Topic 6: Parallel and Perpendicular

for use after

The Shapes of Algebra

Investigation 2

In the diagram below, lines O,

m

,

n

, and

p

are parallel lines. The other two

lines are

Angles

ACI

and

CEK

are in corresponding positions

at the vertices

C

and

E

. Each is in the “top-right” or “north-east” position

at their vertices. Because of their corresponding positions, they are called

Corresponding angles are congruent to each other

if they are formed by a transversal intersecting parallel lines.Angles

BAC

and

XAY

are

Vertical angles are always congruent to each other.

When parallel lines are cut by non-parallel transversals, similar triangles

are formed. In this figure, triangle

ABC

is similar to triangle

ADE

.

Corresponding sides of these similar triangles form equal ratios.

For example:

=

Exercises

For Exercises 1–6, use the diagram above.

1. a. List five other pairs of vertical angles in the diagram.

b. List five other pairs of corresponding angles in the diagram.

2. What other segments form equal ratios? Explain.

3. What angles are congruent to &

CEK

? Explain.

4. What angles are congruent to &

BDJ

? Explain.

5. Why are &

EDL

and &

BDJ

congruent?

6. What other triangle is similar to triangle

ABC

? Explain.

length of AB

length of AD

length of AC

length of AE

?

m

n

p

E

C

B

A

XY

D

F

H

J

L

G

I

K

M

vertical angles.

corresponding angles.

transversals.

---

For Exercises 7–13, use the diagram below. Lines

a, b

, and

c

are parallel.

7. What are the measures of &

PMH

and &

x

?

8. What are the measures of &

FGH

and &

GHF

?

9. What is the measure of &

PGH

?

10. What is the measure of &

y

?

11. What is the length of

?

12. What is the measure of &

GFH

and &

w

?

13. What is the length of

?

Remember that lines are parallel if their slopes are equal and lines are

perpendicular if their slopes are negative reciprocals of each other.

Sample

Are the lines

y

= 2

x

+ 8 and 3 = 2

x

-

y

parallel?

Rewrite the second equation.

3 = 2

x

-

y

y

+ 3 = 2

x

-

y

+

y

y

+ 3 = 2

x

y

= 2

x

- 3

The slope of this line is 2, which is also the slope of the first line. The slopes

are equal, so the lines are parallel.

GH

GP

w

x

P

G

M

F

d

a

b

c

H

65°

y

47°

5 cm

10 cm

4 cm

15.3 cm

---

Sample

Are the lines

y

= 2

x

+ 8 and 7 =

x

-

y

perpendicular?

Rewrite the second equation.

7 =

x

-

y

y

+ 7 =

x

-

y

+

y

y

+ 7 =

x

y

=

x

- 7

The slope of this line is . The slope of the first line is 2. The slopes

are not negative reciprocals of each other, so the lines are not

perpendicular.

Determine whether each pair of lines is parallel, perpendicular, or

neither.

14.

y

= 5

x

- 7

15.

y

=

x

- 0.5

y

+ 5

x

= 12

y

+

x

= 0.25

16.

y

=

x

-

17. 2

y

= 6

x

- 72

y

-

x

=

y

- 3

x

= 15

18.

y

+

x

= 12

19. 5

x

-

y

= 12

y

-

x

= 12

5

y

+

x

= 35

5

6

1

2

3

4

1

2

1

3

1

3

1

3

1

3

1

3

1

3

---

Topic 6: Parallel and Perpendicular

Guided Instruction

Mathematical Goals

•

Investigate parallel and perpendicular algebraically and geometrically

•

Apply properties of angle pairs formed by parallel lines and transversals

•

Understand properties of the ratio of segments when parallel lines are cut

by transversals

Materials

•

Labsheet 6.1

Vocabulary

•

transversals

•

corresponding

angles

•

vertical angles

At a Glance

The terminology of corresponding angles and vertical angles may be new to

students. Corresponding angles are congruent only if they are formed by a

transversal intersecting parallel lines. If they are formed by a transversal

intersecting non-parallel lines, then the corresponding angles are

not

cougruent. Vertical angles are always congruent. Students use these facts to

identify pairs of congruent angles. Before students begin to solve the

exercises, you may want to help students remember what they know about

proportional reasoning and similar triangles.

Because angles are named with triads of letters, many of the angles in the

figure on the first page have multiple names. The letter representing the vertex

of the angle will be the same for each of the names. For example, &

GAC

can

also be named &

GAE

and &

GAM.

The answers given below use only one

name for each angle. Students may use different names for the angles.

Students may have some difficulty identifying five pairs of vertical and

corresponding angles in Exercise 1. You may want to ask students to work

with a partner or in small groups to find all of these pairs of angles. One

strategy that may help students find corresponding angles is to imagine that

one of the transversals is removed from the figure. Students might actually

cover up one of the transversals with a finger.

For the topic introduction, ask:

•

What does perpendicular mean?

•

What does parallel mean?

•

Can you name angles

ACI

or

CEK

in more than one way?

•

What does it mean for ratios to be equal?

•

Suppose that one of the transversals is removed. Is it easier to find pairs

of corresponding angles?

For the section after Exercise 13, ask:

•

What are negative reciprocals?

•

What is the slope of a line?

PACING

1 day

---

Assignment Guide for Topic 6

Core

1–19

Answers to Topic 6

Exercises

1. a.

Any five of the following will do:

&

XAY

and &

BAC

, &

XAF

and &

GAC

,

&

YAG

and &

FAB

, &

ACB

and &

ICE

,

&

ACI

and &

BCE

, &

ABC

and &

HBD

,

&

ABH

and &

CBD

, &

CEK

and &

DEM

,

&

KEM

and &

CED

, &

BDE

and &

JDL

,

&

BDJ

and &

LDE

b.

Any five of the following will do:

&

XAG

and &

ACI

, &

XAG

and &

CEK

,

&

ACI

and &

CEK

&

YXA

and &

GAC

, &

YXA

and &

ICE

,

&

YXA

and &

KEM

, &

GAC

and &

ICE

,

&

GAC

and &

KEM

, &

ICE

and &

KEM

&

XAF

and &

ACB

, &

XAF

and &

CED

,

&

ACB

and &

CED

&

FAC

and &

BCE

, &

FAC

and &

DEM

,

&

BCE

and &

DEM

&

YAG

and &

ABC

, &

YAG

and &

BDE

,

&

ABC

and &

BDE

&

GAB

and &

CBD

, &

GAB

and &

EDL

,

&

CBD

and &

GAB

&

XYA

and &

FAB

, &

XYA

and &

HBD

,

&

XYA

and &

JDL

, &

FAB

and &

HBD

,

&

FAB

and &

JDL

, &

HBD

and &

JDL

&

YAF

and &

ABH

, &

YAF

and &

BDJ

,

&

ABH

and &

BDJ

2.

Proportions can be written in multiple ways,

so the answers below are samples.

:

=

:

=

:

:

=

:

=

:

:

=

:

=

:

Corresponding parts of similar triangles are

congruent. The transversals create three

similar triangles: triangle

AXY

, triangle

ACB

,

and triangle

AED

.

3.

&

CEK

is congruent to &

ACI

and &

XAG

(corresponding angles), and &

DEM

, &

BCE

,

and &

FAC

(vertical angles for the first three

angles).

4.

&

BDJ

is congruent to &

ABH

and &

YAF

(corresponding angles), and &

EDL

, &

CBD

,

and &

GAB

(vertical angles for the first three

angles).

5.

They are vertical angles.

6.

Triangle

AED

is similar, because the angles

are congruent. Triangle

AYX

is similar,

because the angles are congruent.

7.

&

PMH

is 478, &

x

is 658.

8.

&

FGH

is 658, &

GHF

is 478.

9.

&

PGH

is 1158.

10.

&

y

is 1338.

11.

is 8 cm.

Note:

#

FGH

, #

FPM

and the

scale factor is 3.

12.

&

GFH

is 688, &

w

is 688.

13.

5.1 cm

14.

Neither; the slopes are +5 and -5.

15.

Perpendicular; the slopes are +1 and -1

(negative reciprocals).

16.

Parallel; the slopes are and (equal).

17.

Parallel; the slopes are 3 and 3 (equal).

18.

Perpendicular; the slopes are 1 and -1

(negative reciprocals).

19.

Perpendicular; the slopes are 5 and

-

1

(negative reciprocals).

5

1

2

1

2

GP

AX AY

AC AB

AE AD

AY XY

AB BC

AD DE

AX XY

AC CB

AE ED

---

Name

Date

Class

Labsheet 6.1

Topic 3

Introduction

Exercises 8–14

w

x

P

G

M

F

d

a

b

c

H

65°

y

47°

5 cm

10 cm

15.3 cm

4 cm

?

m

n

p

E

C

B

A

XY

D

F

H

J

L

G

I

K

M

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