnat

e in

teri

or a

ngle

s•

alte

rnat

e ex

teri

or a

ngle

s

Als

o,co

nsec

utiv

e in

teri

or a

ngle

s ar

e su

pple

men

tary

.

In t

he

figu

re,m

!2

!75

.Fin

d t

he

mea

sure

s of

th

e re

mai

nin

g an

gles

.m

!1

!10

5!

1 an

d !

2 fo

rm a

line

ar p

air.

m!

3 !

105

!3

and

!2

form

a li

near

pai

r.m

!4

!75

!4

and

!2

are

vert

ical

ang

les.

m!

5 !

105

!5

and

!3

are

alte

rnat

e in

teri

or a

ngle

s.m

!6

!75

!6

and

!2

are

corr

espo

ndin

g an

gles

.m

!7

!10

5!

7 an

d !

3 ar

e co

rres

pond

ing

angl

es.

m!

8 !

75!

8 an

d !

6 ar

e ve

rtic

al a

ngle

s.

In t

he

figu

re,m

!3

!10

2.F

ind

th

e m

easu

re o

f ea

ch a

ngl

e.

1.!

510

22.

!6

78

3.!

11 1

024.

!7

102

5.!

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26.

!14

78

In t

he

figu

re,m

!9

!80

an

d m

!5

!68

.Fin

d t

he

mea

sure

of

eac

h a

ngl

e.

7.!

1210

08.

!1

80

9.!

410

010

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80

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768

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1611

2

wv

p q

12 3

4

65

78

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12

1413

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m n

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4

65

78

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12

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p

m n

12 3

4 65

78

Exam

ple

Exam

ple

Exercis

esExercis

es

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w-H

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2G

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eom

etry

Alg

ebra

an

d A

ng

le M

easu

res

Alg

ebra

can

be

used

to

find

unk

now

n va

lues

in a

ngle

s fo

rmed

by

a tr

ansv

ersa

l and

par

alle

l lin

es.

If m

!1

!3x

"15

,m!

2 !

4x#

5,m

!3

!5y

,an

d m

!4

!6z

"3,

fin

d x

and

y.

p||q

,so

m!

1 !

m!

2 be

caus

e th

ey a

re

corr

espo

ndin

g an

gles

.

3x"

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4x#

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"15

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x#

515

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20 !

x

pq

r s

12

34

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Ang

les

and

Para

llel L

ines

NA

ME

____

____

____

____

____

____

____

____

____

____

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ATE

____

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RIO

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___

3-2

3-2

Exam

ple

Exam

ple

r||s

,so

m!

2 !

m!

3 be

caus

e th

ey a

re

corr

espo

ndin

g an

gles

.

m!

2 !

m!

375

!5y

$7 55 $!

$5 5y $

15 !

y

Exercis

esExercis

es

Fin

d x

and

yin

eac

h f

igu

re.

1.2.

x!

15;y

!19

x!

6;y

!24

3.4.

x!

11;y

!10

x!

10;y

!25

Fin

d x

,y,a

nd

zin

eac

h f

igu

re.

5.6.

x!

74;y

!37

;z!

25

x!

30;y

!15

;z!

150

2x$

2y$

90$

x$

z$

2y$

106$

x$( 4

z "

6) $

( 5x

# 2

0)$

3 x$

2 y$ 4y

$

( 11x

" 4

) $

( 13y

# 5

) $( 5y

" 5

) $

5x$

( 15x

" 3

0)$

( 3y

" 1

8)$

10x$

90$

( 5x

# 5

) $( 6

y #

4) $

( 4x

" 1

0)$



Skil

ls P

ract

ice

Ang

les

and

Para

llel L

ines

NA

ME

____

____

____

____

____

____

____

____

____

____

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ATE

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RIO

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3-2

3-2

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3G

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etry

Lesson 3-2

In t

he

figu

re,m

!2

!70

.Fin

d t

he

mea

sure

of

each

an

gle.

1.!

370

2.!

511

0

3.!

811

04.

!1

110

5.!

411

06.

!6

70

In t

he

figu

re,m

!7

!10

0.F

ind

th

e m

easu

re o

f ea

ch a

ngl

e.

7.!

910

08.

!6

80

9.!

880

10.!

280

11.!

510

012

.!11

100

In t

he

figu

re,m

!3

!75

an

d m

!10

!11

5.F

ind

th

e m

easu

re

of e

ach

an

gle.

13.!

210

514

.!5

105

15.!

710

516

.!15

115

17.!

1465

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965

Fin

d x

and

yin

eac

h f

igu

re.

19.

x!

28,y

!47

20.

x!

10,y

!15

Fin

d m

!1

in e

ach

fig

ure

.

21.

8022

.72

140$

32$

160

$ 1 20$

7x$

( 8x

# 1

0)$

( 6y

" 2

0)$

5x$

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) $

x

w

zy

12

43

65

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910 1112

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q

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87

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lenc

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w-H

ill13

4G

lenc

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eom

etry

In t

he

figu

re,m

!2

!92

an

d m

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!74

.Fin

d t

he

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sure

of

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Fin

d x

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re.

7.8.

x!

14,y

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28,y

!23

Fin

d m

!1

in e

ach

fig

ure

.

9.10

.

130

98

11.P

RO

OF

Wri

te a

par

agra

ph p

roof

of T

heor

em 3

.3.

Giv

en: !

|| m,m

|| nP

rove

:!1

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12S

ampl

e pr

oof:

It is

giv

en t

hat

!|| m

,so

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8 by

the

Alte

rnat

eE

xter

ior

Ang

les

Theo

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.Sin

ce it

is g

iven

tha

t m

|| n,

!8

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y th

e C

orre

spon

ding

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les

Pos

tula

te.

Ther

efor

e,!

1 "

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,sin

ce c

ongr

uenc

e of

ang

les

is t

rans

itive

.

12.F

ENC

ING

A d

iago

nal b

race

str

engt

hens

the

wir

e fe

nce

and

prev

ents

it

fro

m s

aggi

ng.T

he b

race

mak

es a

50°

angl

e w

ith

the

wir

e as

sho

wn.

Fin

d y.

130

50$

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34

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8

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k

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100$

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# 1

0)$

n

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rs

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8

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Pra

ctic

e (A

vera

ge)

Ang

les

and

Para

llel L

ines

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-2

3-2



An

swer

s

Rea

din

g t

o L

earn

Math

emati

csA

ngle

s an

d Pa

ralle

l Lin

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-2

3-2

©G

lenc

oe/M

cGra

w-H

ill13

5G

lenc

oe G

eom

etry

Lesson 3-2

Pre-

Act

ivit

yH

ow c

an a

ngl

es a

nd

lin

es b

e u

sed

in

art

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 3-

2 at

the

top

of

page

133

in y

our

text

book

.•

Your

tex

tboo

k sh

ows

a pa

inti

ng t

hat

cont

ains

tw

o pa

ralle

l lin

es a

nd a

tran

sver

sal.

Wha

t is

the

nam

e fo

r !

1 an

d !

2?co

rres

pond

ing

angl

es•

Wha

t is

the

rel

atio

nshi

p be

twee

n th

ese

two

angl

es?

They

are

con

grue

nt.

Rea

din

g t

he

Less

on

1.C

hoos

e th

e co

rrec

t w

ord

to c

ompl

ete

each

sen

tenc

e.

a.If

tw

o pa

ralle

l lin

es a

re c

ut b

y a

tran

sver

sal,

then

alt

erna

te e

xter

ior

angl

es a

re

(con

grue

nt/c

ompl

emen

tary

/sup

plem

enta

ry).

b.If

tw

o pa

ralle

l lin

es a

re c

ut b

y a

tran

sver

sal,

then

cor

resp

ondi

ng a

ngle

s ar

e

(con

grue

nt/c

ompl

emen

tary

/sup

plem

enta

ry).

c.If

par

alle

l lin

es a

re c

ut b

y a

tran

sver

sal,

then

con

secu

tive

inte

rior

ang

les

are

(con

grue

nt/c

ompl

emen

tary

/sup

plem

enta

ry).

d.In

a p

lane

,if

a lin

e is

per

pend

icul

ar t

o on

e of

tw

o pa

ralle

l lin

es,t

hen

it is

(par

alle

l/per

pend

icul

ar/s

kew

) to

the

oth

er.

Use

th

e fi

gure

for

Exe

rcis

es 2

an

d 3

.

2.a.

Nam

e fo

ur p

airs

of

vert

ical

ang

les.

!1

and

!3,

!2

and

!4,

!5

and

!7,

!6

and

!8

b.N

ame

all a

ngle

s th

at f

orm

a li

near

pai

r w

ith

!7.

!6,

!8

c.N

ame

all a

ngle

s th

at a

re c

ongr

uent

to

!1.

!3,

!6,

!8

d.N

ame

all a

ngle

s th

at a

re c

ongr

uent

to

!4.

!2,

!5,

!7

e.N

ame

all a

ngle

s th

at a

re s

uppl

emen

tary

to

!3.

!2,

!4,

!5,

!7

f.N

ame

all a

ngle

s th

at a

re s

uppl

emen

tary

to

!2.

!1,

!3,

!6,

!8

3.W

hich

con

clus

ion(

s) c

ould

you

mak

e ab

out

lines

uan

d v

if m

!4

!m

!1?

B,D

A.

t|| u

B.

t⊥

uC

.v

⊥u

D.

v⊥

tE

.v

|| t

Hel

pin

g Y

ou

Rem

emb

er4.

How

can

you

use

an

ever

yday

mea

ning

of

the

adje

ctiv

e al

tern

ate

to h

elp

you

rem

embe

rth

e ty

pes

of a

ngle

pai

rs f

or t

wo

lines

and

a t

rans

vers

al?

Sam

ple

answ

er:O

ne m

eani

ng o

f al

tern

ate

is “

obta

ined

by

switc

hing

bac

kan

d fo

rth

from

one

thi

ng t

o an

othe

r.”Th

e an

gle

pair

s in

thi

s le

sson

all

use

angl

es w

ith d

iffer

ent

vert

ices

,and

tho

se w

hose

nam

es c

onta

in t

head

ject

ive

alte

rnat

eca

n be

loca

ted

in a

fig

ure

by s

witc

hing

fro

m o

ne s

ide

of t

he t

rans

vers

al t

o th

e ot

her.

The

pair

s w

hose

nam

es d

o no

t in

clud

e th

ew

ord

alte

rnat

ear

e fo

und

on t

he s

ame

side

of

the

tran

sver

sal.

t

u

v

1 234

6578

perp

endi

cula

r

supp

lem

enta

ry

cong

ruen

t

cong

ruen

t

©G

lenc

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cGra

w-H

ill13

6G

lenc

oe G

eom

etry

Mor

e O

ptic

al Il

lusi

ons

In d

raw

ings

,dia

gona

l lin

es m

ay c

reat

e th

e ill

usio

n of

dep

th.

For

exam

ple,

the

figu

re a

t th

e ri

ght

can

be t

houg

ht o

f as

pi

ctur

ing

a fl

at f

igur

e or

a c

ube.

The

opt

ical

illu

sion

s on

thi

s pa

ge in

volv

e de

pth

perc

epti

on.

An

swer

eac

h q

ues

tion

.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-2

3-2

1.H

ow m

any

cube

s do

you

see

in t

hedr

awin

g?5

or 6

3.D

oes

the

draw

ing

show

a v

iew

fro

m

the

top

or t

he b

otto

m o

f th

e st

airs

?

Ans

wer

s m

ay v

ary.

2.C

an t

his

figu

re s

how

an

actu

al

obje

ct?

no

4.W

hich

line

seg

men

t is

long

er,

A!B!

orC !

D!?

Mea

sure

to

chec

k yo

ur a

nsw

er.

A!B!

A BC

D

5.W

hich

per

son

in t

he d

raw

ing

at t

he r

ight

app

ears

to

be

talle

st?

Mea

sure

to

chec

k yo

ur a

nsw

er.

The

pers

on a

t th

e ri

ght

appe

ars

talle

st,

but

all a

re t

he s

ame

size

.

6.D

raw

tw

o m

ore

obje

cts

the

sam

e si

ze o

n th

e fi

gure

at

the

rig

ht.D

oes

one

appe

ar la

rger

tha

n th

e ot

her?

Ans

wer

s w

ill v

ary.



Stu

dy

Gu

ide

and I

nte

rven

tion

Slo

pes

of L

ines

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-3

3-3

©G

lenc

oe/M

cGra

w-H

ill13

7G

lenc

oe G

eom

etry

Lesson 3-3

Slo

pe

of

a Li

ne

The

slo

pe m

of a

line

con

tain

ing

two

poin

ts w

ith

coor

dina

tes

(x1,

y 1)

and

(x2,

y 2)

is g

iven

by

the

form

ula

m!

$y x2 2

# #

y x1 1$

,whe

re x

1%

x 2.

Fin

d t

he

slop

e of

eac

h l

ine.

For

line

p,le

t (x

1,y 1

) be

(1,

2) a

nd (x

2,y 2

) be

(#2,

#2)

.

m!

$y x2 2

# #

y x1 1$

!$# #

2 2# #

2 1$

or $4 3$

For

line

q,le

t (x

1,y 1

) be

(2,

0) a

nd (x

2,y 2

) be

(#

3,2)

.

m!

$y x2 2

# #

y x1 1$

!$ #2 3# #

0 2$

or #

$2 5$

Det

erm

ine

the

slop

e of

th

e li

ne

that

con

tain

s th

e gi

ven

poi

nts

.

1.J(

0,0)

,K(#

2,8)

#4

2.R

(#2,

#3)

,S(3

,#5)

#%2 5%

3.L

(1,#

2),N

(#6,

3)#

%5 7%4.

P(#

1,2)

,Q(#

9,6)

#%1 2%

5.T

(1,#

2),U

(6,#

2)0

6.V

(#2,

10),

W(#

4,#

3)%1 23 %

Fin

d t

he

slop

e of

eac

h l

ine.

7.A

B!"

#3

8.C

D!"

##

2

9.E

M!"

#un

defin

ed10

.AE

!"#

0

11.E

H!"

#%2 5%

12.B

M!"

##

%1 2%

x

y

O

C( –

2, 2

)

A( –

2, –

2) H

( –1,

–4)

B( 0

, 4)

M( 4

, 2)

E( 4

, –2)

D( 0

, –2)

x

y

O

( 1, 2

)

( –2,

–2)( –

3, 2

)

( 2, 0

)

Exam

ple

Exam

ple

Exercis

esExercis

es

©G

lenc

oe/M

cGra

w-H

ill13

8G

lenc

oe G

eom

etry

Para

llel a

nd

Per

pen

dic

ula

r Li

nes

If y

ou e

xam

ine

the

slop

es o

f pa

irs

of p

aral

lel l

ines

and

the

slop

es o

f pa

irs

of p

erpe

ndic

ular

line

s,w

here

nei

ther

line

in e

ach

pair

is v

erti

cal,

you

will

dis

cove

r th

e fo

llow

ing

prop

erti

es.

Tw

o lin

es h

ave

the

sam

e sl

ope

if a

nd o

nly

if t

hey

are

para

llel.

Tw

o lin

es a

re p

erpe

ndic

ular

if a

nd o

nly

if t

he p

rodu

ct o

f th

eir

slop

es is

#1.

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Slo

pes

of L

ines

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-3

3-3

Fin

d t

he

slop

e of

a l

ine

par

alle

l to

th

e li

ne

con

tain

ing

A(#

3,4)

and

B(2

,5).

Fin

d th

e sl

ope

of A

B! "

# .U

se (#

3,4)

for

(x1,

y 1)

and

use

(2,5

) fo

r (x

2,y 2

).

m!

$y x2 2

# #

y x1 1$

!$ 2

5 ## (#

4 3)$

or $1 5$

The

slo

pe o

f an

y lin

e pa

ralle

l to

AB

!"#

mus

t be

$1 5$ .

Fin

d t

he

slop

e of

a l

ine

per

pen

dic

ula

r to

PQ

! "#

for

P(#

2,#

4) a

nd

Q(4

,3).

Fin

d th

e sl

ope

of P

Q! "

# .U

se (

#2,

#4)

for

(x

1,y 1

) an

d us

e (4

,3)

for

(x2,

y 2).

m!

$y x2 2

# #

y x1 1$

!$3 4

# #

( (# #

4 2) )$

or $7 6$

Sinc

e $7 6$

&##

$6 7$ $!#

1,th

e sl

ope

of a

ny li

ne

perp

endi

cula

r to

PQ

!"# m

ust

be #

$6 7$ .

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exercis

esExercis

es

Det

erm

ine

wh

eth

er M

N!"

#an

d R

S!"

# are

pa

rall

el,p

erpe

nd

icu

lar,

or n

eith

er.

1.M

(0,3

),N

(2,4

),R

(2,1

),S

(8,4

)2.

M(#

1,3)

,N(0

,5),

R(2

,1),

S(6

,#1)

para

llel

perp

endi

cula

r

3.M

(#1,

3),N

(4,4

),R

(3,1

),S

(#2,

2)4.

M(0

,#3)

,N(#

2,#

7),R

(2,1

),S

(0,#

3)

neith

erpa

ralle

l

5.M

(#2,

2),N

(1,#

3),R

(#2,

1),S

(3,4

)6.

M(0

,0),

N(2

,4),

R(2

,1),

S(8

,4)

perp

endi

cula

rne

ither

Fin

d t

he

slop

e of

MN

!"#

and

th

e sl

ope

of a

ny

lin

e p

erp

end

icu

lar

to M

N!"

# .

7.M

(2,#

4),N

(#2,

#1)

8.M

(1,3

),N

(#1,

5)

#%3 4% ;

%4 3%#

1;1

9.M

(4,#

2),N

(5,3

)10

.M(2

,#3)

,N(#

4,1)

5;#

%1 5%#

%2 3% ;%3 2%



An

swer

s

Skil

ls P

ract

ice

Slo

pes

of L

ines

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-3

3-3

©G

lenc

oe/M

cGra

w-H

ill13

9G

lenc

oe G

eom

etry

Lesson 3-3

Det

erm

ine

the

slop

e of

th

e li

ne

that

con

tain

s th

e gi

ven

poi

nts

.

1.S

(#1,

2),W

(0,4

)2

2.G

(#2,

5),H

(1,#

7)#

4

3.C

(0,1

),D

(3,3

)%2 3%

4.J(

#5,

#2)

,K(5

,#4)

#%1 5%

Fin

d t

he

slop

e of

eac

h l

ine.

5.N

P!"

#6.

TW

!"#

%3 4%#

2

7.a

line

para

llel t

o T

W!"

#8.

a lin

e pe

rpen

dicu

lar

to N

P!"

#

#2

#%4 3%

Det

erm

ine

wh

eth

er A

B!"

#an

d M

N!"

#ar

e pa

rall

el,p

erpe

nd

icu

lar,

or n

eith

er.

9.A

(0,3

),B

(5,#

7),M

(#6,

7),N

(#2,

#1)

10.A

(#1,

4),B

(2,#

5),M

(#3,

2),N

(3,0

)pa

ralle

lne

ither

11.A

(#2,

#7)

,B(4

,2),

M(#

2,0)

,N(2

,6)

12.A

(#4,

#8)

,B(4

,#6)

,M(#

3,5)

,N(#

1,#

3)pa

ralle

lpe

rpen

dicu

lar

Gra

ph

th

e li

ne

that

sat

isfi

es e

ach

con

dit

ion

.

13.s

lope

!3,

cont

ains

A(0

,1)

14.s

lope

!#

$3 2$ ,co

ntai

ns R

(#4,

5)

15.c

onta

ins

Y(3

,0),

para

llel t

o D

J!"

#16

.con

tain

s T

(0,#

2),p

erpe

ndic

ular

to

CX

!"#

wit

h D

(#3,

1) a

nd J

(3,3

)w

ith

C(0

,3)

and

X(2

,#1)

T( 0

, –2)

C( 0

, 3)

X( 2

, –1)x

y OD

( –3,

1)

J(3,

3)

Y( 3

, 0)

x

y O

R( –

4, 5

)

x

y O

A( 0

, 1)

x

y O

x

y ON

TW

P

©G

lenc

oe/M

cGra

w-H

ill14

0G

lenc

oe G

eom

etry

Det

erm

ine

the

slop

e of

th

e li

ne

that

con

tain

s th

e gi

ven

poi

nts

.

1.B

(#4,

4),R

(0,2

)#

%1 2%2.

I(#

2,#

9),P

(2,4

)%1 43 %

Fin

d t

he

slop

e of

eac

h l

ine.

3.L

M!"

#4.

GR

!"#

%2 3%#

%2 5%

5.a

line

para

llel t

o G

R!"

#6.

a lin

e pe

rpen

dicu

lar

to P

S!"

#

#%2 5%

#%1 2%

Det

erm

ine

wh

eth

er K

M!"

#an

d S

T!"

#ar

e pa

rall

el,p

erpe

nd

icu

lar,

or n

eith

er.

7.K

(#1,

#8)

,M(1

,6),

S(#

2,#

6),T

(2,1

0)8.

K(#

5,#

2),M

(5,4

),S

(#3,

6),T

(3,#

4)ne

ither

perp

endi

cula

r

9.K

(#4,

10),

M(2

,#8)

,S(1

,2),

T(4

,#7)

10.K

(#3,

#7)

,M(3

,#3)

,S(0

,4),

T(6

,#5)

para

llel

perp

endi

cula

r

Gra

ph

th

e li

ne

that

sat

isfi

es e

ach

con

dit

ion

.

11.s

lope

!#

$1 2$ ,co

ntai

ns U

(2,#

2)12

.slo

pe !

$4 3$ ,co

ntai

ns P

(#3,

#3)

13.c

onta

ins

B(#

4,2)

,par

alle

l to

FG

!"#

14.c

onta

ins

Z(#

3,0)

,per

pend

icul

ar t

o E

K!"

#

wit

h F

(0,#

3) a

nd G

(4,#

2)w

ith

E(#

2,4)

and

K(2

,#2)

15. P

RO

FITS

Aft

er T

ake

Tw

o be

gan

rent

ing

DV

Ds

at t

heir

vid

eo s

tore

,bus

ines

s so

ared

.B

etw

een

2000

and

200

3,pr

ofit

s in

crea

sed

at a

n av

erag

e ra

te o

f $1

2,00

0 pe

r ye

ar.T

otal

prof

its

in 2

003

wer

e $4

6,00

0.If

pro

fits

con

tinu

e to

incr

ease

at

the

sam

e ra

te,w

hat

will

the

tota

l pro

fit

be in

200

9?$1

18,0

00

K( 2

, –2)

E( –

2, 4

)

Z( –

3, 0

)

x

y

O

F( 0

, –3)

B( –

4, 2

)

G( 4

, –2)

x

y

O

P( –

3, –

3)

x

y OU

( 2, –

2)x

y O

x

y O

R

G

S

P

M

L

Pra

ctic

e (A

vera

ge)

Slo

pes

of L

ines

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-3

3-3



Rea

din

g t

o L

earn

Math

emati

csS

lope

s of

Lin

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-3

3-3

©G

lenc

oe/M

cGra

w-H

ill14

1G

lenc

oe G

eom

etry

Lesson 3-3

Pre-

Act

ivit

yH

ow i

s sl

ope

use

d i

n t

ran

spor

tati

on?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 3-

3 at

the

top

of

page

139

in y

our

text

book

.•

If y

ou a

re d

rivi

ng u

phill

on

a ro

ad w

ith

a 4%

gra

de,h

ow m

any

feet

will

the

road

ris

e fo

r ev

ery

1000

hor

izon

tal f

eet

trav

eled

? 40

ft

•If

you

are

dri

ving

dow

nhill

on

a ro

ad w

ith

a 7%

gra

de,h

ow m

any

met

ers

will

the

roa

d fa

ll fo

r ev

ery

500

met

ers

trav

eled

? 35

m

Rea

din

g t

he

Less

on

1.W

hich

exp

ress

ions

can

be

used

to

repr

esen

t th

e sl

ope

of t

he li

ne c

onta

inin

g po

ints

(x1,

y 1)

and

(x2,

y 2)?

Ass

ume

that

no

deno

min

ator

is z

ero.

A,C

,F

A.$

' 'y x$

B.$h vor ei rz to icn at la rl ir su en

$C

.$y x2 2

# #

y x1 1$

D.$c ch ha an ng ge e

i in nx y

$

E.$

y x2 1

# #

y x1 2$

F.$y x1 1

# #

y x2 2$

G.

$x y2 2

# #

x y1 1$

H.$y y2 1

# #

x x2 1$

2.M

atch

the

des

crip

tion

of

a lin

e fr

om t

he f

irst

col

umn

wit

h th

e de

scri

ptio

n of

its

slop

efr

om t

he s

econ

d co

lum

n.

Typ

e of

Lin

eSl

ope

a.a

hori

zont

al li

neii

i.a

nega

tive

num

ber

b.a

line

that

ris

es f

rom

left

to

righ

tiv

ii.0

c.a

vert

ical

line

iiiii

i.un

defi

ned

d.a

line

that

fal

ls f

rom

left

to

righ

ti

iv.

a po

siti

ve n

umbe

r

3.F

ind

the

slop

e of

eac

h lin

e.

a.a

line

para

llel t

o a

line

wit

h sl

ope

$3 4$%3 4%

b.a

line

perp

endi

cula

r to

the

x-a

xis

unde

fined

slo

pe

c.a

line

perp

endi

cula

r to

a li

ne w

ith

slop

e 5

#%1 5%

d.a

line

para

llel t

o th

e x-

axis

0

e.y-

axis

unde

fined

slo

pe

Hel

pin

g Y

ou

Rem

emb

er

4.A

goo

d w

ay t

o re

mem

ber

som

ethi

ng is

to

expl

ain

it t

o so

meo

ne e

lse.

Supp

ose

your

fri

end

thin

ks t

hat

perp

endi

cula

r lin

es (

if n

eith

er li

ne is

ver

tica

l) h

ave

slop

es t

hat

are

reci

proc

als

of e

ach

othe

r.H

ow c

ould

you

exp

lain

to

your

fri

end

that

thi

s is

inco

rrec

t an

dgi

ve h

er a

goo

d w

ay t

o re

mem

ber

the

corr

ect

rela

tion

ship

?

Sam

ple

answ

er:I

n or

der

for

two

lines

(ne

ither

one

ver

tical

) to

mee

t at

righ

t an

gles

,one

mus

t go

upw

ard

from

left

to

righ

t an

d th

e ot

her

mus

t go

dow

nwar

d,so

the

ir s

lope

s m

ust

have

opp

osite

sig

ns.R

ecip

roca

ls h

ave

the

sam

e si

gn.T

he p

rodu

ct o

f th

e sl

opes

mus

t be

#1,

not

1.R

emem

ber:

The

slop

es a

re o

ppos

itere

cipr

ocal

s.

©G

lenc

oe/M

cGra

w-H

ill14

2G

lenc

oe G

eom

etry

Slo

pes

and

Poly

gons

In c

oord

inat

e ge

omet

ry,t

he s

lope

s of

tw

o lin

es d

eter

min

e if

the

line

s ar

epa

ralle

l or

perp

endi

cula

r.T

his

know

ledg

e ca

n be

use

ful w

hen

wor

king

wit

hpo

lygo

ns.

1.T

he c

oord

inat

es o

f th

e ve

rtic

es o

f a

tria

ngle

are

A

(#6,

4),B

(8,6

),an

d C

(4,#

4).G

raph

"A

BC

.

2.J,

K,a

nd L

are

mid

poin

ts o

f A!B!

,B!C!

,and

A!C!

,re

spec

tive

ly.F

ind

the

coor

dina

tes

of J

,K,a

nd L

.D

raw

"JK

L.

J(1,

5),K

(6,1

),L

(#1,

0)

3.W

hich

seg

men

ts a

ppea

r to

be

para

llel?

A!B!

and

L!K!,B!

C!an

d J!L!

,A!C!

and

J!K!4.

Show

tha

t th

e se

gmen

ts n

amed

in E

xerc

ise

3 ar

e pa

ralle

l by

find

ing

the

slop

es o

f al

l six

seg

men

ts.

A!B!

:%1 7% ;

L!K!:%

1 7% ;B!

C!:%

5 2% ;J!L!

:%5 2% ;

A!C!

:#%4 5% ;

J!K!:#

%4 5%

Th

e co

ord

inat

es o

f th

e ve

rtic

es o

f ri

ght

"P

QR

are

give

n.F

ind

th

e sl

ope

of e

ach

sid

e of

th

e tr

ian

gle.

Th

en n

ame

the

hyp

oten

use

.

5.P

(5,1

)Q

(1,#

1)R

(#2,

5)6.

P(#

2,#

3)Q

(5,1

)R

(2,3

)

slop

e of

P!Q!

!%1 2%

slop

e of

P!Q!

!%4 7%

slop

e of

Q!R!

!#

2sl

ope

of Q!

R!!

#%2 3%

slop

e of

P!R!

!#

%4 7%sl

ope

of P!

R!!

%3 2%

hypo

tenu

se:P!

R!hy

pote

nuse

:P!Q!

Th

e co

ord

inat

es o

f qu

adri

late

ral

PQ

RS

are

give

n.G

rap

h q

uad

rila

tera

l P

QR

San

dfi

nd

th

e sl

opes

of

the

dia

gon

als.

Sta

te w

het

her

th

e d

iago

nal

s ar

e p

erp

end

icu

lar.

7.P

(#2,

6),Q

(4,0

),R

(1,#

4),S

(#5,

2)8.

P(0

,6),

Q(3

,0),

R(#

4,#

2),S

(#5,

4)

P!R!:#

%1 30 %;S!

Q!:#

%2 9% ;no

P!R!:2

;S!Q!

:#%1 2% ;

yes

S

P

Q

R

x

y

O

S

P

Q

R

x

y

O

A

B

C

K

J

Lx

y

O

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-3

3-3



An

swer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Equ

atio

ns o

f Lin

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-4

3-4

©G

lenc

oe/M

cGra

w-H

ill14

3G

lenc

oe G

eom

etry

Lesson 3-4

Wri

te E

qu

atio

ns

of

Lin

esYo

u ca

n w

rite

an

equa

tion

of

a lin

e if

you

are

giv

en a

ny o

fth

e fo

llow

ing:

•th

e sl

ope

and

the

y-in

terc

ept,

•th

e sl

ope

and

the

coor

dina

tes

of a

poi

nt o

n th

e lin

e,or

•th

e co

ordi

nate

s of

tw

o po

ints

on

the

line.

If m

is t

he s

lope

of

a lin

e,b

is it

s y-

inte

rcep

t,an

d (x

1,y 1

) is

a p

oint

on

the

line,

then

:•

the

slop

e-in

terc

ept

form

of t

he e

quat

ion

is y

!m

x"

b,•

the

poi

nt-

slop

e fo

rmof

the

equ

atio

n is

y#

y 1!

m(x

#x 1

).

Wri

te a

n e

quat

ion

in

slop

e-in

terc

ept

form

of

the

lin

e w

ith

slop

e #

2 an

d y

-in

terc

ept

4.y

!m

x"

bS

lope

-inte

rcep

t for

m

y!

#2x

"4

m!

#2,

b!

4

The

slo

pe-i

nter

cept

for

m o

f th

e eq

uati

on o

fth

e lin

e is

y!

#2x

"4.

Wri

te a

n e

quat

ion

in

poi

nt-

slop

e fo

rm o

f th

e li

ne

wit

h s

lop

e #

%3 4%th

at c

onta

ins

(8,1

).

y#

y 1!

m(x

#x 1

)P

oint

-slo

pe fo

rm

y#

1 !

#$3 4$ (

x#

8)m

!#

$3 4$ , (

x 1, y

1) !

(8, 1

)

The

poi

nt-s

lope

for

m o

f th

e eq

uati

on o

f th

elin

e is

y#

1 !

#$3 4$ (

x#

8).

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exercis

esExercis

es

Wri

te a

n e

quat

ion

in

slo

pe-

inte

rcep

t fo

rm o

f th

e li

ne

hav

ing

the

give

n s

lop

e an

d

y-in

terc

ept.

1.m

:2,y

-int

erce

pt:#

32.

m:#

$1 2$ ,y-

inte

rcep

t:4

y!

2x#

3y

!#

%1 2% x"

4

3.m

:$1 4$ ,

y-in

terc

ept:

54.

m:0

,y-i

nter

cept

:#2

y!

%1 4% x"

5y

!#

2

5.m

:#$5 3$ ,

y-in

terc

ept:

$1 3$6.

m:#

3,y-

inte

rcep

t:#

8

y!

#%5 3% x

"%1 3%

y!

#3x

#8

Wri

te a

n e

quat

ion

in

poi

nt-

slop

e fo

rm o

f th

e li

ne

hav

ing

the

give

n s

lop

e th

atco

nta

ins

the

give

n p

oin

t.

7.m

!$1 2$ ,

(3,#

1)8.

m!

#2,

(4,#

2)

y"

1 !

%1 2% (x

#3)

y"

2 !

#2(

x#

4)

9.m

!#

1,(#

1,3)

10.m

!$1 4$ ,

(#3,

#2)

y#

3 !

#(x

"1)

y"

2 !

%1 4% (x

"3)

11.m

!#

$5 2$ ,(0

,#3)

12.m

!0,

(#2,

5)

y"

3 !

#%5 2% x

y#

5 !

0

©G

lenc

oe/M

cGra

w-H

ill14

4G

lenc

oe G

eom

etry

Wri

te E

qu

atio

ns

to S

olv

e Pr

ob

lem

sM

any

real

-wor

ld s

itua

tion

s ca

n be

mod

eled

usin

g lin

ear

equa

tion

s.

Don

na

offe

rs c

omp

ute

r se

rvic

es t

o sm

all

com

pan

ies

in h

er c

ity.

Sh

ech

arge

s $5

5 p

er m

onth

for

mai

nta

inin

g a

web

sit

e an

d $

45 p

er h

our

for

each

serv

ice

call

.Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Equ

atio

ns o

f Lin

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-4

3-4

Exam

ple

Exam

ple

a.W

rite

an

equ

atio

n t

ore

pre

sen

t th

e to

tal

mon

thly

cos

t C

for

mai

nta

inin

g a

web

sit

ean

d f

or h

hou

rs o

fse

rvic

e ca

lls.

For

each

hou

r,th

e co

stin

crea

ses

$45.

So t

he r

ate

ofch

ange

,or

slop

e,is

45.

The

y-in

terc

ept

is lo

cate

d w

here

ther

e ar

e 0

hour

s,or

$55

.C

!m

h"

b!

45h

"55

b.D

onn

a m

ay c

han

ge h

er c

osts

to

rep

rese

nt

them

by t

he

equ

atio

n C

!25

h"

125,

wh

ere

$125

is

the

fixe

d m

onth

ly f

ee f

or a

web

sit

e an

d t

he

cost

per

hou

r is

$25

.Com

par

e h

er n

ew p

lan

to

the

old

on

e if

a c

omp

any

has

5%1 2%

hou

rs o

f se

rvic

e ca

lls.

Un

der

w

hic

h p

lan

wou

ld D

onn

a ea

rn m

ore?

Fir

st p

lan

For

5$1 2$

hour

s of

ser

vice

Don

na w

ould

ear

n

C!

45h

"55

!45

#5$1 2$ $"55

!24

7.5

"55

or

$302

.50

Sec

ond

Pla

n

For

5$1 2$

hour

s of

ser

vice

Don

na w

ould

ear

n

C!

25h

"12

5 !

25(5

.5)

"12

5

!13

7.5

"12

5 or

$26

2.50

Don

na w

ould

ear

n m

ore

wit

h th

e fi

rst

plan

.

Exercis

esExercis

es

For

Exe

rcis

es 1

–4,u

se t

he

foll

owin

g in

form

atio

n.

Jerr

i’s c

urre

nt s

atel

lite

tele

visi

on s

ervi

ce c

harg

es a

flat

rat

e of

$34

.95

per

mon

th fo

r th

e ba

sic

chan

nels

and

an

addi

tion

al $

10 p

er m

onth

for

each

pre

miu

m c

hann

el.A

com

peti

ng s

atel

lite

tele

visi

on s

ervi

ce c

harg

es a

flat

rat

e of

$39

.99

per

mon

th fo

r th

e ba

sic

chan

nels

and

an

addi

tion

al $

8 pe

r m

onth

for

each

pre

miu

m c

hann

el.

1.W

rite

an

equa

tion

in s

lope

-int

erce

pt f

orm

that

mod

els

the

tota

l mon

thly

cos

t fo

rea

ch s

atel

lite

serv

ice,

whe

re p

is t

henu

mbe

r of

pre

miu

m c

hann

els.

Cur

rent

ser

vice

:C!

10p

"34

.95

Com

petin

g se

rvic

e:C

!8p

"39

.99

3.A

thi

rd s

atel

lite

com

pany

cha

rges

a f

lat

rate

of

$69

for

all c

hann

els,

incl

udin

g th

epr

emiu

m c

hann

els.

If J

erri

wan

ts t

o ad

da

four

th p

rem

ium

cha

nnel

,whi

ch s

ervi

cew

ould

be

leas

t ex

pens

ive?

the

thir

d co

mpa

ny

2.If

Jer

ri w

ants

to

incl

ude

thre

e pr

emiu

mch

anne

ls in

her

pac

kage

,whi

ch s

ervi

cew

ould

be

less

,her

cur

rent

ser

vice

or

the

com

peti

ng s

ervi

ce?

com

petin

g se

rvic

e

4.W

rite

a d

escr

ipti

on o

f ho

w t

he f

ee f

or t

henu

mbe

r of

pre

miu

m c

hann

els

is r

efle

cted

in t

he e

quat

ion.

The

fee

for

the

num

ber

of p

rem

ium

cha

nnel

sre

pres

ents

the

rat

e of

cha

nge,

or s

lope

,of

the

equa

tion.



Skil

ls P

ract

ice

Equ

atio

ns o

f Lin

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-4

3-4

©G

lenc

oe/M

cGra

w-H

ill14

5G

lenc

oe G

eom

etry

Lesson 3-4

Wri

te a

n e

quat

ion

in

slo

pe-

inte

rcep

t fo

rm o

f th

e li

ne

hav

ing

the

give

n s

lop

e an

d

y-in

terc

ept.

1.m

:#4,

y-in

terc

ept:

32.

m:3

,y-i

nter

cept

:#8

y!

#4x

"3

y!

3x#

8

3.m

:$3 7$ ,

(0,1

)4.

m:#

$2 5$ ,(0

,#6)

y!

%3 7% x"

1y

!#

%2 5% x#

6

Wri

te e

quat

ion

s in

poi

nt-

slop

e fo

rm a

nd

slo

pe-

inte

rcep

t fo

rm o

f th

e li

ne

hav

ing

the

give

n s

lop

e an

d c

onta

inin

g th

e gi

ven

poi

nt.

5.m

:2,(

5,2)

6.m

:#3,

(2,#

4)

y#

2 !

2(x

#5)

,y!

2x#

8y

"4

!#

3(x

#2)

,y!

#3x

"2

7.m

:#$1 2$ ,

(#2,

5)8.

m:$

1 3$ ,(#

3,#

8)

y#

5 !

#%1 2% (

x"

2),y

!#

%1 2% x"

4y

"8

!%1 3% (

x"

3),y

!%1 3% x

#7

Wri

te a

n e

quat

ion

in

slo

pe-

inte

rcep

t fo

rm f

or e

ach

lin

e.

9.r

y!

x"

310

.sy

!#

2x"

2

11.t

y!

3x#

312

.uy

!%1 3% x

#5

13.t

he li

ne p

aral

lel t

o lin

e r

that

con

tain

s (1

,#1)

y!

x#

2

14.t

he li

ne p

erpe

ndic

ular

to

line

sth

at c

onta

ins

(0,0

)y

!%1 2% x

Wri

te a

n e

quat

ion

in

slo

pe-

inte

rcep

t fo

rm f

or t

he

lin

e th

at s

atis

fies

th

e gi

ven

con

dit

ion

s.

15.m

!6,

y-in

terc

ept

!#

216

.m!

#$5 3$ ,

y-in

terc

ept

!0

y!

6x#

2y

!#

%5 3% x

17.m

!#

1,co

ntai

ns (

0,#

6)18

.m!

4,co

ntai

ns (

2,5)

y!

#x

#6

y!

4x#

3

19.c

onta

ins

(2,0

) a

nd (

0,10

)20

.x-i

nter

cept

is #

2,y-

inte

rcep

t is

#1

y!

#5x

"10

y!

#%1 2% x

#1

x

y

O

r

s

t

u

©G

lenc

oe/M

cGra

w-H

ill14

6G

lenc

oe G

eom

etry

Wri

te a

n e

quat

ion

in

slo

pe-

inte

rcep

t fo

rm o

f th

e li

ne

hav

ing

the

give

n s

lop

e an

d

y-in

terc

ept.

1.m

:$2 3$ ,

y-in

terc

ept:

#10

2.m

:#$7 9$ ,

#0,#

$1 2$ $3.

m:4

.5,(

0,0.

25)

y!

%2 3% x#

10y

!#

%7 9% x#

%1 2%y

!4.

5x"

0.25

Wri

te e

quat

ion

s in

poi

nt-

slop

e fo

rm a

nd

slo

pe-

inte

rcep

t fo

rm o

f th

e li

ne

hav

ing

the

give

n s

lop

e an

d c

onta

inin

g th

e gi

ven

poi

nt.

4.m

:$3 2$ ,

(4,6

)5.

m:#

$6 5$ ,(#

5,#

2)

y#

6 !

%3 2% (x

#4)

,y!

%3 2% xy

"2

!#

%6 5% (x

"5)

,y!

#%6 5% x

#8

6.m

:0.5

,(7,

#3)

7.m

:#1.

3,(#

4,4)

y"

3 !

0.5(

x#

7),

y#

4 !

#1.

3(x

"4)

,y

!0.

5x#

6.5

y!

#1.

3x#

1.2

Wri

te a

n e

quat

ion

in

slo

pe-

inte

rcep

t fo

rm f

or e

ach

lin

e.

8.b

y!

#x

#5

9.c

y!

#%2 5% x

"4

10.p

aral

lel t

o lin

e b,

cont

ains

(3,

#2)

y!

#x

"1

11.p

erpe

ndic

ular

to

line

c,co

ntai

ns (#

2,#

4)y

!%5 2% x

"1

Wri

te a

n e

quat

ion

in

slo

pe-

inte

rcep

t fo

rm f

or t

he

lin

e th

at s

atis

fies

th

e gi

ven

con

dit

ion

s.

12.m

!#

$4 9$ ,y-

inte

rcep

t !

213

.m!

3,co

ntai

ns (

2,#

3)

y!

#%4 9% x

"2

y!

3x#

9

14.x

-int

erce

pt is

#6,

y-in

terc

ept

is 2

15.x

-int

erce

pt is

2,y

-int

erce

pt is

#5

y!

%1 3% x"

2y

!%5 2% x

#5

16.p

asse

s th

roug

h (2

,#4)

and

(5,

8)17

.con

tain

s (#

4,2)

and

(8,

#1)

y!

4x#

12y

!#

%1 4% x"

1

18.C

OM

MU

NIT

Y E

DU

CA

TIO

NA

loca

l com

mun

ity

cent

er o

ffer

s se

lf-d

efen

se c

lass

es f

orte

ens.

A $

25 e

nrol

lmen

t fe

e co

vers

sup

plie

s an

d m

ater

ials

and

ope

n cl

asse

s co

st $

10ea

ch.W

rite

an

equa

tion

to

repr

esen

t th

e to

tal c

ost

of x

self

-def

ense

cla

sses

at

the

com

mun

ity

cent

er.

C!

10x

"25

x

y

O

c

b

Pra

ctic

e (A

vera

ge)

Equ

atio

ns o

f Lin

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-4

3-4



An

swer

s

Rea

din

g t

o L

earn

Math

emati

csE

quat

ions

of L

ines

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-4

3-4

©G

lenc

oe/M

cGra

w-H

ill14

7G

lenc

oe G

eom

etry

Lesson 3-4

Pre-

Act

ivit

yH

ow c

an t

he

equ

atio

n o

f a

lin

e d

escr

ibe

the

cost

of

cell

ula

rte

lep

hon

e se

rvic

e?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 3-

4 at

the

top

of

page

145

in y

our

text

book

.If

the

rat

es f

or y

our

cellu

lar

phon

e pl

an a

re d

escr

ibed

by

the

equa

tion

inyo

ur t

extb

ook,

wha

t w

ill b

e th

e to

tal c

harg

e (e

xclu

ding

tax

es a

nd f

ees)

for

am

onth

in w

hich

you

use

50

min

utes

of

air

tim

e? $

23.4

5

Rea

din

g t

he

Less

on

1.Id

enti

fy w

hat

each

for

mul

a re

pres

ents

.

a.y

#y 1

!m

(x#

x 1)

poin

t-sl

ope

form

of

an e

quat

ion

b.m

!$y x2 2

# #

y x1 1$

slop

e of

a li

ne

c.y

!m

x"

bsl

ope-

inte

rcep

t fo

rm o

f an

equ

atio

n

2.W

rite

the

poi

nt-s

lope

for

m o

f th

e eq

uati

on f

or e

ach

line.

a.lin

e w

ith

slop

e #

$1 2$co

ntai

ning

(#2,

5)y

#5

!#

%1 2% (x

"2)

b.lin

e co

ntai

ning

(#4.

5,#

6.5)

and

par

alle

l to

a lin

e w

ith

slop

e 0.

5y

"6.

5 !

0.5(

x"

4.5)

3.W

hich

one

of

the

follo

win

g co

rrec

tly

desc

ribe

s th

e y-

inte

rcep

t of

a li

ne?

CA

.the

y-c

oord

inat

e of

the

poi

nt w

here

the

line

inte

rsec

ts t

he x

-axi

s

B.t

he x

-coo

rdin

ate

of t

he p

oint

whe

re t

he li

ne in

ters

ects

the

y-a

xis

C.t

he y

-coo

rdin

ate

of t

he p

oint

whe

re t

he li

ne c

ross

es t

he y

-axi

s

D.t

he x

-coo

rdin

ate

of t

he p

oint

whe

re t

he li

ne c

ross

es t

he x

-axi

s

E.t

he r

atio

of

the

chan

ge in

y-c

oord

inat

es t

o th

e ch

ange

in x

-coo

rdin

ates

4.F

ind

the

slop

e an

d y-

inte

rcep

t of

eac

h lin

e.

a.y

!2x

#7

slop

e !

2;y-

inte

rcep

t !

#7

b.x

"y

!8.

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Lesson 3-5

Pre-

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at t

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d th

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trod

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o L

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n 3-

5 at

the

top

of

page

151

in y

our

text

book

.

How

can

the

wor

kers

who

are

str

ipin

g th

e pa

rkin

g sp

aces

in a

par

king

lot

chec

k to

see

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he s

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the

spac

es a

re p

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ple

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sea

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uare

or

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r de

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lay

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enta

ry/s

uppl

emen

tary

/con

grue

nt),

then

the

line

s ar

e

para

llel.

e.If

tw

o co

plan

ar li

nes

are

cut

by a

tra

nsve

rsal

so

that

alt

erna

te in

teri

or a

ngle

s ar

e

cong

ruen

t,th

en t

he li

nes

are

(per

pend

icul

ar/p

aral

lel/s

kew

).

2.W

hich

of

the

follo

win

g co

ndit

ions

ver

ify

that

p|| q

?A

,C,F

,GA

.!6

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11 "

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6 an

d !

7 ar

e su

pple

men

tary

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15

G.!

7 an

d !

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re s

uppl

emen

tary

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.!4

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pin

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ou

Rem

emb

er

3.A

goo

d w

ay t

o re

mem

ber

som

ethi

ng n

ew is

to

draw

a p

ictu

re.H

ow c

an a

ske

tch

help

you

to r

emem

ber

the

Para

llel P

ostu

late

? S

ampl

e an

swer

:Dra

w a

line

with

a r

uler

or

stra

ight

edge

and

cho

ose

apo

int

not

on t

he li

ne.T

ry t

o dr

aw li

nes

thro

ugh

the

poin

t th

at a

re p

aral

lel

to t

he li

ne y

ou o

rigi

nally

dre

w.Y

ou w

ill s

ee t

hat

ther

e is

exa

ctly

one

way

to d

o th

is.

p

t

q r

12 9

10

1213

14

1516

11

34 6

57

8

para

llel

supp

lem

enta

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Th

e re

ason

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eces

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to

com

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te t

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foll

owin

g p

roof

are

scra

mbl

ed u

p b

elow

.To

com

ple

te t

he

pro

of,n

um

ber

the

reas

ons

to m

atch

th

e co

rres

pon

din

g st

atem

ents

.

Giv

en:C !

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init

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d "

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s.D

efin

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n of

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pend

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ines

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5.A!

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CP

CT

C8

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AB

D"

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DE

In a

pla

ne,i

f tw

o lin

es a

re c

ut b

y a

tran

sver

sal

so t

hat

a pa

ir o

f co

rres

pond

ing

angl

es is

cong

ruen

t,th

en t

he li

nes

are

para

llel.

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orem

7-5

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8.!

1 "

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en6

9.A!

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L7

AC

BE

D

5 31

42

6

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-5

3-5



An

swer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Perp

endi

cula

rs a

nd D

ista

nce

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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ATE

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PE

RIO

D__

___

3-6

3-6

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Lesson 3-6

Dis

tan

ce F

rom

a P

oin

t to

a L

ine

Whe

n a

poin

t is

no

t on

a li

ne,t

he d

ista

nce

from

the

poi

nt t

o th

e lin

e is

the

le

ngth

of

the

segm

ent

that

con

tain

s th

e po

int

and

ispe

rpen

dicu

lar

to t

he li

ne.

Dra

w t

he

segm

ent

that

rep

rese

nts

th

e d

ista

nce

from

Eto

AF

!"# .

Ext

end

AF

! "# .

Dra

w E!

G!⊥

AF

!"# .

E !G!

repr

esen

ts t

he d

ista

nce

from

Eto

AF

!"# .

Dra

w t

he

segm

ent

that

rep

rese

nts

th

e d

ista

nce

in

dic

ated

.

1.C

to A

B! "

#2.

Dto

AB

!"#

3.T

to R

S!"

#4.

Sto

PQ

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5.S

to Q

R!"

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Sto

RT

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T

S

R

T

S

P

Q

X

RT

S

PQ

X

URS

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X

A

CD

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A

C

BX

AF

G

BE

AF

BE

Pdi

stan

ce b

etw

een

M a

nd P

Q!"#

Q

M

Exam

ple

Exam

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Exercis

esExercis

es

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Dis

tan

ce B

etw

een

Par

alle

l Lin

esT

he d

ista

nce

betw

een

para

llel l

ines

is t

he le

ngth

of a

seg

men

t th

at h

as a

n en

dpoi

nt o

n ea

ch li

ne a

nd is

per

pend

icul

ar t

o th

em.P

aral

lel l

ines

are

ever

ywhe

re e

quid

ista

nt,

whi

ch m

eans

tha

t al

l suc

h pe

rpen

dicu

lar

segm

ents

hav

e th

esa

me

leng

th.

Fin

d t

he

dis

tan

ce b

etw

een

th

e p

aral

lel

lin

es !

and

mw

hos

eeq

uat

ion

s ar

e y

!2x

"1

and

y!

2x#

4,re

spec

tive

ly.

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Perp

endi

cula

rs a

nd D

ista

nce

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

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RIO

D__

___

3-6

3-6

Exam

ple

Exam

ple

Dra

w a

line

pth

roug

h (0

,1)

that

ispe

rpen

dicu

lar

to !

and

m.

Lin

e p

has

slop

e #

$1 2$an

d y-

inte

rcep

t 1.

An

equa

tion

of p

is y

!#

$1 2$ x"

1.T

he p

oint

of

inte

rsec

tion

for

pan

d !

is (

0,1)

.

x

y O

mp

!

( 0, 1

)

x

y O

m!

To f

ind

the

poin

t of

inte

rsec

tion

of p

and

m,

solv

e a

syst

em o

f eq

uati

ons.

Lin

e m

:y

!2x

#4

Lin

e p:

y!

#$1 2$ x

"1

Use

sub

stit

utio

n.

2x#

4 !

#$1 2$ x

"1

4x#

8 !

#x

"2

5x!

10x

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titu

te 2

for

xto

fin

d th

e y-

coor

dina

te.

y!

#$1 2$ x

"1

!#

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"1

!#

1 "

1 !

0

The

poi

nt o

f in

ters

ecti

on o

f pan

d m

is (

2,0)

.

Use

the

Dis

tanc

e Fo

rmul

a to

fin

d th

edi

stan

ce b

etw

een

(0,1

) an

d (2

,0).

d!

%(x

2#

!x 1

)2"

!(y

2#

!y 1

)2!

!%

(2 #

0!

)2"

(!

0 #

1!

)2 !!

%5!

The

dis

tanc

e be

twee

n !

and

mis

%5!

unit

s.

Exercis

esExercis

esF

ind

th

e d

ista

nce

bet

wee

n e

ach

pai

r of

par

alle

l li

nes

.

1.y

!8

2.y

!x

"3

3.y

!#

2xy

!#

3y

!x

#1

y!

#2x

#5

11%

8!%

5!



Skil

ls P

ract

ice

Perp

endi

cula

rs a

nd D

ista

nce

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

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RIO

D__

___

3-6

3-6

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etry

Lesson 3-6

Dra

w t

he

segm

ent

that

rep

rese

nts

th

e d

ista

nce

in

dic

ated

.

1.B

to A

C!"

#2.

Gto

EF

!"#

3.Q

to S

R!"

#

Con

stru

ct a

lin

e p

erp

end

icu

lar

to !

thro

ugh

K.T

hen

fin

d t

he

dis

tan

ce f

rom

Kto

!.

4.5.

%13!

3%2!

Fin

d t

he

dis

tan

ce b

etw

een

eac

h p

air

of p

aral

lel

lin

es.

6.y

!7

7.x

!#

68.

y!

3xy

!#

1x

!5

y!

3x"

10

811

%10!

9.y

!#

5x10

.y!

x"

911

.y!

#2x

"5

y!

#5x

"26

y!

x"

3y

!#

2x#

5

%26!

3%2!

2%5!x

y O

!

K

x

y

O

!

K

S

PQ

RD

EF

GA

B

C

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Dra

w t

he

segm

ent

that

rep

rese

nts

th

e d

ista

nce

in

dic

ated

.

1.O

to M

N!"

#2.

Ato

DC

!"#

3.T

to V

U!"

#

Con

stru

ct a

lin

e p

erp

end

icu

lar

to !

thro

ugh

B.T

hen

fin

d t

he

dis

tan

ce f

rom

Bto

!.

4.5.

4%2!

%13!

Fin

d t

he

dis

tan

ce b

etw

een

eac

h p

air

of p

aral

lel

lin

es.

6.y

!#

x7.

y!

2x"

78.

y!

3x"

12y

!#

x#

4y

!2x

#3

y!

3x#

18

2%2!

2%5!

3%10!

9.G

raph

the

line

y!

#x

"1.

Con

stru

ct a

per

pend

icul

ar

segm

ent

thro

ugh

the

poin

t at

(#2,

#3)

.The

n fi

nd t

he

dist

ance

fro

m t

he p

oint

to

the

line.

3 %2 !

10.C

AN

OEI

NG

Bro

nson

and

a f

rien

d ar

e go

ing

to c

arry

a c

anoe

acr

oss

a fl

at f

ield

to

the

bank

of

a st

raig

ht c

anal

.Des

crib

e th

e sh

orte

st p

ath

they

can

use

.S

ampl

e an

swer

:The

sho

rtes

t pa

th w

ould

be

a pe

rpen

dicu

lar

segm

ent

from

whe

re t

hey

are

to t

he b

ank

of t

he c

anal

.

( –2,

–3)

y !

#x

" 1

x

y

O

x

y O

!B

x

y O

! B

T

VW

SU

AB

CD

MN

O

Pra

ctic

e (A

vera

ge)

Perp

endi

cula

rs a

nd D

ista

nce

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-6

3-6



An

swer

s

Rea

din

g t

o L

earn

Math

emati

csPe

rpen

dicu

lars

and

Dis

tanc

e

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-6

3-6

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lenc

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9G

lenc

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eom

etry

Lesson 3-6

Pre-

Act

ivit

yH

ow d

oes

the

dis

tan

ce b

etw

een

par

alle

l li

nes

rel

ate

to h

angi

ng

new

sh

elve

s?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 3-

6 at

the

top

of

page

159

in y

our

text

book

.

Nam

e th

ree

exam

ples

of

situ

atio

ns in

hom

e co

nstr

ucti

on w

here

it w

ould

be

impo

rtan

t to

con

stru

ct p

aral

lel l

ines

.S

ampl

e an

swer

:opp

osite

wal

ls o

f a

room

,pla

nks

of h

ardw

ood

floor

ing,

tops

and

bot

tom

s of

cab

inet

s

Rea

din

g t

he

Less

on

1.F

ill in

the

bla

nk w

ith

a w

ord

or p

hras

e to

com

plet

e ea

ch s

ente

nce.

a.T

he d

ista

nce

from

a li

ne t

o a

poin

t no

t on

the

line

is t

he le

ngth

of

the

segm

ent

to t

he li

ne f

rom

the

poi

nt.

b.T

wo

copl

anar

line

s ar

e pa

ralle

l if

they

are

eve

ryw

here

.

c.In

a p

lane

,if

two

lines

are

bot

h eq

uidi

stan

t fr

om a

thi

rd li

ne,t

hen

the

two

lines

are

to

eac

h ot

her.

d.T

he d

ista

nce

betw

een

two

para

llel l

ines

mea

sure

d al

ong

a pe

rpen

dicu

lar

to t

he t

wo

lines

is a

lway

s .

e.To

mea

sure

the

dis

tanc

e be

twee

n tw

o pa

ralle

l lin

es,m

easu

re t

he d

ista

nce

betw

een

one

of t

he li

nes

and

any

poin

t on

the

.

2.O

n ea

ch f

igur

e,dr

aw t

he s

egm

ent

that

rep

rese

nts

the

dist

ance

indi

cate

d.

a.P

to !

b.D

to A

B!"

#

c.E

to F

G! "

#d.

Uto

RV

! "#

Hel

pin

g Y

ou

Rem

emb

er3.

A g

ood

way

to

rem

embe

r a

new

wor

d is

to

rela

te it

to

wor

ds t

hat

use

the

sam

e ro

ot.U

seyo

ur d

icti

onar

y to

fin

d th

e m

eani

ng o

f th

e L

atin

roo

t ae

quus

.Lis

t th

ree

wor

ds o

ther

tha

neq

ual a

nd e

quid

ista

nt t

hat

are

deri

ved

from

thi

s ro

ot a

nd g

ive

the

mea

ning

of

each

.S

ampl

e an

swer

:Aeq

uus

mea

ns e

ven,

fair,

or e

qual

.Equ

inox

mea

ns o

ne o

fth

e tw

o tim

es o

f yea

r w

hen

day

and

nigh

t are

of e

qual

leng

th. E

quity

mea

nsbe

ing

just

or

fair.

Equ

ival

entm

eans

bei

ng e

qual

in v

alue

or

mea

ning

.

TV

RS

U

E

FG

A

DC

B

P

!

othe

r lin

e

the

sam

e

para

llel

equi

dist

ant

perp

endi

cula

r

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ill16

0G

lenc

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eom

etry

Para

llelis

m in

Spa

ceIn

spa

ce g

eom

etry

,the

con

cept

of

para

llelis

m m

ust

be

exte

nded

to

incl

ude

two

plan

es a

nd a

line

and

a p

lane

.

Def

init

ion

:T

wo

plan

es a

re p

aral

lel i

f an

d on

ly if

the

y do

not

inte

rsec

t.D

efin

itio

n:

A li

ne a

nd a

pla

ne a

re p

aral

lel i

f an

d on

ly

if t

hey

do n

ot in

ters

ect.

Thu

s,in

spa

ce,t

wo

lines

can

be

inte

rsec

ting

,par

alle

l,or

sk

ew w

hile

tw

o pl

anes

or

a lin

e an

d a

plan

e ca

n on

ly b

ein

ters

ecti

ng o

r pa

ralle

l.In

the

fig

ure

at t

he r

ight

, t$

M,

t$P,

P|| H

,and

"an

d n

are

skew

.

The

fol

low

ing

five

sta

tem

ents

are

the

orem

s ab

out

para

llel p

lane

s.

Th

eore

m:

Tw

o pl

anes

per

pend

icul

ar t

o th

e sa

me

line

are

para

llel.

Th

eore

m:

Tw

o pl

anes

par

alle

l to

the

sam

e pl

ane

are

para

llel.

Th

eore

m:

A li

ne p

erpe

ndic

ular

to

one

of t

wo

para

llel p

lane

s is

pe

rpen

dicu

lar

to t

he o

ther

.T

heo

rem

:A

pla

ne p

erpe

ndic

ular

to

one

of t

wo

para

llel p

lane

s is

pe

rpen

dicu

lar

to t

he o

ther

.T

heo

rem

:If

tw

o pa

ralle

l pla

nes

each

inte

rsec

t a

thir

d pl

ane,

then

th

e tw

o lin

es o

f in

ters

ecti

on a

re p

aral

lel.

Use

th

e fi

gure

giv

en a

bove

for

Exe

rcis

es 1

–10.

Sta

te y

esor

no

to t

ell

wh

eth

er t

he

stat

emen

t is

tru

e.

1.M

&Pye

s2.

"&n

no3.

M&H

yes

4."

&Pye

s5.

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tye

s

6.n

&Hye

s7.

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&Hno

9.M

$t

yes

10.

t $H

yes

Mak

e a

smal

l sk

etch

to

show

th

at e

ach

sta

tem

ent

is f

alse

.

11.I

f tw

o lin

es a

re p

aral

lel t

o th

e sa

me

12.I

f tw

o pl

anes

are

par

alle

l,th

en a

ny li

ne

plan

e,th

en t

he li

nes

are

para

llel.

in o

ne p

lane

is p

aral

lel t

o an

y lin

e in

the

othe

r pl

ane.

13.I

f tw

o lin

es a

re p

aral

lel,

then

any

pla

ne14

.If

two

lines

are

par

alle

l,th

en a

ny p

lane

co

ntai

ning

one

of

the

lines

is p

aral

lel t

oco

ntai

ning

one

of

the

lines

is p

aral

lel t

o an

y pl

ane

cont

aini

ng t

he o

ther

line

.th

e ot

her

line.

t

!M

nP

H

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-6

3-6


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