Unit D Parallel and Perpendicular Lines -...
-
Upload
truonghanh -
Category
Documents
-
view
212 -
download
0
Transcript of 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
Unit D Parallel and Perpendicular Lines
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
Baltimore County Public Schools Division of Curriculum and Instruction
Unit D Parallel and Perpendicular Lines
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
Baltimore County Public Schools Division of Curriculum and Instruction
Unit D Parallel and Perpendicular Lines
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
Baltimore County Public Schools Division of Curriculum and Instruction
Unit D Parallel and Perpendicular Lines
Geometry 2011 D-1 Lines and Angles
D-4
Teacher
Preparation
Materials
Text Holt Geometry, 2011 pp. 146-151
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 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.
Baltimore County Public Schools Division of Curriculum and Instruction
Unit D Parallel and Perpendicular Lines
Geometry 2011 D-1 Lines and Angles
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
Baltimore County Public Schools Division of Curriculum and Instruction
Unit D Parallel and Perpendicular Lines
Geometry 2011 D-1 Lines and Angles
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
Baltimore County Public Schools Division of Curriculum and Instruction
Unit D Parallel and Perpendicular Lines
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
Baltimore County Public Schools Division of Curriculum and Instruction
Unit D Parallel and Perpendicular Lines
Geometry 2011 D-2 Parallel and Perpendicular Lines
D-8
Teacher
Preparation
Materials
Text Holt Geometry, 2011 pp. 155-161
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
Technology Geometer’s Sketchpad
PowerPoint
www.proofblocks.com
Teaching Suggestions
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 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 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.
Baltimore County Public Schools Division of Curriculum and Instruction
Unit D Parallel and Perpendicular Lines
Geometry 2011 D-2 Parallel and Perpendicular Lines
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.
Baltimore County Public Schools Division of Curriculum and Instruction
Unit D Parallel and Perpendicular Lines
Geometry 2011 D-2 Parallel and Perpendicular Lines
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)
Baltimore County Public Schools Division of Curriculum and Instruction
Unit D Parallel and Perpendicular Lines
Geometry 2011 D-2 Parallel and Perpendicular Lines
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
Strategies Accelerate-Review-Reteach
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.
Enrichment-Extension
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
Baltimore County Public Schools Division of Curriculum and Instruction
Unit D Parallel and Perpendicular Lines
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
Baltimore County Public Schools Division of Curriculum and Instruction
Unit D Parallel and Perpendicular Lines
Geometry 2011 D-3 Proving Lines Parallel
D-13
Teacher
Preparation
Materials
Text Holt Geometry, 2011 pp. 162-169
Materials uncooked spaghetti
response boards
dry erase markers and erasers
Supplements Acces 4 Module
ProofBlocks©
Technology N/A
Teaching Suggestions
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 students identified and described geometric relationships between
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)
Baltimore County Public Schools Division of Curriculum and Instruction
Unit D Parallel and Perpendicular Lines
Geometry 2011 D-3 Proving Lines Parallel
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.
Baltimore County Public Schools Division of Curriculum and Instruction
Unit D Parallel and Perpendicular Lines
Geometry 2011 D-3 Proving Lines Parallel
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.
Enrichment-Extension
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
Baltimore County Public Schools Division of Curriculum and Instruction
Unit D Parallel and 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)°
Baltimore County Public Schools Division of Curriculum and Instruction
Unit D Parallel and Perpendicular Lines
Geometry 2011 D-4 Perpendicular Lines
D-17
Teacher
Preparation
Materials
Text Holt Geometry, 2011 pp. 172-178
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
Technology Geometer’s Sketchpad
Teaching Suggestions
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?
Baltimore County Public Schools Division of Curriculum and Instruction
Unit D Parallel and Perpendicular Lines
Geometry 2011 D-4 Perpendicular Lines
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.
Baltimore County Public Schools Division of Curriculum and Instruction
Unit D Parallel and Perpendicular Lines
Geometry 2011 D-4 Perpendicular Lines
D-19
Differentiation
Strategies Accelerate-Review-Reteach
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.
Enrichment-Extension
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
Baltimore County Public Schools Division of Curriculum and Instruction
Unit D Parallel and Perpendicular Lines
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
Text Holt Geometry, 2011 pp. 183-197
Materials response boards
dry erase markers and erasers
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
Baltimore County Public Schools Division of Curriculum and Instruction
Unit D Parallel and Perpendicular Lines
Geometry 2011 D-5 Slopes of Lines, Lines in the Coordinate Plane
D-21
Technology Microsoft Office PowerPoint
ActivInspire
Student Response System
Teaching Suggestions
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
Baltimore County Public Schools Division of Curriculum and Instruction
Unit D Parallel and Perpendicular Lines
Geometry 2011 D-5 Slopes of Lines, Lines in the Coordinate Plane
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
Strategies Accelerate-Review-Reteach
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’.
Baltimore County Public Schools Division of Curriculum and Instruction
Unit D Parallel and Perpendicular Lines
Geometry 2011 D-5 Slopes of Lines, Lines in the Coordinate Plane
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.
Enrichment-Extension
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
Geometry 2011 RS D-1
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
Geometry 2011 RS D-1a
D-27
8 765
4 321
8 765
4 321
Geometry 2011 RS D-2
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
[]//[]
Geometry 2011 RS D-4
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?
Geometry 2011 RS D-4a
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.
Geometry 2011 RS D-5
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
Geometry 2011 RS D-5a
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.
2. Graph the original line and the line
perpendicular to the original.
Equation of parallel line:
Equation of perpendicular line:
3. Graph the original line and the line
intersecting the original.
4. Graph the original line and the line
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
Answers may vary. Sample answer:
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
Answers may vary. Sample answer:
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