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 Instructional Focus DocumentGeometry
 TITLE : Unit 07: Relationships of Two- and Three-Dimensional Figures SUGGESTED DURATION : 13 days

#### Unit Overview

This unit bundles student expectations that address exploring geometric relationships between sides, angles, and diagonals of various quadrilaterals to justify the type of quadrilateral; investigating patterns to make conjectures and generalizations about polygons; and identifying two-dimensional cross-sections of three-dimensional figures as well as identifying three-dimensional objects generated by rotations of two-dimensional shapes. 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, in Grade 3, students used attributes to recognize rhombuses, parallelograms, trapezoids, rectangles, and squares as examples of quadrilaterals and drew examples of quadrilaterals that do not belong to any of these subcategories. In Grade 5, students classified two-dimensional figures in a hierarchy of sets using graphic organizers based on their attributes and properties as well as investigated patterns to make conjectures about geometric relationships, including diagonals of quadrilaterals and interior and exterior angles of polygons. In Grade 7, students used patterns to analyze linear relationships. In Grade 8, students used patterns to analyze proportional and non-proportional linear relationships. In Algebra I Units 01 – 04, students studied linear functions and equations to represent data.

During this unit, students define types of quadrilaterals with a focus on identifying the characteristics (including sides, angles, and diagonal relationships) of parallelograms, rectangles, rhombi, and squares. Students use the characteristics of the quadrilateral to prove quadrilaterals are parallelograms, rectangles, rhombi, or squares using coordinate geometry, two column proofs, paragraph proofs, and flow charts. Students apply quadrilateral relationships to solve real-world problems involving lengths of sides, angles, and midpoints. Students define and identify polygons, including regular polygons by number of sides. Students use tabular, graphical, and symbolic generalization to develop formulas for interior and exterior angles in terms of number of sides. Students extend and apply the properties of quadrilaterals and other polygons and interior and exterior angle theorems to determine lengths of sides, diagonals, midpoints and all angle measures. Students identify shapes of two-dimensional cross sections of prisms, pyramids, cylinders, cones, and spheres. Students explore and identify three-dimensional objects generated by rotations of two dimensional shapes.

After this unit, in Geometry Units 08 – 09, students will use the concepts of polygons to investigate and solve problems involving properties and measurement of two- and three-dimensional figures. The concepts in this unit will be applied in subsequent mathematics courses.

This unit is supporting the development of Texas College Career Readiness Standards (TxCCRS): III. Geometric Reasoning A1, A2, B2, C1, D1; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

According to The National Council of Teachers of Mathematics (2000) this approach to studying quadrilaterals supports the need for all students to be able to analyze characteristics and properties of geometric shapes and develop mathematical arguments about geometric relationships. Students in grades 9 – 12 should specifically be able to investigate relationships among classes of two-dimensional geometric objects, make and test conjectures about them, and solve problems. At the conclusion of this unit, students are asked to create a graphic organizer to help classify quadrilaterals. The TxCCRS cites many skills related to the communication and representation of mathematical ideas. According to the National Council of Teachers of Mathematics (NCTM) in Focus in High School Mathematics: Reasoning and Sense Making (2009), the key elements of reasoning and sense making with geometry must include multiple representations of functions. In this unit, students gather data from geometric figures and organize this information into tables, graphs or diagrams. This leads to the development of symbolic expressions and verbal descriptions. A variety of representations helps make relationships more understandable to more students than working with symbolic representations alone. These approaches serve as the basis for this unit on polygons and circle. TxCCRS cites many skills related to the communication and representation of mathematical ideas. According to the National Council of Teachers of Mathematics (2012), using diagrams and constructions to interpret and communicate geometric relationships is essential in geometry. Using definitions of figures to characterize figures in terms of their properties is another essential in geometry. In geometry, the “proving process involves working with diagrams, variation and invariance, conjectures, and definitions.” (p. 92)

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
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.
National Council of Teachers of Mathematics. (2012). Developing essential understanding of Geometry for Teaching Mathematics in Grades 9 – 12. Reston, VA: National Council of Teachers of Mathematics, Inc.

#### OVERARCHING UNDERSTANDINGS and QUESTIONS

Algebraic and geometric relationships can be used to describe mathematical and real-world patterns.

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

Geometric and spatial reasoning are necessary to describe and analyze geometric relationships in mathematics and the real-world.

• Why are geometric and spatial reasoning necessary in the development of an understanding of geometric relationships?
• Why is it important to visualize and use diagrams to effectively communicate/illustrate geometric relationships?

Attributes and properties of two- and three-dimensional geometric shapes are foundational to developing geometric and measurement reasoning.

• Why is it important to compare and contrast attributes and properties of two- and three-dimensional geometric shapes?
• How does analyzing the attributes and properties of two- and three-dimensional geometric shapes develop geometric and measurement reasoning?

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

• Slope/Rate of Change

Geometric Reasoning

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

Measurement Reasoning

• Angle Measures
• Length

Associated Mathematical Processes

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

Geometric relationships exist between the sides, angles, and diagonals of various classifications of quadrilaterals.

• How can the distance formula be used to classify quadrilaterals?
• How can the slope formula be used to classify quadrilaterals?
• How can the midpoint formula be used to classify quadrilaterals?
• How can the distance, slope and midpoint formula be applied to solve problems relating to quadrilaterals?

Diagrams can be used to visualize and illustrate geometric relationships and aid in solving problems.

• Why are diagrams necessary for visualizing the geometric relationships found in the problem situation?
• How are diagrams used to organize information from the problem situation?
• How do diagrams aid in calculations when solving problems?
• Why is the coordinate plane used to diagram two-dimensional figures?

Determining and comparing the measurements of the attributes of two-dimensional figures are foundational to classifying two-dimensional figures and analyzing the relationships in two-dimensional figures.

• For which attributes of a two-dimensional figure can the measures be determined?
• How are the measures of the attributes determined?
• Why can the measures of the attributes be used to classify two-dimensional figures?

Coordinate geometry can be used to prove a quadrilateral is a square, rectangle, parallelogram, or rhombus.

• How can the distance formula be used to prove the classification of a quadrilateral?
• How can the slope formula be used to prove the classification of a quadrilateral?
• How can the midpoint formula be used to prove the classification of a quadrilateral?
• What relationships can be used when proving quadrilaterals are parallelograms, rectangles, squares, or rhombi?
 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

• Equations
• Patterns/Rules

Functions

• Independent/Dependent Variables
• Patterns/Rules

Geometric Reasoning

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

Measurement Reasoning

• Angle Measures
• Length

Associated Mathematical Processes

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

A geometric relationship exists between the number of sides and interior angles of a regular polygon and sides and exterior angles of a regular polygon.

• How can patterns in attributes of geometric figures be analyzed to describe generalizations about geometric relationships?
• How can the sum of the interior angles of a regular polygon be used to identify a polygon?
• How can the measure of each interior angle of a regular polygon be found using the number of sides?
• What geometric relationships can be found between the number of sides and the measures of the interior angles of a regular polygon?
• What geometric relationships can be found between the number of sides and the measures of the exterior angles of a regular polygon?
• What geometric relationships can be found between the number of sides and the number of diagonals in a polygon?

Determining and comparing the measurements of the attributes of two-dimensional figures are foundational to classifying two-dimensional figures and analyzing the relationships in two-dimensional figures.

• For which attributes of a two-dimensional figure can the measures be determined?
• How are the measures of the attributes determined?
• How can the measures of the attributes of two-dimensional figures be used to analyze relationships within the figure?
 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.

Geometric Reasoning

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

Measurement Reasoning

• Angle Measures
• Length

Associated Mathematical Processes

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

Models and diagrams can be used to visualize and illustrate geometric relationships and aid in solving problems.

• Why are models and diagrams necessary for visualizing the geometric relationships found in the problem situation?
• How are models and diagrams used to organize information from the problem situation?
• How do models and diagrams aid in calculations when solving problems?
• Why is the coordinate plane used to diagram two-dimensional figures?

Determining and comparing the measurements of the attributes of two-dimensional figures are foundational to classifying two-dimensional figures and analyzing the relationships in two-dimensional figures.

• Why does the analysis of cross sections of three-dimensional figures build an understanding of both the two-dimensional shapes formed and the original three-dimensional figure?
• Why does the analysis of three-dimensional figures generated by the rotation of two-dimensional figures build an understanding of both the three-dimensional figure formed and the original two-dimensional figure?

Cross sections of three-dimensional figures can be circles curved surface figures or polygons.

• Why does the cross section of a three-dimensional figure always yield a two-dimensional shape?
• What shapes can be seen in the cross section of a cube?
• What shapes can be seen in the cross section of a square pyramid?
• What shapes can be seen in the cross section of a cylinder?
• What shapes can be seen in the cross section of a cone?
• What shapes can be seen in the cross section of a sphere?
• How can knowing the cross section of a three-dimensional figure help in its identification?

Three-dimensional figures are formed when rotating two dimensional shapes around the x- or y- axes.

• What three-dimensional figure is formed when a triangle is rotated about the x- or y-axis?
• What three-dimensional figure is formed when a rectangle is rotated about the x- or y-axis?
• What three-dimensional figure is formed when a pyramid is rotated about the x- or y-axis?
• How are the surfaces of polyhedral changed when they are rotated on an axis?
• What three-dimensional figure is formed when a cone is rotated about the x- or y-axis?
• What three-dimensional figure is formed when a sphere is rotated about the x- or y-axis?

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think the slopes of perpendicular lines are just negatives or opposites of each other or reciprocals of each other rather than being both opposite reciprocals of each other.
• Some students may incorrectly apply the distance formula even though they know the properties needed to classify quadrilaterals.
• Some students may confuse a statement and its converse. Students may understand that if a figure is a square, then it is also a rectangle; however, knowing a figure is a rectangle does not necessarily imply that it is a square.

#### Unit Vocabulary

• Congruent segments – line segments whose lengths are equal
• Conjecture – statement believed to be true but not yet proven
• Exterior angle of a polygon – angle on the outside of a polygon formed by the side of a polygon and an extension of its adjacent side
• Interior angle of a polygon – angle on the inside of a polygon formed by pairs of adjacent sides
• Irregular polygon – polygon that is not equilateral or equiangular
• Midpoint of a line segment – the point halfway between the endpoints of a line segment
• Parallel lines – lines in the same plane that never intersect. Slopes of parallel lines are equal, my2 = my1.
• Perpendicular lines – lines that intersect at a 90° angle to form right angles. Slopes of perpendicular lines are opposite reciprocals,  my2.
• Regular polygon – a convex polygon in which all sides are congruent (equilateral) and all angles are congruent (equiangular)

Related Vocabulary

 Concave polygon Cone Convex polygon Coordinate geometry proofs Cross sections of three-dimensional figures Cylinder Diagonals of quadrilaterals Distance formula of a line segment Kite Midpoint formula of a line segment Parallelogram Prism Polygon Pyramid Quadrilateral Rectangle Rhombus Rotations of two-dimensional figures Semi-circle Slope formula of a line segment Sphere Square Sum of the exterior angles of a polygon Sum of the interior angles of a polygon Three-dimensional figures Trapezium Trapezoid Two-dimensional figures
Unit Assessment Items System Resources Other Resources

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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.2 Coordinate and transformational geometry. The student uses the process skills to understand the connections between algebra and geometry and uses the one- and two-dimensional coordinate systems to verify geometric conjectures. The student is expected to:
G.2B

Derive and use the distance, slope, and midpoint formulas to verify geometric relationships, including congruence of segments and parallelism or perpendicularity of pairs of lines.

Use

THE DISTANCE, SLOPE, AND MIDPOINT FORMULAS TO VERIFY GEOMETRIC RELATIONSHIPS, INCLUDING CONGRUENCE OF SEGMENTS AND PARALLELISM OR PERPENDICULARITY OF PAIRS OF LINES

Including, but not limited to:

• Distance formula: d =
• Formula for slope of a line: m =
• Formula for midpoint of a line segment on a coordinate plane:
• Congruent segments – line segments whose lengths are equal
• Midpoint of a line segment – the point halfway between the endpoints of a line segment
• Parallel lines – lines in the same plane that never intersect. Slopes of parallel lines are equal, my2my1.
• Perpendicular lines – lines that intersect at a 90° angle to form right angles. Slopes of perpendicular lines are opposite reciprocals, my2 = – .
• Equation of a line
• Slope-intercept form, y = mx + b,
• Point-slope form, yy1 = m(xx1)
• Standard form, Ax + By = C
• Coordinate geometry proofs
• Identification of polygons
• Trapezium
• Kite
• Trapezoid
• Parallelogram
• Rectangle
• Rhombus
• Square

Note(s):

• Grade 8 introduced and applied the Pythagorean Theorem to determine the distance between two points on a coordinate plane.
• Grade 8 introduced slope as  or .
• Algebra I addressed determining equations of lines using point-slope form, slope intercept form, and standard form.
• Algebra I wrote equations of lines that contain a given point and are parallel or perpendicular to a given line.
• 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.
• C1 – Make connections between geometry and algebra.
• 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 DIAGONALS OF QUADRILATERALS, INTERIOR AND EXTERIOR ANGLES OF POLYGONS, 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)
• Isosceles trapezoid: Diagonals are congruent.
• Parallelogram: Diagonals bisect each other.
• Rectangle: Diagonals are congruent and exhibit all properties of a parallelogram.
• Rhombus: Diagonals bisect angles and are perpendicular to each other and exhibit all properties of a parallelogram.
• Square: Exhibits all the properties of a parallelogram, rectangle, and rhombus
• Interior and exterior angles of a polygon
• Interior angle of a polygon – angle on the inside of a polygon formed by pairs of adjacent sides
• Exterior angle of a polygon – angle on the outside of a polygon formed by the side of a polygon and an extension of its adjacent side
• Relationship between interior and exterior angles (one pair per vertex)
• Concave and convex polygons
• Sum of the interior angles of a polygon: sum = 180(n – 2) degrees, where n is the number of sides
• Regular polygon – a convex polygon in which all sides are congruent (equilateral) and all angles are congruent (equiangular)
• Measure of each interior angle:  measure = , where n is the number of sides
• Irregular polygon – polygon that is not equilateral or equiangular
• Sum of the exterior angles of a polygon (one angle per vertex): sum = 360o
• Regular polygon
• Measure of each exterior angle:  measure =

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.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.6E Prove a quadrilateral is a parallelogram, rectangle, square, or rhombus using opposite sides, opposite angles, or diagonals and apply these relationships to solve problems.

Prove

A QUADRILATERAL IS A PARALLELOGRAM, RECTANGLE, SQUARE, OR RHOMBUS USING OPPOSITE SIDES, OPPOSITE ANGLES, OR DIAGONALS

Including, but not limited to:

• Identification of type of quadrilateral
• Parallelogram
• Rectangle
• Square
• Rhombus
• Comparisons by coordinate proofs
• Opposite sides
• Opposite angles
• Diagonals

Apply

RELATIONSHIPS IN QUADRILATERALS TO SOLVE PROBLEMS

Including, but not limited to:

• Mathematical problem situations involving quadrilaterals
• Real-world problem situations involving quadrilaterals
• Parallelogram
• Rectangle
• Square
• Rhombus

Note(s):

• Geometry introduces coordinate proofs of conjectures about figures.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• III. Geometry
• 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.10 Two-dimensional and three-dimensional figures. The student uses the process skills to recognize characteristics and dimensional changes of two- and three-dimensional figures. The student is expected to:
G.10A Identify the shapes of two-dimensional cross-sections of prisms, pyramids, cylinders, cones, and spheres and identify three-dimensional objects generated by rotations of two-dimensional shapes.

Identify

THE SHAPES OF TWO-DIMENSIONAL CROSS-SECTIONS OF PRISMS, PYRAMIDS, CYLINDERS, CONES, AND SPHERES

Including, but not limited to:

• Three-dimensional figures
• Prisms
• Cylinders
• Pyramids
• Cones
• Spheres
• Description of cross sectional intersections
• Verbal
• Pictorial
• Representations of cross sections
• Point
• Line segment
• Region of a plane (polygon or circle)

Identify

THREE-DIMENSIONAL OBJECTS GENERATED BY ROTATIONS OF TWO-DIMENSIONAL SHAPES

Including, but not limited to:

• Rectangle
• Triangle
• Semi-circle
• Trapezoid
• Rotation about x-axis and y-axis

Note(s):