Geometry Unit 1 Parallel & Perpendicular Lines
Enduring understanding (Big Idea): Students will be able to apply and understand the structure (properties) and
relationships between points, lines, planes, and angles.
Essential Questions:
1) What are the undefined terms in Geometry?
2) How can you describe the attributes of a segment or angle?
3) How do you prove that 2 lines are parallel or perpendicular?
BY THE END OF THIS UNIT:
Students will know…
1) distance and midpoint formulas
2) angle relationships formed by intersecting lines
Vocabulary:
1) Undefined terms
2) transversal
3) special angles formed by intersecting lines
Students will be able to…
Find the lengths of a line segment
Find the midpoint of a line segment
Determine angle relationships of intersecting lines
Write linear equations of parallel & perpendicular lines
Interpret slopes of parallel and perpendicular lines
Prove lines parallel or perpendicular
Unit Resources
Learning Task:
Parallel and Perpendicular Lines using Patty Paper
Parallel lines cut by a transversal using cabri jr
Mathematical Practices in Focus:
TI: Points, Lines, and Planes
2) Model with mathematics
Performance Task: Pearson Text Online (Ch 1
performance task)
3) Attend to precision
1) Make sense of problems and persevere in solving them
Unit Review Game: Geometry BINGO
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
.
Geometry Unit 1 Parallel & Perpendicular Lines
CORE CONTENT
Cluster Title: Apply properties, definitions and theorems of angles and lines
Standard 7.G.5. Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write
and solve simple equations for an unknown angle in a figure.
Concepts and Skills to Master
Define and understand properties of angle relationships, adjacent angles, linear pairs, vertical angles, supplementary angles and
complementary angles.
Complete simple proofs with special angle pairs.
Use properties of supplementary, complementary, vertical and adjacent angles to solve for unknown angles in a figure
Write and solve equations based on a diagram of intersecting lines with some known angle measures.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
Solve multi-step algebraic equations.
Academic Vocabulary
Supplementary angles, Complementary angles, Vertical angles, Linear pairs of angles, Angle bisector, intersecting lines
Suggested Instructional Strategies
Resources
Have students draw intersecting lines and compare opposite angle
Textbook Correlation:
measures and adjacent angle measures with a compass.
Use patty paper or a reflecting device (mira) to explore relationships
1-5 exploring angle pairs
among the measures of angles formed by intersecting lines.
Draw two intersecting lines. Measure one angle and find the measures
of the others without measuring.
Provide explicit examples of supplementary, complementary, vertical,
and adjacent angles and demonstrate their relationships, including
adjacent angles that are neither supplementary nor complementary.
Sample Formative Assessment Tasks
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
.
Geometry Unit 1 Parallel & Perpendicular Lines
Skill-based task
I. Pearson Chapter one Performance task, use Task 1
or Task 4
II. Solve for x and y:
Problem Task
If 𝑚∠𝐵=102° and 𝑚∠𝐿=120°, find every other angle measure, explaining
how you found each.
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
.
Geometry Unit 1 Parallel & Perpendicular Lines
CORE CONTENT
Cluster Title: Experiment with transformations in the plane
Standard G.CO.1 Know precise definitions of angle, circle, perpendicular lines, parallel lines, and line segment, based on the
undefined notions of point, line, distance along a line, and distance around a circular arc.
Concepts and Skills to Master
Define angle, circle, perpendicular line, parallel line, and line segment
Use precise definitions to identify and model an angle, circle, perpendicular line, parallel line, and line segment
Demonstrate mathematical notation for each term.
Apply the segment addition postulate and the angle addition postulate.
Apply properties of lines and transversals. Identify special angle pairs.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
Solve algebraic equations; Understanding the undefined terms: point, line, and plane; Understand distance is a non-negative quantity
Academic Vocabulary
Angle, circle, perpendicular line, parallel line, line segment, distance, arc
Suggested Instructional Strategies
Have students write their own understanding of a given term
Give students formal and informal definitions of each term and compare
them
Develop precise definitions through use of examples and non-examples
Discuss the importance of having precise definitions
Start line segment addition with integers, move to labeled segments then
expressions.
Do the same with angle addition postulate.
Sample Formative Assessment Tasks
Skill-based task
State the definition of a circle
Resources
Textbook Correlation:
1-2 points, lines and plane
1-3 measuring segments
1-4 measuring angles
1-5 exploring angle pairs
3-1 lines and angles
Problem Task
Identify real-life examples of each term using definitions.
Pearson Chapter one Performance task, use Task 1 or Task 4
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
.
Geometry Unit 1 Parallel & Perpendicular Lines
CORE CONTENT
Cluster Title: Prove Geometric Theorems
Standard: G.CO.9 -- Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel
lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are
exactly those equidistant from the segment’s endpoints.
Concepts and Skills to Master
Identify parallel, perpendicular, and skew lines and parallel planes
Classify/determine angle relationships formed by two lines and a transversal
Prove that vertical angles are congruent.
Prove that when parallel lines are cut by a transversal, pairs of alternate interior angles are congruent, pairs of alternate exterior angles are congruent, and
pairs of same-side interior angles are supplementary.
When given parallel lines, use properties of parallel lines to calculate angle measures and solve for variables
When given angle measures, determine which lines, if any, are parallel
SUPPORTS FOR TEACHERS
Critical Background Knowledge
Solve multi-step equations; know properties of supplementary, complementary, vertical, & adjacent angles (7.G.5);
Academic Vocabulary
Vertical angles, Parallel lines & planes, Transversal, Alternate Interior/Exterior angles (AIA and AEA), Corresponding angles (CA), Same-side Interior
angles (SSI), Skew lines
Suggested Instructional Strategies
-Use multiple formats to write justifications: narrative paragraphs, flow diagrams, two-column
format, and diagrams without words.
-When teaching skew lines, highlight all segments in a rectangular prism that are not skew; thus all
other segments are skew.
-Use dynamic geometry software to explore angle relationships
-Connect angle relationships to the creation of tessellation patterns
-Formal and informal proofs do not have to be introduced at this point. Students can argue with
justification by completing short answer/open-ended questions such as “John says that consecutive
interior angles are congruent when lines are parallel. How would you convince John that that these
angles are actually supplementary?” Focus on the validity of the underlying reasoning of the
justifications
-The “Build a City” project or similar activity is suggested as a real-world application follow-up
Resources
Textbook Correlation: 2-6, 3-1, 3-2, 3-3, 3-4
Online Teacher Resource Center 3-1 Game: Name It –
Claim It (Suggestion, enlarge the four diagrams on an 8 x
10 and allow students more room to play with their die or
number cube)
Online Teacher Resource Center 3-2 Performance
Task Activity (Suggestion, enlarge the four diagrams on
an 8 x 10 and allow students more room to play with their
die or number cube)
Angle Hunter Video
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
.
Geometry Unit 1 Parallel & Perpendicular Lines
assignment to assess student understanding.
Sample Formative Assessment Tasks
Skill-based task
Lines a and b are parallel.
Solve for x.
Solve for y.
Find the measures of angles 1-5.
Based on this information, are lines c and d
parallel? In complete sentences, explain why or
why not.
Problem Task
Marie is building a sandbox in her back yard.
Only equipped with the tools to measure angles, how can Marie determine whether
both pairs of opposite sides are parallel?
If both pairs of opposite sides are parallel, and one of the angles measures 85
degrees, what are the measures of the remaining angles?
Find as many angle relationships as possible in this pattern:
Have students plan and draw the street map for a fictional city
according to the directions listed below.
I. Appearance
II.
Suppose a || b and c || d.
1. If m6 = 50, then find m11.
2. If m2 = 70, then find m6.
- You will make a detailed drawing of your rough sketch
- Buildings must be placed according to the directions on the following page
- Appropriate building names must be placed on “signs” on or near the buildings
(e.g. Bank of America)
- Your name should be written in the lower right hand corner of your drawing
II. Required Items
- At the top of the drawing, place the name of the city and give it a population #
- 6 parallel streets – all named (be creative)
- 2 transversal streets that are not parallel to each other– named
4. If m4= 45, then find m12.
- Traffic lights or stop signs at 4 intersections
- gas station and restaurant at congruent alternate exterior angles
5. Which angle could you show is congruent to 11 to prove a || b?
- your house and your school at supplementary same side interior angles
- the courthouse and bank at non-congruent alternate interior angles
6. What relationship between 6 and 11 shows c || d?
- a department store and a place of worship at congruent corresponding angles
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications
resources and
for this
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by CMS alternate
teacher leaders.
Standards
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listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form- aareasonable
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structure/object of your choice that is skew to some part of the city
.
3. If m7= 110, then find m10.
Geometry Unit 1 Parallel & Perpendicular Lines
Teacher Created Argumentation Tasks (W1-MP3&6)
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
.
Geometry Unit 1 Parallel & Perpendicular Lines
CORE CONTENT
Cluster Title: Make geometric constructions
Standard: G.CO.12 -- Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string,
reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment;
bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line
parallel to a given line through a point not on the line.
Concepts and Skills to Master
Perform the following constructions using a variety of tools and methods: copying a segment, copying an angle, bisecting a segment,
bisecting an angle, constructing perpendicular lines, including the perpendicular bisector of a line segment, and constructing a line
parallel to a given line through a point not on the line.
Explain why these constructions result in the desired objects.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
Definitions of the following terms: circle, bisector, perpendicular, and parallel
Academic Vocabulary
Segment, Angle, Bisect, Perpendicular, Parallel, Circle, Construction, Transversal, Alternate Interior/Exterior angles, Corresponding
angles, Same-side Interior angles, Skew lines, Parallel planes
Suggested Instructional Strategies
Resources
Completing constructions can be used as a follow up activity or as an investigation.
If used as an investigation, provide students with open-ended follow up questions to
help them draw accurate conclusions. For example, after giving students instructions
for constructing an angle bisector, pose the following: “Fold your patty paper along
this line. What can you conclude about each angle? Which of the vocabulary terms
from this section have you just constructed?”
Have students explore how to make a variety of constructions using different tools. Ask
students to justify how they know their method results in the desired construction.
Discuss the underlying principles that different tools rely on to produce desired
Textbook Correlation: 1-6, 3-6
Investigation Activity (from CMS Curriculum Guide)
http://math.springbranchisd.com/high/classes/alge
bra_one/Laying%20The%20Foundation/Lessons/
Parallel%20and%20Perpendicular%20Lines%201
87-188.pdf
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
.
Geometry Unit 1 Parallel & Perpendicular Lines
constructions (e.g. compass:circle; miro: reflections)
Sample Formative Assessment Tasks
Skill-based task
Using patty/tracing paper, pencil, straight edge, and
compass, complete the following constructions
Parallel lines
Perpendicular lines
Perpendicular bisector
Angle bisector
Problem Task
Jessica is studying architecture at the University of North Carolina
at Charlotte. For homework, she must find the center point of a
regular pentagon by connecting all of the angle bisectors.
Unfortunately, Jessica has her straight edge, but has lost her
protractor. What step by step instructions would you give to
Jessica to help her complete the assignment?
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
.
Geometry Unit 1 Parallel & Perpendicular Lines
CORE CONTENT
Cluster Title: Use coordinates to prove simple geometric theorems algebraically
Standard G.GPE.5: Prove the slope criteria for parallel and perpendicular lines; use them to solve geometric problems (e.g.,
find the equation of a line parallel or perpendicular to a given line that passes through a given point)
Concepts and Skills to Master
Prove that the slopes of parallel lines are equal
Prove that the product of the slopes of perpendicular lines is -1.
Use slope criteria for parallel and perpendicular lines to solve geometric problems.
Write the equation of a line parallel or perpendicular to a given line, passing through a given point.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
Graph parallel and perpendicular lines using transformations; write the equation of a line through a specific point.
Academic Vocabulary
Parallel, perpendicular, reciprocal
Suggested Instructional Strategies
Resources
Allow students to explore and make conjectures about
Textbook Correlation:
relationships between lines and segments using a variety
of methods.
Discuss the role of algebra in providing a precise means of
representing a visual image.
Sample Formative Assessment Tasks
Skill-based task
Find an equation of a line perpendicular to y = 3x – 4 that
Problem Task
Verify that the distance between two parallel lines is constant.
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
.
Geometry Unit 1 Parallel & Perpendicular Lines
passes through (3, 4)
Justify your answer.
CORE CONTENT
Cluster Title: Use coordinates to prove simple geometric theorems algebraically
Standard G.GPE.6: Find the point on a directed line segment between two given points that partitions the segment
Concepts and Skills to Master
Use coordinate geometry to divide a segment into a given ratio
Determine the midpoint of a segment given two endpoints
Determine the endpoint of a segment given a midpoint and one endpoint
SUPPORTS FOR TEACHERS
Critical Background Knowledge
Academic Vocabulary
Coordinates, a:b ratio, directed line segment
Suggested Instructional Strategies
Sample Formative Assessment Tasks
Skill-based task
A segment with endpoints A(3,2) and B(6,11) is
partitioned by a point C such that AC and CB form a 2:1
ratio. Find C.
Resources
Textbook Correlation:
Problem Task
A point B(4, 2) on a segment with endpoints A(2, -1) and C(x, y)
partitions the segment in a 1:3 ratio. Find x and y.
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
.
Geometry Unit 1 Parallel & Perpendicular Lines
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
.