LeSSon Ready to Go On? Skills Intervention 3-1x-x Lines ... Pairs of Angles ... D. Alternate...

Find these vocabulary words in the lesson and the Multilingual Glossary.

Identifying Types of Lines and PlanesIdentify each of the following.

A. Skew segments do not lie in the same ;

they are not and do not .

Name two segments in the figure that are skew.

B. Perpendicular segments intersect at a angle. Name a pair of perpendicular

segments in the figure.

C. Parallel lines are and do not . Name a pair of

parallel segments in the figure.

Classifying Pairs of AnglesGive an example of each angle pair.

A. Corresponding angles lie on the same of the

transversal and are on the same of the other

two lines. In the figure, /3 and / are corresponding angles.

B. Same-side interior angles lie on the side of the transversal

and are the other two lines.

In the figure, /1 and / are same-side interior angles.

C. Alternate exterior angles lie on sides of the transversal and are

the other two lines. In the figure, /5 and / are alternate exterior angles.

D. Alternate interior angles lie on sides of the transversal and are

the other two lines. In the figure, /3 and / are alternate interior angles.

Ready to Go On? Skills InterventionLines and Angles

Vocabulary

parallel lines perpendicular lines skew lines parallel planes

transversal corresponding angles alternate interior angles

alternate exterior angles same-side interior angles

O

D

N

C

P L M

E

B A

4 8

7 1

s r

t 5 3 2 6

Name Date Class

Copyright © by Holt, Rinehart and Winston. 29 Holt McDougal GeometryAll rights reserved.

LeSSon

x-x


LeSSon

3-1

CS10_G_MERG710365_C03SIa.indd 29 4/4/11 2:34:14 PM


Identifying Types of Lines and PlanesIdentify each of the following.

A. Skew segments do not lie in the same ;

they are not and do not .

Name two segments in the figure that are skew.

B. Perpendicular segments intersect at a angle. Name a pair of perpendicular

segments in the figure.

C. Parallel lines are and do not . Name a pair of

parallel segments in the figure.

Classifying Pairs of AnglesGive an example of each angle pair.

A. Corresponding angles lie on the same of the

transversal and are on the same of the other

two lines. In the figure, /3 and / are corresponding angles.

B. Same-side interior angles lie on the side of the transversal

and are the other two lines.

In the figure, /1 and / are same-side interior angles.

C. Alternate exterior angles lie on sides of the transversal and are

the other two lines. In the figure, /5 and / are alternate exterior angles.

D. Alternate interior angles lie on sides of the transversal and are

the other two lines. In the figure, /3 and / are alternate interior angles.

Ready to Go On? Skills InterventionLines and Angles

Vocabulary

parallel lines perpendicular lines skew lines parallel planes

transversal corresponding angles alternate interior angles

alternate exterior angles same-side interior angles

O

D

N

C

P L M

E

B A

4 8

7 1

s r

t 5 3 2 6

Name Date Class

between

plane

parallel intersect

Sample answer: _

MN and _

AB

right

Sample answer: _

AB and _

LA

coplanar intersect

Sample answer: _

MN and _

BC

side

side

8

same

between

2

opposite outside

7

opposite

1


LeSSon

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LeSSon

3-1

CS10_G_MERG710365_C03SIa.indd 29 4/4/11 2:34:14 PM

Using the Corresponding Angles PostulateFind m/RST.

Since two lines in the figure are and cut

by a transversal, the pairs of corresponding

are .

Write an equation relating the measures of the given angles.

(5x 1 27)8 5

Solve the equation. x 5

To find m/RST, the value of x into the expression .

Find the measure of /RST.

Finding Angle MeasuresFind each angle measure.

A. m/DEC

Since the two labeled angles are on sides

of the transversal and are the other

two lines, they are angles.

Since the lines in the figure are parallel, the labeled angles are .

Write an equation relating the measures of the angles.


To find m/DEC, the value of x into the expression

.

Find the measure of /DEC.

B. m/DEF

m/DEC and m/DEF form a so the sum of their

measures is 8.

Subtract from to find m/DEF.

m/DEF 5 2 107 5

Ready To Go On? Skills InterventionAngles Formed by Parallel Lines and Transversals

R (5x + 27)°

(8x + 6)°S T

C

D

F E

(20x + 7)°

(24x – 13)°

Name Date Class


Lesson

x-xLesson

3-2


CS10_G_MERG710365_C03SIb.indd 30 4/4/11 2:20:16 PM

Using the Corresponding Angles PostulateFind m/RST.

Since two lines in the figure are and cut

by a transversal, the pairs of corresponding

are .

Write an equation relating the measures of the given angles.

(5x 1 27)8 5


To find m/RST, the value of x into the expression .

Find the measure of /RST.

Finding Angle MeasuresFind each angle measure.

A. m/DEC

Since the two labeled angles are on sides

of the transversal and are the other

two lines, they are angles.

Since the lines in the figure are parallel, the labeled angles are .

Write an equation relating the measures of the angles.


To find m/DEC, the value of x into the expression

.

Find the measure of /DEC.

B. m/DEF

m/DEC and m/DEF form a so the sum of their

measures is 8.

Subtract from to find m/DEF.

m/DEF 5 2 107 5

Ready To Go On? Skills InterventionAngles Formed by Parallel Lines and Transversals

R (5x + 27)°

(8x + 6)°S T

C

D

F E

(20x + 7)°

(24x – 13)°

Name Date Class

180

738180

180107

linear pair

1078

24x 2 13 or 20x 1 7

substitute

5

20x 1 7 5 24x 2 13

congruent

alternate interior

inside

opposite

628

(8x 1 6) or (5x 1 27)substitute

7

(8x 1 6)8

congruent

angles

parallel


Lesson

x-xLesson

3-2


CS10_G_MERG710365_C03SIb.indd 30 4/4/11 2:20:16 PM

Using the Converse of the Corresponding Angles PostulateUse the given information to show that p i q.

Given: m/2 5 (12x 2 25)8 and m/8 5 (9x 1 2)8; x 5 9

Substitute the value of x into each expression.

m/2 5 12( ) 2 25 5 ; m/8 5 9( ) 1 2 5

Does m/2 5 m/8?

Since m/2 m/8, by the Property of Congruence.

Since the angles formed by two coplanar lines cut by a

transversal are , p i q.

Determining Whether Lines are ParallelUsing the given information and the diagram to show that p i q.

A. /1 /7

What type of angles are /1 and /7?

If two coplanar lines are cut by a transversal so that a pair of

angles are congruent, then the two lines are . Since /1 /7, i .

B. m/3 5 m/5


Since m/3 5 m/5, . Since , p i q by the Converse of

the .

Proving Lines ParallelWrite a paragraph proof to show that

_ RS i _

QT .Given: m/R 5 1318, m/Q 5 498Prove:

_ RS i _

QT

Since m/R 5 1318 and m/Q 5 498, /R and /Q are angles by

the definition of angles. Since /R and /Q lie on the same side

of two coplanar lines cut by a transversal, they are angles.

By the Converse of the Angles Theorem, when same-side angles

are , then the two lines are parallel, so _

RS i _

QT .

Ready To Go On? Skills InterventionProving Lines Parallel

p q

1 2 3

4 5 8 7

6

R

49°

131°S

T Q

Name Date Class

p q

1 2 3

4 5 8 7

6


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LeSSon

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CS10_G_MERG710365_C03SIc.indd 31 4/4/11 2:21:28 PM

Using the Converse of the Corresponding Angles PostulateUse the given information to show that p i q.

Given: m/2 5 (12x 2 25)8 and m/8 5 (9x 1 2)8; x 5 9

Substitute the value of x into each expression.

m/2 5 12( ) 2 25 5 ; m/8 5 9( ) 1 2 5

Does m/2 5 m/8?

Since m/2 m/8, by the Property of Congruence.

Since the angles formed by two coplanar lines cut by a

transversal are , p i q.

Determining Whether Lines are ParallelUsing the given information and the diagram to show that p i q.

A. /1 /7


If two coplanar lines are cut by a transversal so that a pair of

angles are congruent, then the two lines are . Since /1 /7, i .

B. m/3 5 m/5


Since m/3 5 m/5, . Since , p i q by the Converse of

the .

Proving Lines ParallelWrite a paragraph proof to show that

_ RS i _

QT .Given: m/R 5 1318, m/Q 5 498Prove:

_ RS i _

QT

Since m/R 5 1318 and m/Q 5 498, /R and /Q are angles by

the definition of angles. Since /R and /Q lie on the same side

of two coplanar lines cut by a transversal, they are angles.

By the Converse of the Angles Theorem, when same-side angles

are , then the two lines are parallel, so _

RS i _

QT .

Ready To Go On? Skills InterventionProving Lines Parallel

p q

1 2 3

4 5 8 7

6

R

49°

131°S

T Q

Name Date Class

p q

1 2 3

4 5 8 7

6

supplementary

supplementary

/5/3/5/3

9 838838

Same-Side

same-side interior

supplementary

Alternate Interior Angles Theorem

Alternate interior angles

qpparallel

alternate exterior

Alternate exterior angles

congruent

corresponding

/8/25

Yes

9


LeSSon

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LeSSon

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CS10_G_MERG710365_C03SIc.indd 31 4/4/11 2:21:28 PM

Ready To Go On? Skills InterventionPerpendicular Lines


Proving Properties of LinesWrite a two-column proof.Given: m/1 5 m/2, b i cProve: d ' b

Plan your proof:

Step 1: Write the given information in the two-column proof.

Step 2: Since it is given that m/1 5 m/2, you know that /1 is

to /2 by the definition of .

Put this information in the two-column proof.

Step 3: If two intersecting lines form a linear pair of angles, then

the lines are . So you know that d c.

Put this information in Step 3 of the two-column proof.

Step 4: It is given that b i c. In Step 3, you proved that d c. You can conclude

that d ' b because of the Theorem.

Complete Step 4 of the two-column proof.

Statements Reasons

1. 1. Given

2. 2.

3. 3.

4. d ' b 4.

Vocabulary

perpendicular bisector distance from a point to a line

d

b

c 2 1

Name Date Class


Lesson

x-x


Lesson

3-4

CS10_G_MERG710365_C03SId.indd 32 4/4/11 2:24:15 PM

Ready To Go On? Skills InterventionPerpendicular Lines


Proving Properties of LinesWrite a two-column proof.Given: m/1 5 m/2, b i cProve: d ' b

Plan your proof:

Step 1: Write the given information in the two-column proof.

Step 2: Since it is given that m/1 5 m/2, you know that /1 is

to /2 by the definition of .

Put this information in the two-column proof.

Step 3: If two intersecting lines form a linear pair of angles, then

the lines are . So you know that d c.

Put this information in Step 3 of the two-column proof.

Step 4: It is given that b i c. In Step 3, you proved that d c. You can conclude

that d ' b because of the Theorem.

Complete Step 4 of the two-column proof.

Statements Reasons

1. 1. Given

2. 2.

3. 3.

4. d ' b 4.

Vocabulary

perpendicular bisector distance from a point to a line

d

b

c 2 1

Name Date Class

Perpendicular Transversal

'

'perpendicular

congruent

congruent angles

congruent

Def. of > /s

Perpendicular Transversal Theorem

Two intersecting lines form a linear pair of > /s, lines are '.

m/1 5 m/2, b i c

d ' c

/1 > /2


Lesson

x-x


Lesson

3-4

CS10_G_MERG710365_C03SId.indd 32 4/4/11 2:24:15 PM

Ready to Go On? Quiz

Lines and Angles

Identify each of the following.

1. a pair of parallel segments

2. a pair of perpendicular segments

3. a pair of skew segments 4. a pair of parallel planes

Give an example of each angle pair.

5. same-side interior angles

6. alternate exterior angles

7. corresponding angles 8. alternate interior angles

Angles Formed by Parallel Lines and Transversals

Find each angle measure.

9. 10.

58°

x°

(9x – 8)°

(7x + 6)°

11.

(15x – 40)°(11x + 4)°

D

B A

E

F

C

1 2 3 4 5 8 7 6

Name Date Class


Section

xAx-x

x-x


Section

3A3-1

3-2

CS10_G_MERG710365_C03QZa.indd 33 3/19/11 7:48:41 PM


Lines and Angles

Identify each of the following.

1. a pair of parallel segments

2. a pair of perpendicular segments

3. a pair of skew segments 4. a pair of parallel planes

Give an example of each angle pair.

5. same-side interior angles

6. alternate exterior angles

7. corresponding angles 8. alternate interior angles

Angles Formed by Parallel Lines and Transversals

Find each angle measure.

9. 10.

58°

x°

(9x – 8)°

(7x + 6)°

11.

(15x – 40)°(11x + 4)°

D

B A

E

F

C

1 2 3 4 5 8 7 6

Name Date Class

/7 and /5; /2 and /4;

/1 and /5 or /8 and /4

/2 and /3 or /7 and /6

_

DE and _

EF

_

DE and _

BA

/1 and /3; /8 and /6 /2 and /6 or /7 and /3

_

EA and _

BC DEF and BAC

588 558

1258

Sample answers given for 1–3.


Section

xAx-x

x-x


Section

3A3-1

3-2


Name Date Class

Proving Lines ParallelUse the given information and the theorems and postulates you have learned to show that a i b.

12. m/3 1 m/6 5 1808

13. /1 /7

14. m/4 5 (7x 2 1)8, m/8 5 (5x 1 31)8, x 5 16

15. m/7 5 m/3

16. Write a paragraph proof to show that _

DC i _

AB .

Perpendicular Lines

17. Complete the two-column proof below.

Given: t ' m, m/1 5 m/2

Prove: n ' t

Statements Reasons

1. t ' m, m/1 5 m/2 1. Given

2. /1 /2 2.

3. 3. Converse of the Alternate Exterior Angles Theorem

4. n ' t 4.

Ready to Go On? Quiz continued

1

a

2

b

8 7 6 5

4 3

C D

A B

72°

108°1

a

2

b

8 7 6

n

m

t

1

2

5 4 3


Section

xAx-x

x-x


Section

3A3-3

3-4


Name Date Class

Proving Lines ParallelUse the given information and the theorems and postulates you have learned to show that a i b.

12. m/3 1 m/6 5 1808

13. /1 /7

14. m/4 5 (7x 2 1)8, m/8 5 (5x 1 31)8, x 5 16

15. m/7 5 m/3

16. Write a paragraph proof to show that _

DC i _

AB .

Perpendicular Lines

17. Complete the two-column proof below.

Given: t ' m, m/1 5 m/2

Prove: n ' t

Statements Reasons

1. t ' m, m/1 5 m/2 1. Given

2. /1 /2 2.

3. 3. Converse of the Alternate Exterior Angles Theorem

4. n ' t 4.


1

a

2

b

8 7 6 5

4 3

C D

A B

72°

108°1

a

2

b

8 7 6

n

m

t

1

2

5 4 3

converse of the Same-Side

interior Angles theorem

converse of the corresponding Angles Postulate

m/4 5 m/8 5 1118,

so /4 > /1; Alternate exterior Angles theorem

/7 > /3 by the def. of >/s;

Alternate interior Angles theorem

Sample answer: 1088 1 728 5 1808; so /C and

/B are supplementary by the definition of

supplementary angles. Since /C and /B are on

the same side of the transversal and between the

other two lines; they are same-side interior

angles. When same-side interior angles are

supplementary the lines are parallel.

Perpendicular transversal theorem

Def. > /s

m i n


Section

xAx-x

x-x


Section

3A3-3

3-4


Finding Angle MeasuresUse the figure at the right and the given information to answer the questions below. s i t, s i r, l i m, n i m

m/1 5 (7x)8

m/2 5 (4x 1 18)8

m/3 5 (11a 1 10b)8

m/4 5 (6a 1 18b)8

m/5 5 (3y )8

m/6 5 (5a 1 2)8

m/7 5 (28b 2 5)8

1. Find the value of x. 2. Find m/1. 3. Find m/2.

4. How are /1 and /3 related? 5. What is m/3?

6. What is m/4? 7. What is the value of a? 8. What is the value of b?

9. Find m/5. 10. Find the value of y.

11. Is n i m? Explain your answer.

12. Write a paragraph proof to show that s i r.

Name Date Class

Ready to Go On? Enrichment

m

n

r s t

6 1 3

4 2

5

7


SeCtioN

xA


SeCtioN

3A

CS10_G_MERG710365_C03ENa.indd 35 3/19/11 4:34:40 AM

Finding Angle MeasuresUse the figure at the right and the given information to answer the questions below. s i t, s i r, l i m, n i m

m/1 5 (7x)8

m/2 5 (4x 1 18)8

m/3 5 (11a 1 10b)8

m/4 5 (6a 1 18b)8

m/5 5 (3y )8

m/6 5 (5a 1 2)8

m/7 5 (28b 2 5)8

1. Find the value of x. 2. Find m/1. 3. Find m/2.

4. How are /1 and /3 related? 5. What is m/3?

6. What is m/4? 7. What is the value of a? 8. What is the value of b?

9. Find m/5. 10. Find the value of y.

11. Is n i m? Explain your answer.

12. Write a paragraph proof to show that s i r.

Name Date Class


m

n

r s t

6 1 3

4 2

5

7

Corresponding Angles Postulate s i r

1388 8 5

6 428 428

definition of congruent angles; /6 > /1. By the Converse of the

Sample answer: m/6 5 5(8) 1 2 5 428; and m/6 5 m/1. By the

Sample answer: No, because m/7 5 28(5) 2 5 5 1358, and m/4 m/7.

428 14

Same-side interior angles 1388


SeCtioN

xA


SeCtioN

3A

CS10_G_MERG710365_C03ENa.indd 35 3/19/11 4:34:40 AM

Name Date Class


Finding the Slope of a LineUse the slope formula to determine the slope of each line.

A. ‹

__ › AB

What is the slope formula? m 5 y 2 2

_________ 2 x 1

What are the coordinates of A? of B?

Substitute the coordinates of A and B into the slope formula to find the slope of

‹

__ › AB .

m 5 3 2 _________

2 6 5 2 _____

B. ‹

__ › BD

What are the coordinates of D?

Substitute the coordinates of B and D into the slope formula to find the slope of ‹

__ › BD .

m 5 23 2

__________ 2 1

5 2 ______

The slope is . What kind of line is ‹

__ › BD ?

Determining Whether Lines are Parallel, Perpendicular, or Neither ‹

___ › LM passes through L(4, 2) and M(0, 24), and

‹

___ › XY passes

through X(22, 5) and Y(2, 21). Use slopes to determine whether the lines are parallel, perpendicular, or neither.

Graph the coordinates and draw each line on the grid at the right. Find the slope of each line by substituting the coordinates into the slope formula.

Slope of ‹

__ › LM 5 y

2 2 y 1 _______ x 2 2 x 1 5 24 2 __________

0 2 5

Slope of ‹

__ › XY 5 2 5 ___________

2 5 _____ 5 _____

Do the lines have the same slope? Are they parallel?

Is the product of the slopes 21? Are the lines perpendicular?

The lines are neither nor .

Ready to Go On? Skills InterventionSlopes of Lines

Vocabulary

rise run slope

A

D C

B

y

2 4 6 –2 –4 O

2

–4

x

2 4 –2 –4 O

2

–2

4

–4


LessoN

x-x


LessoN

3-5

CS10_G_MERG710365_C03SIe.indd 36 4/4/11 2:35:24 PM

Name Date Class


Finding the Slope of a LineUse the slope formula to determine the slope of each line.

A. ‹

__ › AB

What is the slope formula? m 5 y 2 2

_________ 2 x 1

What are the coordinates of A? of B?

Substitute the coordinates of A and B into the slope formula to find the slope of

‹

__ › AB .

m 5 3 2 _________

2 6 5 2

_____

B. ‹

__ › BD

What are the coordinates of D?

Substitute the coordinates of B and D into the slope formula to find the slope of ‹

__ › BD .

m 5 23 2

__________ 2 1

5 2 ______

The slope is . What kind of line is ‹

__ › BD ?

Determining Whether Lines are Parallel, Perpendicular, or Neither ‹

___ › LM passes through L(4, 2) and M(0, 24), and

‹

___ › XY passes

through X(22, 5) and Y(2, 21). Use slopes to determine whether the lines are parallel, perpendicular, or neither.

Graph the coordinates and draw each line on the grid at the right. Find the slope of each line by substituting the coordinates into the slope formula.

Slope of ‹

__ › LM 5 y

2 2 y 1 _______ x 2 2 x 1 5 24 2

__________

0 2 5

Slope of ‹

__ › XY 5 2 5 ___________

2 5

_____

5 _____

Do the lines have the same slope? Are they parallel?

Is the product of the slopes 21? Are the lines perpendicular?

The lines are neither nor .

Ready to Go On? Skills InterventionSlopes of Lines

Vocabulary

rise run slope

A

D C

B

y

2 4 6 –2 –4 O

2

–4

x

2 4 –2 –4 O

2

–2

4

–4

2324

26

42

22221

31

60

11

25

y 1 x 2

perpendicularparallel

NoNo

NoNo

Verticalundefined

(1, 23)

(1, 3)(6, 1)

6 __ 4 or 3 __

2


LessoN

x-x


LessoN

3-5

CS10_G_MERG710365_C03SIe.indd 36 4/4/11 2:35:24 PM


Writing Equations of Lines

A. Write the equation of the line with slope 3 through (21, 4) in point-slope form.

What is point-slope form? y 2 y 1 5

Substitute for m, for x 1 and for y 1 :

B. Write the equation of the line through points (26, 2) and (3, 24) in slope-intercept form.

What is slope-intercept form? y 5

Substitute 22 ___ 3 for m, 26 for x, and 2 for y, and then simplify to find b.

5 _____ (26) 1 b

b 5 Write the equation in slope-intercept form.

Graphing LinesGraph the line. y 1 2 5 2 1 __ 2 (x 2 3)

The equation is given in form. The slope of the line

is . The line goes through the point . Plot the

point and then rise and run to find another point. Draw a line connecting the two points.

Classifying Pairs of LinesDetermine whether the lines are parallel, intersect, or coincide.y 5 3 __

2 x 1 4 and 3x 2 2y 5 6

The slope of the first line is and the y-intercept is . Solve the second equation for y to rewrite it in slope-intercept form.

The slope of the second line is and the y-intercept is .

The slopes lines are and the y-intercepts are . The lines

are .

Name Date Class

Ready to Go On? Skills InterventionLines in the Coordinate Plane

Vocabulary

point-slope form slope-intercept form

2 4 –2 –4 O

2

–2

4

–4

Use the slope formula to find the slope. m 5 y 2 2

_________ 2 x 1

5 24 2 ___________ 2 (26)

5 2 ______

5 2

______


LeSSon

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LeSSon

3-6

CS10_G_MERG710365_C03SIf.indd 37 4/4/11 2:36:19 PM


Writing Equations of Lines

A. Write the equation of the line with slope 3 through (21, 4) in point-slope form.

What is point-slope form? y 2 y 1 5

Substitute for m, for x 1 and for y 1 :

B. Write the equation of the line through points (26, 2) and (3, 24) in slope-intercept form.

What is slope-intercept form? y 5

Substitute 22 ___ 3 for m, 26 for x, and 2 for y, and then simplify to find b.

5 _____

(26) 1 b

b 5 Write the equation in slope-intercept form.

Graphing LinesGraph the line. y 1 2 5 2 1 __ 2 (x 2 3)

The equation is given in form. The slope of the line

is . The line goes through the point . Plot the

point and then rise and run to find another point. Draw a line connecting the two points.

Classifying Pairs of LinesDetermine whether the lines are parallel, intersect, or coincide.y 5 3 __

2 x 1 4 and 3x 2 2y 5 6

The slope of the first line is

and the y-intercept is . Solve the second equation for y to rewrite it in slope-intercept form.

The slope of the second line is

and the y-intercept is .

The slopes lines are and the y-intercepts are . The lines

are .

Name Date Class

Ready to Go On? Skills InterventionLines in the Coordinate Plane

Vocabulary

point-slope form slope-intercept form

2 4 –2 –4 O

2

–2

4

–4

Use the slope formula to find the slope. m 5 y 2 2

_________ 2 x 1

5 24 2 ___________ 2 (26)

5 2 ______

5 2

______

Sample answers:

y 1

x 2 2

369

23

22

223

2

paralleldifferentthe same

23

4

mx 1 b

y 2 4 5 3(x 1 1)4213

m(x 2 x 1 )

221(3, 22)

point-slope

y 5 3 __ 2 x 2 3

3 _ 2

3 _ 2

1 __ 2

y 5 22 ___ 3 x 1 (22)

2


LeSSon

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CS10_G_MERG710365_C03SIf.indd 37 4/4/11 2:36:19 PM

Name Date Class

Slopes of Lines Use the slope formula to determine the slope of each line.

1. ‹

__ › AD

2. ‹

__ › AB

3. ‹

__ › AC

4. ‹

__ › DB

Find the slope of the line through the given points.

5. R(4, 7) and S(22, 0) 6. C(0, 24) and D(5, 9)

7. H(3, 5) and I(24, 2) 8. S(26, 1) and T(3, 26)

Graph each pair of lines and use their slopes to determine if they are parallel, perpendicular, or neither.

9. ‹

___ › CD and

‹

__ › AB for A(21, 0), B(1, 5), 10.

‹

__ › LM and

‹

___ › MN for L(23, 2), M(21, 5),

C(4, 5), and D(22, 4) N(2, 3), and P(1, 25)

2 4 –2 –4 O

2

–2

4

–4

2 4 –2 –4 O

2

–2

4

–4

11. ‹ __ › PR and ‹ __ › PS for P(2, 21), Q(2, 1), 12. ‹ ___ › GH and ‹ __ › FJ for F(23, 2), G(22, 5) R(23, 1), and S(22, 22) H(2, 4), and J(2, 1)

4 –2 –4 O

2

–2

4

–4

2 4 –2 –4 O

2

–2

4

–4


x

y

2 4 6 –2 O

2

–2

6

–4 B

D

C

A


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CS10_G_MERG710365_C03QZb.indd 38 3/19/11 7:54:25 PM

Name Date Class

Slopes of Lines Use the slope formula to determine the slope of each line.

1. ‹

__ › AD

2. ‹

__ › AB

3. ‹

__ › AC

4. ‹

__ › DB

Find the slope of the line through the given points.

5. R(4, 7) and S(22, 0) 6. C(0, 24) and D(5, 9)

7. H(3, 5) and I(24, 2) 8. S(26, 1) and T(3, 26)

Graph each pair of lines and use their slopes to determine if they are parallel, perpendicular, or neither.

9. ‹

___ › CD and

‹

__ › AB for A(21, 0), B(1, 5), 10.

‹

__ › LM and

‹

___ › MN for L(23, 2), M(21, 5),

C(4, 5), and D(22, 4) N(2, 3), and P(1, 25)

2 4 –2 –4 O

2

–2

4

–4

2 4 –2 –4 O

2

–2

4

–4

11. ‹ __ › PR and ‹ __ › PS for P(2, 21), Q(2, 1), 12. ‹ ___ › GH and ‹ __ › FJ for F(23, 2), G(22, 5) R(23, 1), and S(22, 22) H(2, 4), and J(2, 1)

4 –2 –4 O

2

–2

4

–4

2 4 –2 –4 O

2

–2

4

–4


x

y

2 4 6 –2 O

2

–2

6

–4 B

D

C

A

Neither Perpendicular

2 7 __ 9

2 1 __ 2

7 __ 2

2 __ 4 or 1 __

2

2 10 ___ 4 or 2 5 __ 2

7 __ 6 13 ___

5

3 __ 7

Neither ParallelCopyright © by Holt, Rinehart and Winston. 38 Holt McDougal GeometryAll rights reserved.

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3B3-5


Lines in the Coordinate PlaneWrite the equation of each line in the given form.

13. the line through (23, 21) and (3, 23) in slope-intercept form

14. the line through (6, 22) with slope 2 3 __ 4 in point-slope form

15. the line with y-intercept 23 through the point (2, 5) in point-slope form

16. the line with x-intercept 24 and y-intercept 2 in slope-intercept form

Graph each line.

17. y 5 3x 2 1 18. y 2 1 5 3 __ 5 (x 1 2) 19. y 5 25

2 4 –2 –4

2

4

–4

2 4 –2 –4 O –2

4

–4

2 4 –2 –4 O

2

–2

4

–4

Write the equation of each line.

20. 21. 22.

x

y

2 –2 –4 O

2

–2

4

–4

x

y

2 –2 –4 O

2

–2

4

x

y

2 4 –2 –4 O

2

–2

4

–4

Determine whether the lines are parallel, intersect, or coincide.

23. y 5 2 1 __ 5 x 1 2 24. 2x 1 3y 5 9 25. y 5 5x 2 3

x 1 5y 5 10 y 5 2 __ 3 x 2 1 y 5 5x 1 1

Name Date Class



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3B3-6


Lines in the Coordinate PlaneWrite the equation of each line in the given form.

13. the line through (23, 21) and (3, 23) in slope-intercept form

14. the line through (6, 22) with slope 2 3 __ 4 in point-slope form

15. the line with y-intercept 23 through the point (2, 5) in point-slope form

16. the line with x-intercept 24 and y-intercept 2 in slope-intercept form

Graph each line.

17. y 5 3x 2 1 18. y 2 1 5 3 __ 5 (x 1 2) 19. y 5 25

2 4 –2 –4

2

4

–4

2 4 –2 –4 O –2

4

–4

2 4 –2 –4 O

2

–2

4

–4

Write the equation of each line.

20. 21. 22.

x

y

2 –2 –4 O

2

–2

4

–4

x

y

2 –2 –4 O

2

–2

4

x

y

2 4 –2 –4 O

2

–2

4

–4

Determine whether the lines are parallel, intersect, or coincide.

23. y 5 2 1 __ 5 x 1 2 24. 2x 1 3y 5 9 25. y 5 5x 2 3

x 1 5y 5 10 y 5 2 __ 3 x 2 1 y 5 5x 1 1

Name Date Class


y 5 2 1 __ 3 x 2 2

y 1 2 5 2 3 __ 4 (x 2 6)

y 2 5 5 4(x 2 2)

y 5 1 __ 2 x 1 2

coincide intersect Parallel

x 5 4 y 5 3 __ 5 x 2 3 y 5 23


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Name Date Class

Slopes and Lengths of Segments

Quadrilateral ABCD has vertices A(25, 3), B(21, 4), C(5, 23) and D(24, 21).

1. Sketch and label the quadrilateral using the grid at the right.

2. Find the slopes of _

AC and _

BD .

3. How are the segments related?

Quadrilateral PQRS has vertices P(2, 3), Q(2, 22), R(22, 25), S(22, 0). Use the information to answer the following questions:


Find the length of each segment.

5. Find PQ. 6. Find QR.

7. Find RS. 8. Find PS.

9. What can you conclude about the side lengths of the quadrilateral?

10. What is the slope of _

PR ? 11. What is the slope of _

QS ?

12. What can you conclude about the diagonals of the quadrilateral?

13. Is the quadrilateral a square? Explain your answer.

14. A triangle has vertices L(2, 8), M(5, 9), and N(4, 2). Write a paragraph proof to show that triangle LMN is a right triangle.


2 4 –2 –4 O

2

–2

4

–4

2 4 –2 –4 O

2

–2

4

–4


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3B

CS10_G_MERG710365_C03ENb.indd 40 3/19/11 4:35:43 AM

Name Date Class

Slopes and Lengths of Segments

Quadrilateral ABCD has vertices A(25, 3), B(21, 4), C(5, 23) and D(24, 21).


2. Find the slopes of _

AC and _

BD .

3. How are the segments related?

Quadrilateral PQRS has vertices P(2, 3), Q(2, 22), R(22, 25), S(22, 0). Use the information to answer the following questions:


Find the length of each segment.

5. Find PQ. 6. Find QR.

7. Find RS. 8. Find PS.

9. What can you conclude about the side lengths of the quadrilateral?

10. What is the slope of _

PR ? 11. What is the slope of _

QS ?

12. What can you conclude about the diagonals of the quadrilateral?

13. Is the quadrilateral a square? Explain your answer.

14. A triangle has vertices L(2, 8), M(5, 9), and N(4, 2). Write a paragraph proof to show that triangle LMN is a right triangle.


2 4 –2 –4 O

2

–2

4

–4

2 4 –2 –4 O

2

–2

4

–4

• •• •

• •

• •

• •

• •

• •

• •

definition, a right triangle is a triangle that has one right angle.

the segments are perpendicular. Perpendicular lines form right angles, and by

right angles.

They are congruent.

_

LN is 23. The product of the slopes is 21, so by the Perpendicular Lines Theorem,

Sample answer: Using the slope formula; the slope of _

LM is 1 __ 3 and the slope of

No; the sides do not meet at

They are perpendicular.

They are perpendicular.

2

5 units 5 units

5 units 5 units

2 3 __ 5 and 5 __ 3

2 1 __ 2


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CS10_G_MERG710365_C03ENb.indd 40 3/19/11 4:35:43 AM

LeSSon Ready to Go On? Skills Intervention 3-1x-x Lines ... Pairs of Angles ... D. Alternate...

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Transcript of LeSSon Ready to Go On? Skills Intervention 3-1x-x Lines ... Pairs of Angles ... D. Alternate...