unit d parallel and perpendicular lines -...

Baltimore County Public Schools Division of Curriculum and Instruction

Unit D Parallel and Perpendicular Lines

Geometry 2011 Unit D Parallel and Perpendicular Lines

D-1


Essential

Question

How can vocabulary and proofs associated with parallel lines improve

logical and critical thinking skills?

Sections

D-1 Lines and Angles (Section 3-1)

D-2 Angles Formed by Parallel Lines and Transversals (Section 3-2)

D-3 Proving Lines Parallel (Section 3-3)

D-4 Perpendicular Lines (Section 3-4)

D-5 Slopes of Lines (Section 3-5)

Lines in the Coordinate Plane (Section 3-6)

Enduring

Knowledge

Big Ideas Alignment

Transformations in the plane can be used to show

congruence.

D-1

Similarity transformations can be used to show similarity

relationships between figures.

D-1,

D-2

D-3

Definitions, postulates, and theorems are used to prove

theorems involving similarity.

D-4

Visualizing relationships between two-dimensional and

three-dimensional objects can help you connect geometric

concepts to real objects.

D-5

Vocabulary alternate exterior angles

alternate interior angles

parallel planes

perpendicular bisector

same-side interior angles

skew lines

transversal

Prerequisite

Skills Solve linear equations.

Copying angles using a compass and straight edge.

Solve quadratic equations (honors and GT).

Write basic geometric proofs including vertical angles and linear pairs.

Estimated Unit

Length A: 16 (8) days H: 16 (8) days GT: 16 (8) days



Geometry 2011 Unit D Parallel and Perpendicular Lines

D-2

Resources and Materials

Text Holt Geometry, 2011

Materials objects of varying shape (i.e. book, cone, jar)

response boards

dry erase markers and erasers

chart paper and markers

sentence strips

patty paper

straightedges

protractors

Supplements Serra, Michael. Patty Paper Geometry

Acces 4 Module

Chapter Resources, Volume 1

Vocabulary Bingo WS D-1

Lines and Transversals Foldable, WS D-1a

Drawing a Parallel, RS D-2

Angle Relationships - Parallel Lines GSP D-2a

ProofBlock© Templates; Template Samples, RS D-2b.

Paving the Way, RS D-4

Parallel Postulate, RS D-4a

Classifying Lines, RS D-5

Classifying Lines, WS D-5a

Technology The Geometer’s Sketchpad

Microsoft Office PowerPoint

ActivInspire

Defined STEM The Science of Sound Production in Different Instruments (Connections),

Section D-2

http://stem.definedlearning.com/connect.cfm?asset_guid=811dbfc0-51af-40c1-841b-61ee3f0c3571&blnSearch=1



Geometry 2011 D-1 Lines and Angles

D-3

D-1 Lines and Angles (Section 3-1)

Objective(s) Identify parallel, perpendicular, and skew lines.

Identify the angles formed by two lines and a transversal.

Alignment CCSC: G.CO.9, G.CO.1 AIM: O-7

Essential

Understanding

The relationships between the angles formed from coplanar lines cut by a

transversal are further explored as the coplanar lines are made parallel.

Assessment Level 1

Identify one pair

of each of the

following:

1. Parallel planes

2. Perpendicular planes

3. Skew lines

[Plane PQR || Plane ABC;

Plane PQR Plane SRC;

QB and

DC

Level 2

Identify one pair of each of the

following:

1. Corresponding angles

2. Alternate interior angles

3. Same side interior angles

[angles 1,9; 2,5; 7,10]

Level 3

Given the diagram above, which of

the following cannot be true?

A. Line l is parallel to line m

B. 2 and 10 are corresponding

angles

C. 5 and 8 are same-side

interior angles

D. 2 and 12 are alternate

exterior angles

[C]

Level 4

Given non-coplanar lines l, m, and n

such that nml , and line k is

parallel and coplanar to line l. What

is the relationship between line k and

lines m and n?

[k will be skew to one of the lines

and perpendicular to the other.]

t

m

l

k

10

12 11

9

8 7

2

65

34

1

12 11

109

87

6

5

4 3

21




D-4

Teacher

Preparation

Materials

Text Holt Geometry, 2011 pp. 146-151

Materials objects of varying shape (i.e. book, cone, jar)

response boards


chart paper and markers

sentence strips

patty paper

straightedges

protractors

Supplements Vocabulary Bingo RS D-1

Lines and Transversals Foldable, RS D-1a

Half Plane Model of Hyperbolic Geometry

Technology Geometer’s Sketchpad

Teaching Suggestions

Background

Information Students may struggle in geometry because of the plethora of

vocabulary terms they need to know. Refer the students to “Study

Strategy: Take Effective Notes” p. 145, for an overview of Cornell

Notes. This type of organized note taking may benefit the students as

they learn more geometric terms, postulates, theorems, etc. Provide the

students with sufficient practice in associating angle relationships. Give

the opportunity to practice identifying and naming pairs of angles with

parallel and non-parallel lines cut by a transversal.

The students identified and described geometric relationships between

the angles formed when parallel lines are cut by a transversal in grades

7 and 8. These angle pairs included alternate interior, alternate exterior,

and corresponding angles. The students are familiar with the words

parallel and perpendicular as well as their respective notations. The

term same-side-interior angles (consecutive interior angles) will be new

learning for the students. Skew lines were covered in section A-1, but

should be reviewed since it is a relatively new concept.

Core

Instructional

Strategies

Level 1

Motivate the students by several displaying objects of varying shape,

such as a ball, book, cone, jar, tissue box, and a tube of toothpaste.

Review the vocabulary terms parallel segments, perpendicular

segments, skew segments, parallel planes, and perpendicular plane, by

inviting several students to the front of the room to point out examples

of each of these terms on the objects. Connect the terms to other real

world representations with p. 150 (41). Begin a vocabulary list on a

piece of large poster paper for the students to reference.

http://www.dynamicgeometry.com/General_Resources/Advanced_Sketch_Gallery/Beyond_Euclid/Half_Plane_Model_of_Hyperbolic_Geometry.html




D-5

Project the rectangular prism from p. 146 for the students to view.

Identify parallel and perpendicular lines and planes, modeling the

correct way to write these relationships symbolically using ⊥ and ||.

Remind the students to write out the word skew because there is no

symbol to represent the word. Give the students response boards, and

call out various pairs of lines and planes (i.e. parallel, perpendicular,

intersecting, skew) shown in the rectangular prism. Ask the students to

write down the relationship between each pair using the correct

notation on the response boards and to hold them up for assessment.

Display the diagram shown on the right. Ask the

students to represent the relationship of lines j and k

symbolically. Allow time for the students to

conclude that the lines are not parallel or

perpendicular and that there is no symbol for

intersecting only. What is the name of line t, which

intersects lines j and k? Trace over the transversal

with a different color, label it ‘transversal’, and add

this term to the vocabulary list.

Write each of the names of the “Angle Pairs Formed by a Transversal”,

p. 147, on separate sentence strips. Ask the students to list the interior

angles on the left side of their response boards and the exterior angles

on the right. Post only the sentence strips with the terms ‘alternate

interior’ and ‘alternate exterior’ on the board. Which pairs of angles do

you think are alternate interior and which are alternate exterior?

Record the angles under the appropriate sentence strips. Repeat this

with ‘corresponding’ and ‘same side interior’ angles. Add each term to

the vocabulary list.

Present the diagram shown on the

right. Ask the students to list all the

corresponding angles on their

response boards. Check for

understanding as they hold up their

boards. Repeat this process with the

remaining angle relationships. (field independent, active)

Provide practice identifying relationships between lines and angle pairs

formed by a transversal with Vocabulary Bingo - Planes, Lines, and

Angles, RS D-1. Give the students a game card and a highlighter. Have

the students complete the game card as explained on the resource sheet.

Project a copy of the diagrams from the game sheet for the students to

view during the game. Play the game by calling out a pair of lines,

angles, or planes. Instruct the students to highlight a column that

represents what is named. Continue to play until a student has five in a

row highlighted. Allow the students to trade game boards for multiple

rounds of play. (field dependent, auditory, active)

BG

I

C

A

D

F

H

t

k

j4

3

8 7

65

21




D-6

Level 2

Extend the recognition of angles and transversals by presenting a

diagram similar to Example 3 on p. 147. Name a pair of angles and ask

the students to identify the transversal and the relationship of the angle

pair. Repeat this with several angle pairs. Emphasize the importance of

choosing the correct transversal. Switch to naming a transversal and

angle relationship for which the students must identify an angle pair

that fits the description. Asses the students by instructing them to

display their answers on response boards. (field independent, auditory)

Level 3

Prepare the students for proofs with 3-1 Practice C (1 – 4). Allow the

students to compare diagrams and conclusions to verify correctness.

Differentiation

Strategies Accelerate-Review-Reteach

Consider using a cardboard box separator as a three dimensional model

of parallel and perpendicular planes.

Use Lesson 3-1 Reading Strategies, Chapter Resources to reinforce the

meanings of symbols and diagram markings. Encourage the students to

use the color coding and letter references as shown on p. 149 (27-29).

Create a life-sized diagram of two lines and a transversal on the floor.

Pair the students and begin by naming an angle relationship. Have

student pairs stand in the appropriate spaces. Add a second transversal

when the students are ready.

Distribute Lines and Transversals Foldable, RS D-1a, to assist with

organizing and identifying the geometry terms. Help them fill in the

correct angle pairs in each box.

Enrichment-Extension

Allow the students to explore in the same way as above using

Geometer’s Sketchpad instead of patty paper and a protractor.

Level 5

Introduce the students to spherical geometry with Lesson 3-1

Challenge, Chapter Resources.

Allow the students to explore the basics of Hyperbolic Geometry using

The Geometer’s Sketchpad Resource Center, Half Plane Model of

Hyperbolic Geometry at Half Plane Model of Hyperbolic Geometry.

Identify and name

points, lines, and

planes, and classify

angles

Identify parallel and

perpendicular lines and

angles formed by lines

and a transversal

Prove and use theorems

about angles formed by

parallel lines and a

transversal

http://www.dynamicgeometry.com/General_Resources/Advanced_Sketch_Gallery/Beyond_Euclid/Half_Plane_Model_of_Hyperbolic_Geometry.html



Geometry 2011 D-2 Parallel and Perpendicular Lines

D-7

D-2 Angles Formed by Parallel Lines and Transversals (Section 3-2)

Objective(s) Prove and use theorems about the angles formed by parallel lines and a

transversal.

Alignment CCSC: G.CO.9 AIM: O-7

Essential

Understanding

The construction and design of many real world objects depends on the

congruency of the angles formed by parallel lines and transversals.

Assessment Level 1

If ba || , state the postulate or

theorem that supports the conclusion

.86

[Corresponding angles postulate]

Level 2

Given BFCE || , find .ABFm

117

Level 3

Draw line q. Draw two lines r and s

so they are both perpendicular and

coplanar to q. Prove r║s.

[See SA]

Level 4

Given: 35 and ml ||

Prove: 113

[See SA]

n

m

l

1211

109

87

65

43

21

ED

C F

B

A

(6x + 33)°

(5x - 7)°

ba

1 2 3 4 5 6 7 8




D-8

Teacher

Preparation

Materials


Materials patty paper

rulers

protractors

colored pencils

transparencies

transparency pens

index cards

tape

Supplements Drawing a Parallel, RS D-2

Angle Relationships – Parallel Lines, GSP D-2a

ProofBlock© Templates, RS D-2b


PowerPoint

www.proofblocks.com


Background

Information As the students advance through the geometry course, the number of

postulates, theorems, converses, definitions, etc. continues to grow at a

steady rate. Keeping track of these statements, remembering them, and

then applying them in problem-solving and proof settings is crucial for

success. Encourage the students to keep an ongoing notebook of

postulates, theorems, converses, definitions, etc., and allow the students

to reference this notebook whenever necessary. Understanding the

difference between a theorem and its converse becomes key in this

section. The distinction between the two statements needs to be made

clear, so that the students are using the statements appropriately in

proofs. This distinction will need to be made even more clear in later

units as the students work with parallelograms and triangles.


the angles formed when parallel lines are cut by a transversal in grades

7 and 8. These angle pairs included alternate interior, alternate exterior,

and corresponding angles. The term same-side-interior angles

(consecutive interior angles) was new learning in the previous section.

The students applied the relationships between the angles to solve for

missing angle measures in the previous middle school grades.

However, working with algebraic expressions to solve for missing

angle measures will be new. The students may be comfortable

identifying the vocabulary and setting up the appropriate equations, but

may arrive at incorrect answers due to algebraic errors. It is important

to provide algebraic examples to help these students polish their

algebra skills; yet, avoid allowing the algebra to overshadow the

geometry.

http://www.proofblocks.com/




D-9

Core

Instructional

Strategies

Level 1

Motivate the students by discussing situations in which objects must be

parallel in order to function properly. What would happen if the chalk

ledge was not parallel to the floor? What if the wheel axles on a car

pointed in different directions? Explain that the functionality of the

objects rely on the angle relationships associated with parallel lines.

Begin the lesson by projecting Drawing a Parallel, RS D-2, and present

the questions at the bottom of the page. Use the responses to remind the

students that the angle pair vocabulary reviewed in the last section is

the same when the lines are parallel. Emphasize the importance of the

markings that indicate parallel lines. (field independent, visual)

Distribute patty paper to pairs of students. Allow for free

exploration on how to construct parallel lines using folding

techniques. Instruct the students to draw a transversal to the

parallel lines using a straightedge. Give each pair of students

a protractor and have the students measure the special angle

relationships and make a conjecture about the relationships

between each type of angle pair.

Level 2

Open the Geometer’s Sketchpad file, Angle Relationships-Parallel

Lines, GSP D-2a. Present the first Sketchpad page, Corresponding

Angles. Ask the students to predict the relationship between the angles,

based on their appearance. Ask, How will moving the transversal affect

the measures of the angles? Click the ‘Move Transversal’ button and

let the students observe the changes. Click the button a second time to

pause the movement. Poll the class for conjectures before clicking the

button ‘angle JCD’. Move the transversal again and direct the students’

attention to the changes of the angle measures as the transversal moves.

Allow the class to formulate their own theorem, as a conditional

statement. Choose a volunteer to record the class’s theorem on the

board or on chart paper to be displayed throughout the class period.

Reveal the ‘formal postulate’ by clicking the button. Ask one student to

read the formal statement aloud. Discuss as a class the differences

between the formal postulate and the class’s statement and make any

necessary adjustments.

Display Example 1A on p. 155 and ask, How can we use the

Corresponding Angles Postulate to find the measure of x? Display

Example 1B and have the students work in a Think-Pair-Share style to

discuss the correct way to set up an equation and find the measure of

the angle. Invite a student to share the correct equation and explain the

process used to arrive at this equation. Present a third algebraic

example similar to 1B. Create a third example for the Honors and GT

students where the students must factor the equation to solve for the

variable and missing angle measures.

http://www.google.com/url?q=http://www.webstaurantstore.com/suffixitem/433P50/BOX.html&sa=X&ei=FHIzTIHsFoG78gabs6jJCw&ved=0CEYQ9gIwAw&usg=AFQjCNESIYw10IxYsfrKMNIwP7aLJjqXOA




D-10

Continue with the Geometer’s Sketchpad activity, Angle Relationships-

Parallel Lines, GSP D-2a. Follow the same procedure to introduce

alternate interior, alternate exterior, and same-side interior angles by

clicking the tabs located at the bottom of the Sketchpad file. Delay

clicking the ‘Angle Sum’ button on the Same Side Interior Angles

page. Allow the students to analyze the measurements of the angle pair

at any given time during the movement of the transversal. (field

independent, visual)

Use the patty paper activity, Explorations Transparencies,

Exploration, Alternate Openers: 3-2, to introduce the angle

relationships and corresponding theorems associated with

parallel lines.

Present algebraic examples for each type of angle pair, incorporating

factoring skills for Honors and GT. Consider using Example 3 on p.

157 for highly able students. Present several exercises as mixed

practice. Consider creating a PowerPoint Presentation with each

exercise on a separate slide. Require the students to identify the

postulate or theorem needed in each exercise, and then set up the

appropriate equation to solve for variables and angle measures.

Extend the angle relationships to applications using exercises 5, 12, 24,

and 30 on pp. 158 – 160. Use Application Practice p. S30 for additional

application exercises.

Level 3

Transition to proofs given parallel lines by posing the question, Which

of the angle relationships could be used to prove the other three angle

relationships? Remind the students that postulates are statements that

are accepted as fact. Design a ProofBlock© for the Corresponding

Angles Postulate using ProofBlock© Templates, RS D-2b.

Present the given and proof statements for the Alternate Interior Angles

Theorem proof, p. 156. Provide a Transitive Property ProofBlocks©

from ProofBlocks©, RS D-5a, from the previous unit C, and the

Vertical Angles ProofBlock© available from the ProofBlocks© Web

site, www.proofblocks.com. Model how to write the input and output

lines between the ProofBlocks© to complete the proof. Invite two

students to the board, one to complete a two-column proof and one to

complete a flowchart proof. Compare the three proof styles so that the

students can use any method when completing proofs.

Assign p. 159 (25, 26), where the students complete a proof for the

Alternate Exterior Angles Theorem and the Same-Side Interior Angles

Theorem. Encourage the students to use any style of proof they feel the

most comfortable with. Select several students to present their proofs

with the intent of having various proof styles and proof solutions

demonstrated. (field independent, sequential)

http://www.proofblocks.com/





D-11

Level 4

Assign problems that apply the theorems and postulate from this

section in a proof type setting, such as pp. 158–161 (27 – 29, 33, 36).

Differentiation


Reinforce the angle relationships using the “Teaching Tips” and the

“Reaching All Learners Activity”, from TE, p. 156. Recall the letter

references on p. 149 (27–29), and encourage the students to find these

letters within the parallel lines.

Display two parallel lines cut by a transversal with all angles labeled.

Give each student an index card with a pair of angles written on it. Put

the headings ‘Congruent’ and ‘Supplementary’ on the board. Instruct

the students to tape their index card under the correct heading. Have the

class verify that all angles are placed under the correct heading and, as

a class, identify the postulate or theorem that explains their placements.

Ease the students into parallel line proofs by providing several proofs

with a few statements and reasons left blank. Allow the students time

to become comfortable with filling in the blanks before asking them to

develop a complete proof.


Present diagrams including a second or third transversal with algebraic

expressions in which the students must determine the measure of

multiple angles. Use p. 161 (39) as a reference.

Introduce the proof of the Alternate Interior Angle Theorem by having

the students draw an appropriate diagram and identify, from the formal

statement, the given information and what needs to be proved.

Consider assigning 3-2 Practice C to pairs of students. Allow groups of

students to collaborate to complete the ten-step proof.

Identify relationships

of angles formed by

lines and transversals

Use and apply

theorems about the

angles formed by

parallel lines and a

transversal

Prove lines parallel

given angle

relationships



Geometry 2011 D-3 Proving Lines Parallel

D-12

D-3 Proving Lines Parallel (Section 3-3)

Objective(s) Use the angles formed by a transversal to prove two lines are parallel.

Alignment CCSC: G.CO.9 AIM: O-7

Essential

Understanding

Establishing lines parallel by the converse theorems is essential in

architecture, landscaping, and carpentry to ensure structures are

appropriately designed.

Assessment Level 1

Write the converse of the

Corresponding Angles Postulate.

[If two lines are cut by a transversal

so that corresponding angles are

congruent, then the two lines are

parallel.]

Level 2

Given 11x , 1326m , and

22103 xm , show that ba || .

Identify the theorem used to justify

your answer.

,36,132221010 mm

Thm. s' Int. Alt.by || ba

Level 3

Justify each step in the flowchart

proof.

Given: 18058 mm

Prove: ml ||

[See SA]

Level 4

Given: 180101 mm

Prove: l||n

[See SA]

ba

1 2 3 4 5 6 7 8

n

m

l

1211

109

87

65

43

21

m

t

l

43

5678

21




D-13

Teacher

Preparation

Materials


Materials uncooked spaghetti

response boards


Supplements Acces 4 Module

ProofBlocks©

Technology N/A


Background

Information As converses of theorems and postulates are introduced, the students

tend to think the content of the lesson is a repeat of the previous

section. They do not differentiate between the two distinctly different

statements. This error becomes evident when the students are

completing proofs. To help them choose the correct postulates and/or

theorems, point out that the given information corresponds to the

hypothesis in a conditional statement. Often times the students have

difficulty moving from the given information to the second step in the

proof. The students may incorrectly assume relationships based on the

lines ‘looking’ parallel. Emphasize that lines are not parallel until they

are proven parallel.


the angles given parallel lines cut by a transversal, in grades 7 and 8.

The students applied the relationships between the angles to solve for

missing angle measures in the previous middle school grades. The

students did not consider the idea of the converses of the parallel line

theorems and postulate. This section will be new learning for the

students, although it may not appear to be new learning to them.

Continually stress the differences between the parallel line theorems,

postulate, and their converses.

Core

Instructional

Strategies

Level 1

Begin by reviewing the types of conditional statements. Present the

Vertical Angles Theorem. Have the students write the theorem as a

conditional statement, and then write the converse, inverse, and

contrapositive. Which statements are true? Which statements will

always have the same truth values? Point out that a conditional

statement and its converse are not logically equivalent and therefore

will not always have the same truth values. (field independent,

auditory)




D-14

Level 2

Display the Corresponding Angles Postulate, and have the students

write the converse. Identify this converse as a postulate, and that it is a

true statement. While postulates do not have to be proven true, how do

you know that the converse of the Corresponding Angles Postulate

must be true? How can you show this? Lay three pieces of uncooked

spaghetti under the document camera, and invite a student up to try and

disprove the Converse of the Corresponding Angles postulate.

Demonstrate how the congruency of the corresponding angles forces

the lines to be parallel.

Present Example 1B, p. 162, and discuss the steps needed to show that

the lines are parallel. Allow time for the students to conclude that both

corresponding angles equal 140°. What can we conclude about the

lines? Why? Stress the importance of stating that the angles are

congruent in order to correctly use the converse of the Corresponding

Angles Postulate. Use the Additional Examples Transparency and

Power Presentation, Example 1, for extra practice.

Split the class into three groups and assign each group a parallel line

theorem. Instruct the students in each group to write the converse of

their theorem and determine its truth value. Invite one student from

each group to the board to write each converse. Remind the students

that converses are not guaranteed true until proven so. Emphasize that

the Converse of the Corresponding Angles is a postulate, and is

accepted as true. Summarize the converses using a chart similar to the

one on p. 163.

Lay three pieces of uncooked spaghetti under the document camera,

and invite students up to attempt to disprove the converses.

Demonstrate how the congruency or supplementary of the angle pairs

force the lines to be parallel. Note that more rigorous proofs are

necessary to prove converses true, and will be done later in this section.

Work through more examples that apply the converses, such as

Examples 2 and 4, pp. 164 – 165. Remind the students that they must

state that the angles are congruent or supplementary before using

converses to conclude that the lines are parallel. Extend the level of

difficulty in the exercises by adding multiple parallel lines and

transversals. Refer to pp. 167 – 169 (30–35, 46–53). Consider using

response boards for a quick assessment of student understanding and to

provide immediate feedback to the students. (field dependent, tactile)

Level 3

Present the ProofBlock© the students created in section D-2 for the

Corresponding Angles Theorem. How will this block need to be

changed for the Converse of the Corresponding Angles Theorem?

Allow the students time to build the ProofBlocks© for each of the

converses of the parallel line theorems.




D-15

Model the proof of the Converse of the Alternate Exterior Angles

Theorem using the ProofBlocks©. Invite students to the board to

complete a two-column proof, a paragraph proof, and a flowchart

proof. Compare and contrast the proof methods, correcting any errors

as a class. Guide the students through the two-column proof of

Example 3, p. 164, andflowchart proof for “Check It Out!”, p. 164 (3).

Assign the two-column proof on p. 167 (22) for the students to

complete independently. Allow the students to modify the format of the

proof to a paragraph, flowchart or ProofBlock© proof. Have the

students partner with another student who chooses the same method of

proof to verify their proof. (field dependent, sequential)

Level 4

Assign pp. 166 – 168 (10, 38, 39) as independent practice with writing

proofs. Consult the Acces 4 Module, Geometry Database, Parallel Lines

section and Geometric Proofs section for additional practice with

proofs and situations involving multiple transversals and parallel lines.

Differentiation

Strategies

Accelerate-Review-Reteach

Use the “Think and Discuss” at the bottom of p. 165 to help the

students understand the difference between the theorems and their

converses.

Distribute Lesson 3-3 Reading Strategies, Chapter Resource to

summarize the four ways to prove lines parallel. Have the students

work through exercises 1 – 6.

Provide several proofs that are partially completed. Allow the students

to practice correctly filling in the blanks before assigning independent

proofs.

Have the students analyze proofs in which there are errors. Allow the

students to work in pairs to identify the errors and rewrite the proofs

correctly.


Introduce proofs with several parallel lines and transversals, such as

p. 169 (4651, 54–56).

Encourage critical thinking by assigning Lesson 3-3 Challenge,

Chapter Resources, Volume 1.

Apply postulate and

theorems to write

proofs given parallel

lines

Verify that lines are

parallel using algebra

and formal proofs

Explore and prove

relationships involving

perpendicular lines



Geometry 2011 D-4 Perpendicular Lines

D-16

D-4 Perpendicular Lines (Section 3-4)

Objective(s) Prove and apply theorems about perpendicular lines.

Alignment CCSC: G.CO.9, G.CO.12 AIM: O-7

Essential

Understanding

The relationships of two lines to a third line can be used to determine if the

two lines are parallel or perpendicular to each other.

Assessment Level 1

Name the shortest segment from

point V to WZ .

VY

Level 2

Find x and y.

15,30 yx

Level 3

Use construction techniques to

verify each theorem below:

1. Through a point not on a line,

there is one and only one line

perpendicular to the given line.

2. Through a point on a line, there

is one and only one line

perpendicular to the given line.

[Construction tools may vary;

See SA]

Level 4

Given: 18021 mm , ac

Prove: ab

[See SA]

YW

Z

V

X

dc

b

a

2

1

(10x + 4y)°

(3x)°




D-17

Teacher

Preparation

Materials


Materials compass straightedge patty paper

Supplements Paving the Way, RS D-4

Parallel Postulate, RS D-4a

Serra, Michael. Patty Paper Geometry

Acces 4 Module



Background

Information Read “Math Background” TE p. 146B for a historical development of

the Parallel Postulate. As done in section D-4 of this curriculum, this

“Math Background” shows the connections between the Parallel

Postulate, perpendicularity, and constructions. Review this reading as a

‘professional learning’ guide to gain insight on how to effectively

present these interrelated concepts to the students.

In Grade 7, the students learned how to construct the perpendicular

bisector of a given line segment. In Grade 8, the students constructed

segments perpendicular to a given segment through a given point. The

students also copied angles using a compass and straightedge in order

to construct congruent triangles.

Core

Instructional

Strategies

Level 1 – 2

Present the scenario from Paving the Way, RS D-4. Provide the

students with construction tools such as protractor, straightedge, patty

paper, and Mira. Let the students to puzzle through the situation

without providing guidance or instructions. Divide the students into

pairs to share their approach to the construction. Facilitate a class

discussion outlining the possible constructions. Bring about the various

techniques for constructing parallel lines, using patty paper and/or a

compass, pp. 163, 170 – 171. Make connections to translations,

pointing out that the angles translate along the horizontal line to create

congruent corresponding angles. Which of the converses discussed

earlier support our constructions, and prove that the lines constructed

are definitely parallel?

Extend the students’ understanding of constructions and parallel lines

with the Parallel Postulate, RS D-4a. Note that the Parallel Postulate

guarantees that for any line l, a parallel line can always be constructed

through a point that is not on l. Why can there be only one parallel line

through the point? Which converse supports this postulate and the

construction?




D-18

Transition to perpendicular lines by modifying the parking lot example

to include parking spaces that are perpendicular to the given horizontal

line. How can we be sure that the parking lines already created are

parallel? Guide the students to the conclusion that the right angles

formed by the perpendicular lines are congruent corresponding angles,

which makes the lines parallel. Have the students use this application to

develop the theorem, “If two coplanar lines are perpendicular to the

same line, then the two lines are parallel to each other.”

Introduce the perpendicular bisector construction as a tool that can be

used to create parallel lines. Guide the students through the

construction as shown on p. 172. Connect this to the patty paper

construction the students used in Unit A. Ask, How could we add to

this construction to get parallel lines? Demonstrate the additional steps

necessary to construct two parallel lines with a perpendicular

transversal, p. 179. (field independent, tactile, active)

Have the students complete Open Investigations 2.2 – 2.5

from Patty Paper Geometry, to discover the Parallel

Postulate and various perpendicular line theorems.

Use the results of the constructions to develop the various

perpendicular theorems in the “Know It Note” at the top of p. 173.

Present the hypotheses of each theorem, and have the students make

the conclusions based on their constructions. Give the students a blank

chart in which to record and summarize the theorems.

Distribute patty paper and instruct the students to complete

Perpendicular Lines Exploration 3-4. Summarize the activity and apply

the definition of the distance from a point to a line using Example 1

and “Check it Out” p. 172 (1).

Level 3

Present the given information and diagram from the Perpendicular

Transversal Theorem Proof on p. 173. What is the shortest segment

from BC to DE ? Is there enough information to conclude that

DEAB ? Informally prove that DEAB .

Model Example 2 on p. 173 to introduce the students to formal proofs

involving perpendicular lines. Present p. 175 (4, 8) and allow the

students to work with the person beside them to complete the proof.

(field dependent, sequential)

Level 4

Give the students the opportunity to write a proof using only the given

information with Lesson 3-4 Additional Example 2, “Check it Out” (2)

on p. 173, exercise 23 on p. 176, and exercise 18 on p. 181.

Consult the Acces 4 Module, Geometry Database, Geometric Proofs

section for additional proofs involving perpendicular lines.





D-19

Differentiation


Help the students complete the graphic organizer from “Think and

Discuss” on p. 174 to help the students translate the theorems from

words to symbols.

Reinforce solving inequalities and the definition of the distance

between a point and a line with Lesson 3-4 Reteach, Chapter

Resources.

Create several short proofs and place the diagram with the statements

and reasons in an envelope. Distribute one envelope to each student

and instruct them to put the statements in logical order and match each

statement with the appropriate reason. Check their answers before

allowing them to switch envelopes with another student.

Review basic vocabulary and the converses from the previous section

using exercises 40-45 on p. 178.


Introduce the circumcenter of a triangle using the Lesson 3-4

Challenge, Chapter Resources.

Consider allowing the students to complete the previous activity using

Geometer’s Sketchpad, instead of a compass and straightedge.

Prove lines parallel

given angle

relationships

Apply theorems

relating parallel and

perpendicular lines

Identify parallel and

perpendicular lines

from their slopes



Geometry 2011 D-5 Slopes of Lines, Lines in the Coordinate Plane

D-20

D-5 Slopes of Lines, Lines in the Coordinate Plane (Sections 3-5, 3-6)

Objective(s) Find the slope of a line.

Use slopes to identify parallel and perpendicular lines.

Graph lines and write their equations in slope-intercept and point-slope

form.

Classify lines as parallel, intersecting, or coinciding.

Alignment CCSC: G.GPE.5 AIM: O-8

Essential

Understanding

Understanding the slopes of parallel and perpendicular lines is essential for

completing coordinate proofs.

Assessment Level 1

Determine whether lines with the

given slopes represent parallel lines

or perpendicular lines.

1. 2

10 and

1

5

2. 4

1 and 4

[parallel, perpendicular]

Level 2

Write the equation of the line with

slope of 2

3through the point (0, -1),

in slope-intercept form.

1

2

3xy

Level 3

Find the equation of the line perpendicular to the line 2 5y x through

the given point P(–2, 3).

4

2

1xy

Teacher

Preparation

Materials


Materials response boards


patty paper

coordinate grid

equation posters

Supplements Power Presentations with PowerPoint

Chapter Resources, Volume 1

Classifying Lines, RS D-5

Classifying Lines – Exploration Sheet, WS D-5a

Classifying Lines – Verification Sheet, WS D-5b




D-21

Technology Microsoft Office PowerPoint

ActivInspire

Student Response System


Background

Information Sections 3-5 and 3-6 have been combined to form the guide section

D-5. The students have studied the majority of the content in this

section in previous mathematics courses. The intent of this section is to

refresh the students’ knowledge, not reteach it. Calculating slope and

writing equations of lines are covered in previous algebra courses. By

the end of the eighth grade, all students have identified the slope and y-

intercept of lines when given a graph. Students may still struggle with

finding the slope and writing the equation of a line. All calculations

and graphing should be done using the graphing calculator. Avoid

calculations and graphing by hand since this content is a review.Note

that the work in this section is to prepare the students for coordinate

proofs in later sections, such as Section E-7 (Introduction to Coordinate

Proofs) and Unit G (Polygons and Quadrilaterals). Understanding the

slopes of parallel and perpendicular lines is essential for this future

work with coordinate proofs.

Core

Instructional

Strategies

Level 1

Review the slope formula with Power Presentations Warm Up 3-5. Do

these equations look familiar? Does anyone remember what this

equation represents? Distribute response boards to the students. Use

the Additional Examples 3-5 Power Presentation to assess the students’

proficiency calculating slope. Summarize the types of slopes by

displaying only the graphs from the graphic organizer on p. 183, and

have the students appropriately name the type of slope shown. (field

independent, global)

Level 2

Have the students find the slope of each line in Example 3A, p. 184.

What does it mean when lines have equal slopes? Have the students

sketch the graphs of the lines for a visual reminder that parallel lines

have equal slopes.

Display Example 3B, p. 184, and ask the students to identify the

relationship between the lines using only their slopes. How can you tell

what kind of lines they are if their slopes are not the same? Does

anyone remember how to tell if lines are perpendicular by their slopes?

Distribute pieces of patty paper and a coordinate grid. Have the

students construct a pair of perpendicular lines. Lay the patty paper on

top of a coordinate grid so that one of the lines has a positive slope.

Have the students identify two points on each line, and calculate the

slopes of the lines. Discuss the relationship between the slopes as a




D-22

class. Extend the concept of opposite reciprocals to the Perpendicular

Lines Theorem. Require the students to multiply the slopes of the lines

on the patty paper to verify that their product is -1. Use “Check it Out”

p. 184 (3a, b, c) to reinforce the concepts reviewed.

Transition to writing equations of lines with Power Presentations

Warm Up 3-6. What information is necessary to use the equation

bmxy ? What is the name of this equation? Review the other

forms of linear equations using the “Know it Note” on p. 190.

Review writing equations of lines, graphing lines, and classifying pair

of lines using Examples 1 – 3 on pp. 191 – 192. Allow the students to

perform all calculations and all graphing using the graphing calculator.

Avoid calculations and graphing by hand since this content is a review

from previous mathematics courses. Assess the students with Power

Presentations, Additional Examples 1 – 3. (field independent, tactile)

Merge the Power Presentations into an ActivInspire Flipchart

Use the Student Response System to assess student

proficiency.

Use the graphing calculator to graph linear equations, and to

explore the slopes of parallel and perpendicular lines using

Explore Parallel and Perpendicular Lines, Technology Lab

3-6, pp. 188 – 189.

Level 3

Provide independent practice using Classifying Lines, RS D-5.

Transfer the equations from Section A onto green sheets of paper and

the equations from Section B onto white sheets of paper. Post the

equations around the room in random order. Instruct the students to

begin at a sheet of green paper and to find the slope and the y–intercept

of the line printed on that paper. Direct the students to then go to at

least four white sheets of paper to find lines that are perpendicular,

parallel, intersecting, and coinciding with the line from the green paper

they chose. Make clear to the students that they may need to visit

several white papers, and find the slope and y-intercept of several

equations, before finding the four lines that satisfy the required given

conditions. Require the students to show the work for each equation

tested on Classifying Lines – Exploration Sheet, WS D-5a, and to

graph the pairs of lines as indicated on Classifying Lines – Exploration

Sheet, WS D-5b. (field dependent, active)

Differentiation


Ask, What movements are necessary for you to leave your seat?

Connect this to the slope ratio by emphasizing the importance of

‘rising’ from your chair before taking a step or ‘running’.




D-23

Use Lesson 3-5 Practice A in Chapter Resources, for a review of slope

basics. Assign Lesson 3-6 Practice B in Chapter Resources for

additional review of point-slope and slope-intercept forms of equations.

Review and relate concepts from Algebra using “Connecting Geometry

to Data Analysis, Scatter Plots and Lines of Best Fit” on p. 198.


Use the Lesson 3-5 Challenge, Chapter Resources, Volume 1 to

connect slopes to classifying coordinate quadrilaterals.

Divide the class into small groups and assign exercises 62-66 on p.

197. Have each group present their solution to exercises 63 and 64.

Apply theorems

involving parallel and

perpendicular lines

Use slopes to identify

parallel and

perpendicular lines

Write the equations of

lines perpendicular or

parallel to a given line

Geometry 2011 RS D-1

D-24

Vocabulary BINGO - Planes, Lines, and Angles

Directions: Fill in the game board with words from the Word Box. Words may be repeated.

Word Box Words for

“Lines” Column

Words for

“Planes” Column

Words for

“Angles” Columns

Pairs for “Name Me”

Column

Skew Parallel Same-Side Interior 4 and 8 4 and 5

Parallel Perpendicular Alternate Exterior 3 and 7 3 and 6

Perpendicular Corresponding 2 and 6 3 and 5

// Alternate Interior 1 and 5 4 and 6

// 2 and 8 1 and 7

GAME BOARD

LINES PLANES ANGLES NAME ME


D-25

Vocabulary BINGO - Planes, Lines, and Angles

Use the diagrams below to choose the correct name or pair of figures from the

Game Board. Mark the corresponding box on the Game Board with a highlighter

or a slash. Five in a row wins, horizontally, vertically, or diagonally.

B C

R

SP

AD

Q

3

2 1

4

5

8

6

7

Geometry 2011 RS D-1a

D-26


D-27

8 765

4 321

8 765

4 321


D-28

Drawing a Parallel

Figure 1 Figure 2

Name a pair of corresponding angles in Figures 1 and 2.

Name a pair of alternate interior angles in Figures 1 and 2.

How are Figure 1 and Figure 2 similar?

How are they different?

m

l

k

87

6 5

43

2 1a

c

b

1615

14 13

1211

10 9

Geometry 2011 RS D-2b

D-29

ProofBlock© Templates

Postulate

[][]

Corresponding

Angles

Postulate []//[]

Theorem

[][]

Alternate

Interior

Angles

Theorem

[]//[]


D-30

Paving the Way

Leslie has been hired to help finish the

resurfacing of a parking lot. Her job is to paint

the lines that designate parking spaces. Before

she arrives to the job site, two spaces are painted

as guidelines.

How can she complete a row of eight parking

spaces, and guarantee that all the lines are

parallel?


D-31

Parallel Postulate

Use construction techniques to

verify the truthfulness of this

postulate.

Through a point P

not on line l, there

is exactly one line

parallel to l.


D-32

Classifying Lines

Section A:

(Place each equation onto a green sheet of paper. Note that all equations have a slope of 3

2 .)

yx

xy

xy

xy

xy

xy

464

232

2

31

)4(2

37

)2(2

34

12

3

xy

xy

yx

xy

xy

yx

2)1(3

623

532

)4(3

24

13

2

464

Section B:

(Place each equation onto a white sheet of paper.)

)1(32

4)1(6

323

32

232

13

2

yx

xy

xy

xy

yx

xy

)4(2

39

)2(2

36

32

3

623

632

xy

xy

xy

yx

xy


D-33

Classifying Lines – Exploration Sheet

My equation: ___________________ slope = _________ y-intercept = _________

Use the space below to show your work for each equation tested. Circle the symbol at the bottom

of the box to indicate the line’s relation to your original line.

Equation:

slope = y-int =

Equation:

slope = y-int =

Equation:

slope = y-int =

Equation:

slope = y-int =

Equation:

slope = y-int =

Equation:

slope = y-int =

Equation:

slope = y-int =

Equation:

slope = y-int =

Equation:

slope = y-int =

x → x → x →

x → x → x →

x → x → x →

(intersecting: x, coincinding: →)

Geometry 2011 RS D-5b

D-34

Classifying Lines – Verification Sheet

Directions: Graph each pair of lines as indicated below. Be sure to label the lines on each graph.

1. Graph the original line and the line

parallel to the original.


perpendicular to the original.

Equation of parallel line:

Equation of perpendicular line:


intersecting the original.


coinciding with the original.

Equation of intersecting line:

Equation of coinciding line:

Geometry 2011 Unit D Supplementary Answers

D-35

Unit D: Parallel and Perpendicular Lines

Supplementary Answers to Assessment Questions

D-2: Level 3

Answers may vary. Sample answer:

D-2: Level 4


Statements Reasons

1. 35 1. Given

2. ||n m 2. Converse of Alternate Interior Angle Theorem

3. ml || 3. Given

4. 5 11 4. Alternate Exterior Angle Theorem

5. 113 5. Transitive Property of Congruence

D-3: Level 3

1. q r 1. Given

q s

2. 1 90m 2. Definition of Perpendicular

2 90m

3. 1 2m m 3. Substitution

4. 1 2 4. Definition of Congruent Angles

5. ||r s 5. Converse of Corresponding

Angles Postulate

18058 mm 18032 mm ml ||

Given Conv. Same-Side Int. Thm.

35 mm

35

Vertical Thm.

28 mm

28

Vertical Thm.

Def. of Def. of

Substitution

r

s

1

Geometry 2011 Unit D Supplementary Answers

D-36

D-3: Level 4


Statements Reasons

1. 180101 mm 1. Given

2. 1 2 180m m 2. Linear Pair Theorem

3. 1 2 1 10m m m m 3. Substitution

4. 2 10m m 4. Subtraction Property of Equality

5. ||l n 5. Converse of Corresponding Angles Postulate

D-4: Level 3

D-4: Level 4

Statements Reasons

1. 18021 mm 1. Given

2. ||b c 2. Same-Side Interior Angle Theorem

3. ac 3. Given

4. ab 4. Perpendicular Transversal Theorem

A

B

unit d parallel and perpendicular lines -...

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