Hello, Guest!
 Instructional Focus DocumentGeometry
 TITLE : Unit 03: Relationships of Lines and Transversals SUGGESTED DURATION : 6 days

Unit Overview

This unit bundles student expectations that address special pairs of angles formed when one or more lines are intersected by a transversal. Constructions and manipulatives are used to explore the geometric relationships, and make conjectures by investigating patterns with a focus on parallel lines cut by a transversal and their related angles. Geometric conjectures are tested developing an awareness of the connections between conjectures, postulates, and theorems. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this unit, students in middle school became familiar with parallel lines in the context of different geometric figures such as trapezoids and parallelograms. In Algebra 1, students approached parallel lines as having the same slope but different y-intercepts, or as lines that do not intersect when analyzing a system of equations with no solutions. In Geometry Unit 02, students determined whether lines in a coordinate plane were parallel based on their slopes.

During this unit, students explore angle relationships formed by one line and one transversal including vertical angles, linear pairs, and adjacent angles. Students construct congruent angles and a line parallel to a given line through a point not on a line using a compass and a straightedge. Students investigate patterns to make conjectures and define angles formed by parallel lines cut by a transversal. Students explore angle relationships formed by two parallel lines and one or more transversal(s) including corresponding angles, same side interior angles, alternate interior angles, and alternate exterior angles. Students use a variety of tools such as patty paper, folding techniques, etc. to investigate these relationships between angle pairs formed when parallel lines are cut by a transversal(s). Students formulate deductive proofs for conjectures about angles formed by parallel lines and transversals and apply these relationships to solve mathematical and real-world problems. Students explore and apply the converse of theorems and postulates for parallel lines cut by a transversal to solve mathematical and real-world problems.

After this unit, in Geometry Unit 05, students will apply alternate interior angles to determine angles of elevation and angles of depression to solve problems. In Geometry Unit 07, students will investigate quadrilaterals and regular polygons using the parallel lines theorems and postulates. In Geometry Unit 04, students will continue to apply deductive proofs to verify congruency and similarity of triangles. These geometric concepts are foundational for the continued development of geometric reasoning in Geometry and subsequent courses in mathematics.

This unit is supporting the development of Texas College Career Readiness Standards (TxCCRS): III. Geometric Reasoning A1, A2, B2, D1, D2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

According to Focus in High School Mathematics: Reasoning and Sense Making (2009), the National Council of Teachers of Mathematics (NCTM) states that students should demonstrate growing levels of formality in their reasoning in the classroom during high school. The first step in this process involves conjecturing about geometric objects, where students should activate their natural inquisitiveness to wonder why something is happening. Students should then construct and evaluate geometric arguments, where students follow up their conjectures with efforts to justify or disprove them.

National Council of Teachers of Mathematics. (2009). Focus in High School Mathematics: Reasoning and Sense Making. Reston, VA: National Council of Teachers of Mathematics, Inc.

OVERARCHING UNDERSTANDINGS and QUESTIONS

Geometric relationships can be used to describe mathematical patterns.

• Why is it important to describe the geometric relationships found in spatial patterns?
• What geometric relationships can be found in patterns?

Geometric systems are axiomatic systems built on undefined terms, defined terms, postulates, and theorems which are fundamental in verifying conjectures through logical arguments.

• What roles do undefined terms, defined terms, postulates, and theorems serve in an axiomatic system?
• How does the investigation of geometric patterns lead to the development of conjectures and postulates?
• How are two-dimensional coordinate systems and algebra used to investigate and verify geometric relationships?
• How are logical arguments applied in the study of geometric relationships and their application in real-world settings?
• How is deductive reasoning used to understand, prove, and apply geometric conjectures and theorems pertaining to geometric relationships?
• How are logical arguments and deductive reasoning used to prove and disprove conditionals and their related statements?
• How can constructions be used to validate conjectures about geometric figures?

Application of attributes and measures of figures can be generalized to describe geometric relationships which can be used to solve problem situations.

• Why are attributes and measures of figures used to generalize geometric relationships?
• How can numeric patterns be used to formulate geometric relationships?
• Why is it important to distinguish measureable attributes?
• How do geometric relationships relate to other geometric relationships?
• Why is it essential to develop generalizations for geometric relationships?
• How are geometric relationships applied to solve problem situations?
Performance Assessment(s) Overarching Concepts
Unit Concepts
Unit Understandings
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic Reasoning

• Patterns/Rules
• Solve

Geometric Reasoning

• Constructions
• Geometric Attributes/Properties
• Geometric Relationships
• Logical Arguments
• Proofs
• Theorems/Postulates/Axioms

Measurement Reasoning

• Angle Measures

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Geometric relationships exist between the angles formed by a transversal that cuts parallel lines.

• How can patterns observed in angles formed by a transversal that intersects parallel lines be used to make conjectures about geometric relationships between the angles?

When lines intersect and when parallel lines are cut by a transversal, special relationships occur among the angles formed.

• What special angle relationships occur when lines intersect?
• What special angle relationships occur when parallel lines are cut by a transversal?

Angle relationships formed when lines are cut by a transversal can be used to prove lines are parallel.

• Are all lines cut by a transversal parallel? Justify your reasoning.
• What angle relationships must occur to prove that lines cut by a transversal are parallel?

Constructions provide insight into geometric relationships.

• How does construction of two parallel lines cut by a transversal validate the geometric relationships that occur between the angles formed?

Developing conjectures, theorems, and postulates is important to the development of logical reasoning.

• How does the investigation of patterns lead to the development of conjectures, theorems, and postulates?
• What is the difference between a postulate and a theorem?

MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that a visual confirmation is enough to determine that lines are parallel rather than verifying this property using geometric relationships.

Underdeveloped Concepts:

• Some students may confuse the vocabulary and terms to describe different angle pairs, such as same-side interior, alternate interior, alternate exterior, etc.

Unit Vocabulary

• Congruent angles – angles whose angle measurement are equal
• Conjecture – statement believed to be true but not yet proven
• Geometric construction – construction of accurate representations of lengths, angles, and geometric figures using only a straight edge and compass
• Parallel line – a line parallel to a given point not on a line
• Postulates(axioms) – statements accepted as true without requiring proof
• Theorems – conjectures that have been proven to be true
• Vertical angles – a pair of non-adjacent, non-overlapping angles formed by two intersecting lines creating angles that are opposite and congruent to each other

Related Vocabulary:

 Adjacent angles Alternate exterior   angles Alternate interior   angles Congruent angles Conjectures Corresponding   angles Converse of   postulates and theorems Linear pair Postulates Proofs Same-side interior   angles Same-side exterior   angles Supplementary   angles Theorems Transversal
Unit Assessment Items System Resources Other Resources

Show this message:

Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Creator if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards (select CCRS from Standard Set dropdown menu)

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Geometry Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TEKS# SE# TEKS Unit Level Specificity

• Bold black text in italics: Knowledge and Skills Statement (TEKS)
• Bold black text: Student Expectation (TEKS)
• Strike-through: Indicates portions of the Student Expectation that are not included in this unit but are taught in previous or future unit(s)
• Blue text: Supporting information / Clarifications from TCMPC (Specificity)
• Blue text in italics: Unit-specific clarification
• Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards (TxCCRS)
G.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
G.1A Apply mathematics to problems arising in everyday life, society, and the workplace.

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE

Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
G.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION

Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
G.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS

Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Real objects
• Manipulatives
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
G.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE

Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Graphs
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
G.1E Create and use representations to organize, record, and communicate mathematical ideas.

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
G.1F Analyze mathematical relationships to connect and communicate mathematical ideas.

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
G.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION

Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
G.4 Logical argument and constructions. The student uses the process skills with deductive reasoning to understand geometric relationships. The student is expected to:
G.4A

Distinguish between undefined terms, definitions, postulates, conjectures, and theorems.

Distinguish

BETWEEN POSTULATES, CONJECTURES, AND THEOREMS

Including, but not limited to:

• Postulates (axioms) – statements accepted as true without requiring proof
• Conjecture – statement believed to be true but not yet proven
• Theorems – conjectures that have been proven to be true
• Representation and notation for undefined and defined terms
• Connections between undefined terms and defined terms
• Connections between definitions, postulates, conjectures, and theorems
• Conjectures are suspected of being true but can be disproved with a counterexample or proven with a logical argument at which point a conjecture becomes a theorem.
• Definitions, postulates, and theorems can be used to support a logical argument that a conjecture is true.

Note(s):

• Geometry introduces the vocabulary of undefined terms, definitions, postulates, conjectures, and theorems.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• III. Geometric Reasoning
• D2 – Understand that Euclidean geometry is an axiomatic system.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
G.5 Logical argument and constructions. The student uses constructions to validate conjectures about geometric figures. The student is expected to:
G.5A

Investigate patterns to make conjectures about geometric relationships, including angles formed by parallel lines cut by a transversal, criteria required for triangle congruence, special segments of triangles, diagonals of quadrilaterals, interior and exterior angles of polygons, and special segments and angles of circles choosing from a variety of tools.

Investigate

PATTERNS TO MAKE CONJECTURES ABOUT GEOMETRIC RELATIONSHIPS, INCLUDING ANGLES FORMED BY PARALLEL LINES CUT BY A TRANSVERSAL CHOOSING FROM A VARIETY OF TOOLS

Including, but not limited to:

• Conjecture – statement believed to be true but not yet proven
• Investigations should include good sample design, valid conjecture, and inductive/deductive reasoning.
• Patterns include numeric and geometric properties
• Utilization of a variety of tools in the investigations (e.g., compass and straightedge, paper folding, manipulatives, dynamic geometry software, technology)
• Angles formed by parallel lines cut by a transversal
• Criteria required for triangle congruence

Note(s):

• Previous grade levels investigated attributes of geometric figures.
• Grade 8 used informal arguments to establish facts about the angle sum and exterior angle of triangles, the angles created when parallel lines are cut by a transversal, and the Angle-Angle criterion for similarity of triangles
• Geometry introduces analyzing patterns in geometric relationships and making conjectures about geometric relationships which may or may not be represented using algebraic expressions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• III. Geometric Reasoning
• A1 – Identify and represent the features of plane and space figures.
• A2 – Make, test, and use conjectures about one-, two-, and three-dimensional figures and their properties.
• B2 – Identify the symmetries in a plane figure.
• D1 – Make and validate geometric conjectures.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
G.5B

Construct congruent segments, congruent angles, a segment bisector, an angle bisector, perpendicular lines, the perpendicular bisector of a line segment, and a line parallel to a given line through a point not on a line using a compass and a straightedge.

Construct

CONGRUENT ANGLES AND A LINE PARALLEL TO A GIVEN LINE THROUGH A POINT NOT ON A LINE USING A COMPASS AND A STRAIGHTEDGE

Including, but not limited to:

• Geometric construction – construction of accurate representations of lengths, angles, and geometric figures using only a straight edge and compass
• Congruent angles – angles whose angle measurements are equal
• Parallel line – line parallel to a given line through a point not on a line

Note(s):

• Previous grade levels investigated attributes of geometric figures.
• Geometry introduces constructions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• III. Geometric Reasoning
• A1 – Identify and represent the features of plane and space figures.
• A2 – Make, test, and use conjectures about one-, two-, and three-dimensional figures and their properties.
• B2 – Identify the symmetries in a plane figure.
• D1 – Make and validate geometric conjectures.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
G.5C

Use the constructions of congruent segments, congruent angles, angle bisectors, and perpendicular bisectors to make conjectures about geometric relationships.

Use

THE CONSTRUCTIONS OF CONGRUENT ANGLES TO MAKE CONJECTURES ABOUT GEOMETRIC RELATIONSHIPS

Including, but not limited to:

• Geometric construction – construction of accurate representations of lengths, angles, and geometric figures using only a straight edge and compass
• Use of various tools
• Compass and straightedge
• Dynamic geometric software
• Patty paper
• Constructions
• Congruent angles
• Conjectures about attributes of figures related to the constructions
• Angle measure and angle addition

Note(s):

• Previous grade levels investigated attributes of geometric figures.
• Geometry introduces the use of constructions to make conjectures about geometric relationships.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• III. Geometric Reasoning
• A1 – Identify and represent the features of plane and space figures.
• A2 – Make, test, and use conjectures about one-, two-, and three-dimensional figures and their properties.
• B2 – Identify the symmetries in a plane figure.
• D1 – Make and validate geometric conjectures.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
G.6 Proof and congruence. The student uses the process skills with deductive reasoning to prove and apply theorems by using a variety of methods such as coordinate, transformational, and axiomatic and formats such as two-column, paragraph, and flow chart. The student is expected to:
G.6A

Verify theorems about angles formed by the intersection of lines and line segments, including vertical angles, and angles formed by parallel lines cut by a transversal and prove equidistance between the endpoints of a segment and points on its perpendicular bisector and apply these relationships to solve problems.

Verify

THEOREMS ABOUT ANGLES FORMED BY THE INTERSECTION OF LINES AND LINE SEGMENTS, INCLUDING VERTICAL ANGLES, AND ANGLES FORMED BY PARALLEL LINES CUT BY A TRANSVERSAL

Including, but not limited to:

• Theorems about angles formed from intersecting lines and line segments
• Vertical angles – a pair of non-adjacent, non-overlapping angles formed by two intersecting lines creating angles that are opposite and congruent to each other
• Angles formed by parallel lines cut by a transversal
• Methods for verification
• Coordinate geometry
• Transformations
• Axiomatic proofs
• Constructions
• Compass and straight edge
• Dynamic geometric software
• Patty paper
• Manipulatives

Apply

THEOREMS ABOUT ANGLES FORMED BY THE INTERSECTION OF LINES AND LINE SEGMENTS TO SOLVE PROBLEMS

Including, but not limited to:

• Application of theorems about angles formed by the intersection of lines and line segments
• Vertical angles
• Angles formed by parallel lines cut by a transversal

Note(s):

• Previous grade levels investigated attributes of geometric figures.
• Grade 8 used informal arguments to establish facts about the angle sum and exterior angle of triangles, the angles created when parallel lines are cut by a transversal, and the Angle-Angle criterion for similarity of triangles.
• Geometry introduces proofs of geometric relationships.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• III. Geometric Reasoning
• A1 – Identify and represent the features of plane and space figures.
• A2 – Make, test, and use conjectures about one-, two-, and three-dimensional figures and their properties.
• B2 – Identify the symmetries in a plane figure.
• D1 – Make and validate geometric conjectures.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections