3.1 n reteaching with practice ame...

3.1 Reteaching with PracticeFor use with pages 129–134

LESSON

16 GeometryChapter 3 Resource Book

Copyright © McDougal Littell Inc. All rights reserved.

NAME _________________________________________________________ DATE ____________

Less

on

3.1

Less

on

3.1

Identify relationships between lines and identify angles

formed by transversals

Identifying Relationships in Space

Think of each segment in the diagram as part of a line. Which of the lines appear to fit the description?

a. parallel to b. skew to

c. parallel to d. Are planes ABE and CDEparallel?

SOLUTION

a. Only is parallel to

b. and are skew to

c. Only is parallel to

d. No, the two planes are not parallel. At the very least, we can see thatthe two planes intersect at point E.

↔BC.

↔AD

↔AB.

↔EC

↔ED

↔AB.

↔CD

↔BC

↔AB

↔AB

A

D

E

B

C

GOAL

EXAMPLE 1

VOCABULARY

Two lines are parallel lines if they are coplanar and do not intersect.

Lines that do not intersect and are not coplanar are called skew lines.

Two planes that do not intersect are called parallel planes.

A transversal is a line that intersects two or more coplanar lines at different points.

When two lines are cut by a transversal, two angles are correspondingangles if they occupy corresponding positions.

When two lines are cut by a transversal, two angles are alternate exterior angles if they lie outside the two lines on opposite sides of the transversal.

When two lines are cut by a transversal, two angles are alternate interior angles if they lie between the two lines on opposite sides of thetransversal.

When two lines are cut by a transversal, two angles are consecutiveinterior angles (or same side interior angles) if they lie between thetwo lines on the same side of the transversal.

Postulate 13 Parallel Postulate If there is a line and a point not on theline, then there is exactly one line through the point parallel to the givenline.

Postulate 14 Perpendicular Postulate If there is a line and a point noton the line, then there is exactly one line through the point perpendicularto the given line.

Reteaching with PracticeFor use with pages 129–134

3.1LESSON

CONTINUED

Geometry 17Chapter 3 Resource Book


NAME _________________________________________________________ DATE ____________Lesso

n 3

.1

EXAMPLE 2

Exercises for Example 1

Think of each segment in the diagram as part of a line.

Fill in the blank with parallel, skew, or perpendicular.

1. and are

2. and are

3. Plane ABC and plane DEF are

4. and are

Think of each segment in the diagram as part of a line.

There may be more than one right answer.

5. Name a line perpendicular to

6. Name a plane parallel to DCH.

7. Name a line parallel to

8. Name a line skew to

Identifying Angle Relationships

List all pairs of angles that fit the description.

a. corresponding

b. alternate exterior

c. alternate interior

d. consecutive interior

SOLUTION

a. and b. and c. and d. and and and and and and and


Complete the statement with corresponding, alternate interior,

alternate exterior, or consectutive interior.

9. and are angles.

10. and are angles.

11. and are angles.

12. and are angles.

13. and are angles.

14. and are angles. �1�5

�5�4

�2�7

�8�1

�6�2

�8�4 6 785

4 123

�5�7�6�8

�6�7�3�7�4�8�4�2�3�2�6�2�5�1�3�1

12

34 5

67

8

↔FG.

↔BC.

↔HD. A

EH

F

B

C

G

D

.↔AB

↔BE

.

.↔CF

↔AD,

↔BE,

.↔CF

↔DE

A

D

E

F

B

C


Lesson

3.1


LESSONNAME _________________________________________________________ DATE ____________

Write different types of proofs and prove results

about perpendicular lines

Comparing Types of Proofs

Write a two-column proof of Theorem 3.1 (a flow proofis provided in Example 2 on page 137 of the text).

Given:

are a linear pair.

Prove:

SOLUTION

Statements Reasons

1. 1. Given

2. 2. Definition of congruent angles3. and are a linear pair 3. Given4. and are supplementary 4. Linear Pair Postulate5. 5. Definition of supplementary angles6. 6. Substitution property of equality7. 7. Distributive property8. 8. Division property of equality9. is a right 9. Definition of right angle

10. 10. Definition of perpendicular lines

Exercises for Example 11. Write a two-column proof of Theorem 3.2. Note that you are asked to complete a

paragraph proof of this theorem in Practice and Applications Exercise 17 on page139. Compare these two proofs.

2. Write a paragraph proof of Theorem 3.3. Note that you are asked to complete a flowproof of this theorem in Practice and Applications Exercise 18 on page 139 and atwo-column proof of this theorem on Exercise 19. Compare these three proofs.

g � h��1

m�1 � 90�2 � �m�1� � 180�

m�1 � m�1 � 180�m�1 � m�2 � 180�

�2�1�2�1

m�1 � m�2

�1 � �2

g � h

�1 � �2, �1 and �2

g

h

1 2

GOAL

EXAMPLE 1

VOCABULARY

A flow proof uses arrows to show the flow of the logical argument.

Theorem 3.1 If two lines intersect to form a linear pair of congruentangles, then the lines are perpendicular.

Theorem 3.2 If two sides of two adjacent acute angles are perpendicu-lar, then the angles are complementary.

Theorem 3.3 If two lines are perpendicular, then they intersect to formfour right angles.



Lesson

3.2



Less

on

3.2


3.2LESSON

CONTINUED

NAME _________________________________________________________ DATE ____________

EXAMPLE 2 Application of the Theorems

Find the value of x.

a. b.

SOLUTION

a. because, by Theorem 3.3, since k and are perpendicular, all four anglesformed are right angles. By definition of a right angle, x is 90.

b. By Theorem 3.3, since m and n are perpendicular, all four angles formed are rightangles. By Theorem 3.2, the angle and the angle are complementary. Thus

so


Find the value of x.

1. 2. 3.

x �

(2x)�x �x �

x �

x � 28.x � 62 � 90,x�62�

�x � 90

m

nx �

62�

k

x �




LESSONNAME _________________________________________________________ DATE ____________

Lesson

3.3

Prove and use results about parallel lines and transversals and use

properties of parallel lines to solve problems

Using Properties of Parallel Lines

Given that find each measure. Tell which postulate ortheorem you use.

a.

b.

c.

d.

SOLUTION

a. Corresponding Angles Postulate

b. Alternate Exterior Angles Theorem

c. Linear Pair Postulate

d. Vertical Angles Theorem


Find each measure given that

1. 2.

3. 4.

5. 6.

7. m�13

m�12m�11

m�10m�9

m�8m�7

m�6 � 67�.

m�5 � 32�

m�4 � 180� � m�3 � 148�

m�3 � 32�

m�2 � 32�

m�5

m�4

m�3

m�2

m�1 � 32�,

GOAL

EXAMPLE 1

VOCABULARY

Postulate 15 Corresponding Angles Postulate If two parallel lines are cutby a transversal, then the pairs of corresponding angles are congruent.

Theorem 3.4 If two parallel lines are cut by a transversal, then the pairsof alternate interior angles are congruent.

Theorem 3.5 If two parallel lines are cut by a transversal, then the pairsof consecutive interior angles are supplementary.

Theorem 3.6 If two parallel lines are cut by a transversal, then the pairsof alternate exterior angles are congruent.

Theorem 3.7 If a transversal is perpendicular to one of two parallellines, then it is perpendicular to the other.


5

3

1

24

67

89

1011

1213


3.3LESSON

CONTINUED



NAME _________________________________________________________ DATE ____________

Less

on

3.3

EXAMPLE 2


Using Properties of Parallel Lines

Use properties of parallel lines to find the value of x.

SOLUTION

Alternate Exterior Angles Theorem

Add.


Use properties of parallel lines to find the value of x.

8. 9.

10. 11.

12. 13.

14. 15.

115�

(9x � 7)�135�

(4x � 3)�

103�

(14x � 7)�

60�

(12x)�

98�

(10x � 2)�

48� (4x )�

53�

(5x � 2)�

140�

(2x � 4)�

x � 63�

�x � 8�� 55�

55�

(x � 8)�


LESSON



NAME _________________________________________________________ DATE ____________

Less

on

3.4

Prove that two lines are parallel and use properties of parallel lines to

solve problems

Proving that Two Lines are Parallel

Prove that lines j and k are parallel.

SOLUTION

Given:

Prove:

Statements Reasons

1. 1. Given

2. 2. Linear Pair Postulate

3. 3. Substitute.4. 4. Subtract.5. 5. Substitute.6. 6. Corresponding Angles Conversej � k

�3 � �1m�3 � 53�m�3 � 127� � 180�

m�3 � m�2 � 180�

m�2 � 127�

m�1 � 53�

j � k

m�2 � 127�

m�1 � 53�

1

32

EXAMPLE 1

VOCABULARY

Postulate 16 Corresponding Angles Converse If two lines are cut by atransversal so that corresponding angles are congruent, then the lines areparallel.

Theorem 3.8 Alternate Interior Angles Converse If two lines are cutby a transversal so that alternate interior angles are congruent, then thelines are parallel.

Theorem 3.9 Consecutive Interior Angles Converse If two lines arecut by a transversal so that consecutive interior angles are supplemen-tary, then the lines are parallel.

Theorem 3.10 Alternate Exterior Angles Converse If two lines are cutby a transversal so that alternate exterior angles are congruent, then thelines are parallel.

NAME _________________________________________________________ DATE ____________

GOAL


3.4LESSON

CONTINUED



NAME _________________________________________________________ DATE ____________Lesso

n 3

.4


NAME _________________________________________________________ DATE ____________

EXAMPLE 2


Prove the statement from the given information.

1. Prove: 2. Prove:

Identifying Parallel Lines

Determine which rays are parallel.

a. Is parallel to

b. Is parallel to

SOLUTION

a. Decide whether

are congruent alternate interior angles, so

b. Decide whether

are alternate interior angles, but they are not

congruent, so are not parallel.


Find the value of x that makes

3. 4. 5. (x � 15)�

(2x)�

a

b(4x)�

(2x)�

a

bx �

(2x � 120)�

a b

a � b.

→PO �

→SQ

�OPS and �PSQ

m�PSQ � 98�

m�OPS � 101�

→PO �

→SQ.

→PN �

→SR.�NPS and �RSP

� 140�

m�RSP � 42� � 98�

� 140�

m�NPS � 39� � 101�

→PN �

→SR.

→SQ?

→PO

→SR?

→PN

39�

42�98�

101�

N

P

O

R

S

Q

2

45�

135�n

o160�

120�

m

n � o� � m





LESSONNAME _________________________________________________________ DATE ____________

Lesson

3.5Use properties of parallel lines and construct parallel lines using

straightedge and compass

Showing Lines are Parallel

Explain how you would show that

a. b.

SOLUTION

a. Because the angle and angle 1 form a linear pair, must equal Thus and the other angle are congruent. Because they are alsocorresponding angles, lines k and are parallel by the Corresponding AnglesConverse postulate.

b. Because the three angles with measures of form a straightline, their sum must be So Thus andtherefore We can now conclude that the angle with the measure isa right angle Therefore, line n is perpendicular to line Since line k is also perpendicular to line n (the angle is indicated), lines kand are parallel by Theorem 3.12.


Explain how you would show

1. 2. 3. k

70�(110� � x)

x �1

kk

45�80�

125�

k � �.

�90�

�.�6 � �15� � 90�.6x�x � 15.

12x � 180,6x � 5x � x � 180.180�.6x�, 5x�, and x�

�

140��1140�.m�140�

k

n

(6x)�(5x)�

x �

k

140�

40�1

k � �.

GOAL

EXAMPLE 1

VOCABULARY

Theorem 3.11 If two lines are parallel to the same line, then they areparallel to each other.

Theorem 3.12 In a plane, if two lines are perpendicular to the sameline, then they are parallel to each other.



3.5LESSON

CONTINUED



NAME _________________________________________________________ DATE ____________

Less

on

3.5

EXAMPLE 2


NAME _________________________________________________________ DATE ____________


Naming Parallel Lines

Determine which lines, if any, must be parallel.

SOLUTION

Lines c and d are parallel because they have congruent corresponding angles.Likewise, lines d and e are parallel because they have congruent correspondingangles. Also, lines c and e are parallel because they are both parallel to the sameline, line d. Because and the angle form a linear pair),and the angle are not congruent. Since and the angle arecorresponding angles that are not congruent, lines a and b are not parallel.Exercises for Example 2

Determine which lines, if any, must be parallel.

4. 5.

6.j k

m

n

45�

40�

50�40�

g

f

eh

i

a b

c

d

100��1100��175��1m�1 � 105�

a

bc

de

75�

100�

1


LESSON



NAME _________________________________________________________ DATE ____________

Less

on

3.6

Find slopes of lines and use slope to identify parallel lines in a coordinate

plane and write equations of parallel lines in a coordinate plane

Finding the Slope of a Line

Find the slope of the line that passes through the points

SOLUTION

Let

The slope of the line is


Find the slope of the line that passes through the given points.

1. 2. 3.

4. 5. 6.

Identifying Parallel Lines

Find the slope of each line. Is

y

x

1

1

(0, 2)

(0, 5)

(�5, 0)

(�2, 0)

a � b?

��18, 5� and �4, 5��7, �5� and �5, 4��8, 3� and �14, 5��8, 12� and �0, �12��3, �1� and ��5, �11��4, 2� and �6, 8�

�4.

� �4

�12�3

�9 � ��3�

0 � 3

m �y2 � y1

x2 � x1

�x1, y1� � �3, �3� and �x2, y2� � �0, 9�.

�3, �3� and �0, 9�.

EXAMPLE 1

VOCABULARY

Postulate 17 Slopes of Parallel Lines In a coordinate plane, twononvertical lines are parallel if and only if they have the same slope.Any two vertical lines are parallel.

NAME _________________________________________________________ DATE ____________

GOAL

EXAMPLE 2


3.6LESSON

CONTINUED



NAME _________________________________________________________ DATE ____________Lesso

n 3

.6

SOLUTION

Find the slope of a. Line a passes through

Find the slope of b. Line b passes through

Compare the slopes. Because have the same slope, they are parallel.


Find the slope of each line. Which lines are parallel?

7. 8.

Writing an Equation of a Parallel Line

Line k has the equation

Line is parallel to and passes through the point Write an equation of

SOLUTION

Find the slope. The slope of k is Because parallel lines have the same slope,the slope of is also

Solve for b. Use

Write an equation. Because an equation of


Write an equation of the line the passes through the given point P

and is parallel to the line with the given equation.

9. 10. 11. P��1, 2�, y �23

x � 2P��5, 2�, y � �x � 9P�10, 3�, y � x � 12

� is y � �x � 6.m � �1 and b � 6,

6 � b

5 � �1 � b

5 � �1�1� � b

y � mx � b

�x, y� � �1, 5� and m � �1.

�1.��1.

�.�1, 5�.k�

y � �x � 4.

y

x

1

1

(3, 4)

(0, �5) (2, �4)

(�3, 1)

y

x

1

1

(5, 1)(�7, 2)

(�7, �2)(5, �1)

a and b

mb �2 � 0

0 � ��2� �22

� 1

��2, 0� and �0, 2�.

ma �5 � 0

0 � ��5� �55

� 1

��5, 0� and �0, 5�.


EXAMPLE 3




LESSONNAME _________________________________________________________ DATE ____________

Lesson

3.7

Use slope to identify perpendicular lines in a coordinate plane and write

equations of perpendicular lines

Deciding Whether Lines are Perpendicular

a. Decide whether are perpendicular.

b. Decide whether the lines are perpendicular.

Line Line k:

SOLUTION

a. Find each slope.

Slope of

Slope of

Multiply slopes to see if the lines are perpendicular.

The product of the slopes is not So, are not perpendicular.

b. Rewrite each equation in slope-intercept form to find the slope.

Line Line k:

Multiply the slopes to see if the lines are perpendicular.

so the lines are perpendicular.�23� � ��

32� � �1,

slope � �32

slope �23

y � �32

x �32

y �23

x �43

�:

↔PQ and

↔QR�1.

34

� ��3� � �94

↔QR �

0 � 31 � 0

��31

� �3

↔PQ �

3 � 00 � ��4� �

34

3x � 2y � 3�: 2x � 3y � �4

y

x

1

�1P(�4, 0)

R(1, 0)

Q(0, 3)

↔PQ and

↔QR

GOAL

EXAMPLE 1

VOCABULARY

Postulate 18 Slopes of Perpendicular Lines In a coordinate plane, twononvertical lines are perpendicular if and only if the product of theirslopes is �1.



3.7LESSON

CONTINUED



NAME _________________________________________________________ DATE ____________

Less

on

3.7

EXAMPLE 2


NAME _________________________________________________________ DATE ____________



Decide whether lines k and are perpendicular.

1. k passes through (3, 2) and 2. k has the equation

passes through has the equation

Writing the Equation of a Perpendicular Line

Line k has equation Find an equation of line that passes

through and is perpendicular to k.

SOLUTION

First determine the slope of For k and to be perpendicular, the product oftheir slopes must equal

Then use to find b.

So, an equation of is


Line j is perpendicular to the line with the given equation and

line j passes through P. Write an equation of line j.

3. 4. 5. x � 5y � 6, P��1, 2�5x � 2y � 3, P�0, �32�4x � 7y � 13, P��2, 6�

y � �32

x �72

�

72

� b

�1 � �32

� �3� � b

y � mx � b

m � �32

and �x, y� � �3, �1�

m� � �32

23

� m� � �1

mk � m� � �1

�1.��.

P�3, �1�

�y �23

x �43

.

x � 2y � �6��0, 2� and �3, 6��

2x � 4y � �3��1, 5�

�




LESSONNAME _________________________________________________________ DATE ____________

Prove that a quadrilateral is a parallelogram and use coordinate geometry

with parallelograms

Using Properties of Parallelograms

Show that andare the vertices of a parallelogram.

SOLUTION

There are many ways to solve this problem.

Method 1 Show that opposite sides havethe same slope, so they are parallel.

Slope of

Slope of CD �2 � 6

�3 � ��2��

�4

�1� 4

AB �4 � 0

3 � 2� 4

D��3, 2�

2

y

x1

C

B

AD

A�2, 0�, B�3, 4�, C��2, 6�,

GOAL

Theorem 6.6If both pairs of opposite sides of a quadrilateral are congruent, then thequadrilateral is a parallelogram.

Theorem 6.7If both pairs of opposite angles of a quadrilateral are congruent, then thequadrilateral is a parallelogram.

Theorem 6.8If an angle of a quadrilateral is supplementary to both of its consecutiveangles, then the quadrilateral is a parallelogram.

Theorem 6.9If the diagonals of a quadrilateral bisect each other, then the quadrilateralis a parallelogram.

Theorem 6.10If one pair of opposite sides of a quadrilateral are congruent and parallel,then the quadrilateral is a parallelogram.

Ways to Prove a Shape is a Parallelogram

• Show that both pairs of opposite sides are parallel.

• Show that both pairs of opposite sides are congruent.

• Show that both pairs of opposite angles are congruent.

• Show that one angle is supplementary to both consecutive angles.

• Show that the diagonals bisect each other.

• Show that one pair of opposite sides are congruent and parallel.

EXAMPLE 1

Lesson

6.3


6.3LESSON

CONTINUED

Slope of

Slope of

and have the same slope, so they are parallel. Similarly,Because opposite sides are parallel, ABCD is a parallelogram.

Method 2 Show that the opposite sides have the same length.

and Because both pairs of opposite sides are congruent, ABCD is a parallelogram.

Method 3 Show that one pair of opposite sides is congruent andparallel. Find the slopes and lengths of and as shown in Methods1 and 2.

Slope of Slope of

and are congruent and parallel, so ABCD is a parallelogram.


Refer to the methods demonstrated in Example 1 to show

that the quadrilateral with the given vertices is a

parallelogram.

1. Show that the quadrilateral with vertices and is a parallelogram using Method 1

from Example 1.

2. Show that the quadrilateral with vertices and is a parallelogram using Method 2

from Example 1.

3. Show that the quadrilateral with vertices and is a parallelogram using Method 3 from

Example 1.

4. Show that the quadrilateral with verticesand is a parallelogram using any of the three

methods demonstrated in Example 1.D�0, 7�C�6, 6�,

B�5, �3�,A��1, �2�,

D�2, 2�C�6, 3�,B�4, �5�,A�0, �6�,

D��1, �5�C�4, �4�,B�1, 2�,A��4, 1�,

D��8, �2�C��7, �6�,B��2, �4�,A��3, 0�,

CDAB

AB � CD � �17

CD � 4AB �

CDAB

BC � DA.AB � CD

DA � ��2 � ��3��2 � �0 � 2�2 � �29

BC � ��2 � 3)2 � �6 � 4�2 � �29

CD � ��3 � ��2��2 � �2 � 6�2 � �17

AB � ��3 � 2�2 � �4 � 0�2 � �17

BC � DA.CDAB

DA �0 � 2

2 � ��3��

�2

5� �

2

5

BC �6 � 4

�2 � 3�

2

�5� �

2

5



NAME _________________________________________________________ DATE ____________

Less

on

6.3

3.1 n reteaching with practice ame...

Documents