section ready to go on? skills intervention 3a 3-1 lines ... · pdf file3-1 lines and angles...

Find these vocabulary words in Lesson 3-1 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.

3AReady to Go On? Skills Intervention3-1 Lines 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

PL M

E

BA

48

71

sr

t532 6

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

Name Date Class

SECTION

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 � 27)� �

Solve the equation. x �

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.

Solve the equation. x �

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 �.

Subtract from to find m�DEF.

m�DEF � � 107 �

3AReady To Go On? Skills Intervention3-2 Angles Formed by Parallel Lines and Transversals

R (5x + 27)°

(8x + 6)°ST

C

D

FE

(20x + 7)°

(24x – 13)°

Name Date Class


SECTION

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

Given: m�2 � (12x � 25)� and m�8 � (9x � 2)�; x � 9

Substitute the value of x into each expression.

m�2 � 12( ) � 25 � ; m�8 � 9( ) � 2 �

Does m�2 � 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 � q.

Determining Whether Lines are ParallelUsing the given information and the diagram to show that p � 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, � .

B. m�3 � m�5

What type of angles are �3 and �5?

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

the .

Proving Lines ParallelWrite a paragraph proof to show that

_ RS � _

QT .Given: m�R � 131�, m�Q � 49�Prove:

_ RS � _

QT

Since m�R � 131� and m�Q � 49�, �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 � _

QT .

3AReady To Go On? Skills Intervention3-3 Proving Lines Parallel

pq

12 3

4 58 7

6

49°

131°S

TQ


Name Date Class

SECTION

pq

12 3

4 58 7

6

Ready To Go On? Skills Intervention3-4 Perpendicular Lines


Proving Properties of LinesWrite a two-column proof.Given: m�1 � m�2, b � 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 � 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 � 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.

3A

Vocabulary

perpendicular bisector distance from a point to a line

d

b

c21

Name Date Class


SECTION

Ready to Go On? Quiz

3-1 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

3-2 Angles Formed by Parallel Lines and Transversals

Find each angle measure.

9. 10.

58°

x°

(9x – 8)°

(7x + 6)°

11.

(15x – 40)°(11x + 4)°

3A

D

BA

E

F

C

1 2 3 458 7 6


Name Date Class

SECTION

Name Date Class


3-3 Proving Lines ParallelUse the given information and the theorems and postulates you have learned to show that a � b.

12. m�3 � m�6 � 180�

13. �1 � �7

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

15. m�7 � m�3

16. Write a paragraph proof to show that _

DC � _

AB .

3-4 Perpendicular Lines

17. Complete the two-column proof below.

Given: t � m, m�1 � m�2

Prove: n � t

Statements Reasons

1. t � m, m�1 � 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

3ASECTION

a

2

b

876 5

4 3

CD

A B

72°

108°

n

m

t

1

2

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

m�1 � (7x)�

m�2 � (4x � 18)�

m�3 � (11a � 10b)�

m�4 � (6a � 18b)�

m�5 � (3y )�

m�6 � (5a � 2)�

m�7 � (28b � 5)�

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 � m? Explain your answer.

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


Name Date Class

Ready to Go On? Enrichment3A

SECTION

�

m

n

r s t

6 1 3

42

5

7

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 � y 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 � 3 � _________

� 6 � � _____

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 � �3 �

__________ � 1

� � ______

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, �4), and

‹

___ › XY passes

through X(�2, 5) and Y(2, �1). 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 � y

2 � y 1 _______ x 2 � x 1 � �4 � __________

0 � �

Slope of ‹

__ › XY � � 5 ___________

� � _____ � _____

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

Is the product of the slopes �1? Are the lines perpendicular?

The lines are neither nor .

Ready to Go On? Skills Intervention3-5 Slopes of Lines3B

SECTION

Vocabulary

rise run slope

A

DC

B

y

2 4 6–2–4 O

2

–4

x

2 4–2–4 O

2

–2

4

–4

22224224 4O24– 4O–2–4 4O24 2 4O24 O

2222

4

–4

O 2222

44

22222222222222222


Writing Equations of Lines

A. Write the equation of the line with slope 3 through (�1, 4) in point-slope form.

What is point-slope form? y � y 1 �

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

B. Write the equation of the line through points (�6, 2) and (3, �4) in slope-intercept form.

What is slope-intercept form? y �

Substitute �2 ___ 3

for m, �6 for x, and 2 for y, and then simplify to find b.

� _____ (�6) � b

b � Write the equation in slope-intercept form.

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

2 x � 4 and 3x � 2y � 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 Intervention3-6 Lines in the Coordinate Plane3B

SECTION

Vocabulary

point-slope form slope-intercept form

2 4–2–4 O

2

–2

4

–4

2O


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

_________ � x 1

� �4 � ___________ � (�6)

� � ______

� �

______

Name Date Class


3-5 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(�2, 0) 6. C(0, �4) and D(5, 9)

7. H(3, 5) and I(�4, 2) 8. S(�6, 1) and T(3, �6)

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

9. ‹

___ › CD and

‹

__ › AB for A(�1, 0), B(1, 5), 10.

‹

__ › LM and

‹

___ › MN for L(�3, 2), M(�1, 5),

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

2 4–2–4 O

2

–2

4

–4

22224224 2 4O24– 2 4O–2–4 2 4O24 2 4O24 O

2222

4

–4

O

2 4–2–4 O

2

–2

4

–4

22224224 2 4O24– 2 4O–2–4 2 4O24 2 4O24 O

2222

4

–4

O

11. ‹

__ › PR and

‹

__ › PS for P(2, �1), Q(2, 1), 12.

‹

___ › GH and

‹

__ › FJ for F(�3, 2), G(�2, 5)

R(�3, 1), and S(�2, �2) H(2, 4), and J(2, 1)

4–2–4 O

2

–2

4

–4

2 4–2–4 O

2

–2

4

–4

Ready to Go On? Quiz3B

SECTION

x

y

2 4 6–2 O

2

–2

6

–4 B

D

C

A

–

44

3-6 Lines in the Coordinate PlaneWrite the equation of each line in the given form.

13. the line through (�3, �1) and (3, �3) in slope-intercept form

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

15. the line with y-intercept �3 through the point (2, 5) in point-slope form

16. the line with x-intercept �4 and y-intercept 2 in slope-intercept form

Graph each line.

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

2 4–2–4

2

4

–4

2 4–2–4

2

4

–4

2 4–2–4 O–2

4

–4

2 4–2–4 O

2

–2

4

–4

22224224 2 4O24– 2 4O–2–4 2 4O24 2 4O24 O

2222

4

–4

O

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

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


Name Date Class

Ready to Go On? Quiz continued

3BSECTION

44

Name Date Class


Slopes and Lengths of Segments

Quadrilateral ABCD has vertices A(�5, 3), B(�1, 4), C(5, �3) and D(�4, �1).

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, �2), R(�2, �5), S(�2, 0). Use the information to answer the following questions:

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

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.

Ready to Go On? Enrichment3B

SECTION

2 4–2–4 O

2

–2

4

–4

2 4–2–4 O

2

–2

4

–4

22224224 2 4O24– 2 4O–2–4 2 4O24 2 4O24 O

2222

4

–4

O

4444

444444

4

44 22

2 4

4

44

44

4444

–44

222222 222222

section ready to go on? skills intervention 3a 3-1 lines ... · pdf file3-1 lines and angles...

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