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 Instructional Focus DocumentGeometry
 TITLE : Unit 04: Relationships of Triangles, including Congruence and Similarity SUGGESTED DURATION : 19 days

#### Unit Overview

Introduction
This unit bundles student expectations that address patterns and properties of triangles, special segments of triangles, congruency of triangles, and similarity of triangles. These geometric relationships are verified using constructions and proofs and used to solve problems. 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 previous grade levels, students studied attributes of triangles and proportionality. In Grade 6, students laid the foundation for the Triangle Inequality Theorem when verifying that a given set of lengths could form a triangle. In Grade 7, students generalized the critical attributes of similarity, including ratios within and between similar shapes. In Grade 8, students used dilations to transform figures, introduced informal arguments to establish facts about the angle sum and exterior angle of triangles, and used the Angle-Angle criterion for similarity of triangles. In Geometry Unit 02, students also investigated parallel and perpendicular lines, including comparison of parallel lines in Euclidean and spherical geometry. Students also studied geometric explorations of distance, midpoint, slope, and parallel and perpendicular lines in a two-dimensional coordinate system. In Geometry Unit 03, students studied angle relationships when parallel lines are cut by a transversal.

During this Unit
Students explore patterns and properties of triangles according to sides and angles (interior and exterior angles) using a variety of tools. Students verify theorems involving the sum of the interior angles of a triangle and theorems involving the base angles of isosceles triangles and apply these geometric relationships to solve mathematical and real-world problems. Students compare geometric relationships between Euclidean and spherical geometries, including the sum of the angles in a triangle. Students use constructions to verify the Triangle Inequality theorem and apply the theorem to solve problems. Students construct angle bisectors, segment bisectors, perpendicular lines, and perpendicular bisectors using a compass and a straightedge in order to investigate patterns and make conjectures about geometric relationships of special segments in triangles (altitudes, angle bisectors, medians, perpendicular bisectors, midsegments). Students verify and formalize properties and theorems of special segments and apply the geometric relationships to solve problems. Students analyze patterns of congruent triangles using a variety of methods to identify congruent figures and their corresponding congruent sides and angles. Students use rigid transformations of triangles and constructions to explore triangle congruency. Students formalize a definition of triangle congruency establishing necessary criterion for congruency, as well as formalize postulates and theorems for triangle congruency (Side-Angle-Side, Angle-Side-Angle, Side-Side-Side, Angle-Angle-Side, and Hypotenuse-Leg). Students apply triangle congruency and corresponding parts of congruent triangles are congruent (CPCTC) to prove two triangles are congruent using a variety of proofs. Students apply triangle congruency theorems and CPCTC to solve problems. Students use dilations of triangles and constructions to investigate and explore similarity. Students formalize a definition of triangle similarity establishing corresponding sides of triangles are proportional and corresponding angles of triangles are congruent. Students formalize postulates and theorems to prove triangles are similar using Apply Angle-Angle similarity and the Triangle Proportionality theorem. Students apply triangle similarity to prove two triangles are similar using a variety of proofs. Students apply triangle similarity theorems and proportional understanding to solve problems.

After this Unit
In Units 05 – 07, students will continue to use logical reasoning and proofs, as well as properties and attributes of triangles to explore and analyze other polygons. The concepts in this unit will also be applied in subsequent mathematics courses.

This unit is supporting the development of Texas College Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1; II. Algebraic Reasoning D1, D2; III. Geometric and Spatial Reasoning A2, A3, B1, B2, C1, D3; V. Statistical Reasoning A1, C2; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, C2, D1, D2; VIII. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

Research
According to the National Council of Teachers of Mathematics (2000), all students in grades 9 – 12 should explore relationships (including congruence and similarity) in two-dimensional geometric figures, make and test conjectures about two-dimensional geometric figures, and solve problems involving two-dimensional geometric figures.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency. (2009). Texas college and career readiness standards. Austin, TX: Author.

 Geometric, spatial, and measurement reasoning are foundational to visualizing, analyzing, and applying relationships within and between scale, shapes, quantities, and spatial relations in everyday life. Why is developing geometric, spatial, and measurement reasoning essential? How does geometric, spatial, and measurement reasoning affect how one sees and works in the world? Logical reasoning can be used to make sense of claims, determine their validity, and construct and communicate arguments. Why is developing logical reasoning in mathematics important and how does this reasoning influence decision making in everyday life? What elements of logical reasoning influence the truth of a statement? How is logical reasoning used to uncover truths and/or make sense of, construct, and determine the validity of arguments and claims?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding and working with variance and invariance within geometry builds flexible algebraic and geometric reasoning and deepens understanding of intrinsic properties of geometric relationships.
• What invariant (unchanging) and variant (changing) relationships exist within …
• and between Euclidean and spherical geometries?
• side lengths and angle measures of triangles?
• How does examining variance and invariance lead to new conjectures and theorems about triangle relationships?
• Deductive reasoning can be used to determine the validity of a conditional statement and its related statements and conjectures about geometric relationships in order to support or refute mathematical claims through the process of proving.
• How is deductive reasoning used to understand, prove, and apply geometric conjectures of triangle relationships?
• How are undefined terms, definitions, and postulates of a geometric system used to determine the validity of a conjecture about geometric relationships in that system?
• How can the converse of theorems and postulates be used to find angle measures in triangles?
• Accurate representations, models, or diagrams within a geometric system allows for visualizing, illustrating, and analyzing geometric relationships to aid in making and validating conjectures about those geometric relationships and is central to geometric thinking.
• What types of problem situations represent triangle relationships?
• How can representations and appropriate geometric language be used to effectively communicate and illustrate geometric relationships of triangles?
• What tools and processes can be used to verify the …
• Triangle Inequality theorem?
• relationship of angles in a triangle, including isosceles triangles?
• Attributes and quantifiable measures of geometric figures can be generalized to describe, determine, and represent algebraic and geometric relationships and be applied to solve problem situations.
• How can understanding triangle relationships be applied when solving problem situations?
• How can measurable attributes related to side lengths and angle measures be distinguished and described in order to generalize geometric relationships of triangles?
• What processes can be used to determine the angle measures in a triangle?
• Logical Arguments and Constructions, Proof and Congruence
• Constructions
• Deductive Reasoning
• Definitions
• Conjectures
• Theorems
• Proofs
• Geometric Relationships
• Congruence
• Equidistance
• Parallelism
• Side relationships
• Angle relationships
• Triangle relationships
• Proportionality
• Geometric Representations
• Angles
• Sides
• Segments
• Triangles
• Geometries
• Euclidean
• Spherical
• Patterns, Operations, and Properties
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 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, spatial, and measurement reasoning are foundational to visualizing, analyzing, and applying relationships within and between scale, shapes, quantities, and spatial relations in everyday life. Why is developing geometric, spatial, and measurement reasoning essential? How does geometric, spatial, and measurement reasoning affect how one sees and works in the world? Logical reasoning can be used to make sense of claims, determine their validity, and construct and communicate arguments. Why is developing logical reasoning in mathematics important and how does this reasoning influence decision making in everyday life? What elements of logical reasoning influence the truth of a statement? How is logical reasoning used to uncover truths and/or make sense of, construct, and determine the validity of arguments and claims?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding and working with variance and invariance within geometry builds flexible algebraic and geometric reasoning and deepens understanding of intrinsic properties of geometric relationships.
• What invariant (unchanging) and variant (changing) relationships exist within side lengths and angle measures of triangles?
• How does examining variance and invariance lead to new conjectures and theorems about relationships between special segments, side lengths and angle measures of triangles?
• Deductive reasoning can be used to determine the validity of a conditional statement and its related statements and conjectures about geometric relationships in order to support or refute mathematical claims through the process of proving.
• How is deductive reasoning used to understand, prove, and apply geometric conjectures about triangle relationships?
• the relationship of special triangle segments?
• special triangle segment intersections?
• triangle congruency?
• triangle similarity?
• Accurate representations, models, or diagrams within a geometric system allows for visualizing, illustrating, and analyzing geometric relationships to aid in making and validating conjectures about those geometric relationships and is central to geometric thinking.
• What types of problem situations represent triangle relationships?
• How can representations and appropriate geometric language be used to effectively communicate and illustrate …
• side and angle relationships?
• special triangle segment relationships?
• special triangle segment intersection properties?
• interior and exterior angle relationships in polygons
• What tools and processes can be used to construct …
• congruent segments?
• congruent angles?
• angle bisectors?
• segment bisectors?
• perpendicular lines?
• perpendicular bisectors?
• How can constructions be used to make and validate conjectures about geometric relationships of special triangles segments and their intersections?
• What conjectures about congruency and similarity can be made and validated by exploring the patterns and properties of triangles?
• Attributes and quantifiable measures of geometric figures can be generalized to describe, determine, and represent algebraic and geometric relationships and be applied to solve problem situations.
• How can understanding triangle relationships be applied when solving problem situations?
• How can measurable attributes related to …
• angle bisectors
• segment and perpendicular bisectors
• side lengths of triangles
• interior and exterior angle measures of polygons
• special triangle segment lengths
• special polygon segment lengths
… be distinguished and described in order to generalize geometric relationships?
• What processes can be used to determine the …
• sum of interior angles?
• sum of exterior angles?
• Logical Arguments and Constructions, Proof and Congruence
• Constructions
• Congruent segments
• Congruent angles
• Segment bisector
• Angle bisector
• Perpendicular lines
• Perpendicular bisector
• Deductive Reasoning
• Undefined terms
• Definitions
• Postulates
• Conjectures
• Theorems
• Geometric Relationships
• Congruence
• Equidistance
• Parallelism
• Perpendicularity
• Side relationships
• Angle relationships
• Triangle relationships
• Proportionality
• Similarity
• Geometric Representations
• Angles
• Sides
• Segments
• Triangles
• Polygons
• Geometries
• Euclidean
• Patterns, Operations, and Properties
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 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, spatial, and measurement reasoning are foundational to visualizing, analyzing, and applying relationships within and between scale, shapes, quantities, and spatial relations in everyday life. Why is developing geometric, spatial, and measurement reasoning essential? How does geometric, spatial, and measurement reasoning affect how one sees and works in the world? Logical reasoning can be used to make sense of claims, determine their validity, and construct and communicate arguments. Why is developing logical reasoning in mathematics important and how does this reasoning influence decision making in everyday life? What elements of logical reasoning influence the truth of a statement? How is logical reasoning used to uncover truths and/or make sense of, construct, and determine the validity of arguments and claims?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding and working with variance and invariance within geometry builds flexible algebraic and geometric reasoning and deepens understanding of intrinsic properties of geometric relationships.
• What invariant (unchanging) and variant (changing) relationships exist within side lengths and angle measures of congruent geometric figures?
• How does examining variance and invariance lead to new conjectures and theorems about congruent geometric figures?
• Deductive reasoning can be used to determine the validity of a conditional statement and its related statements and conjectures about geometric relationships in order to support or refute mathematical claims through the process of proving.
• How is deductive reasoning used to understand, prove, and apply geometric conjectures about congruent geometric figures?
• congruent geometric figures?
• the relationship of special triangle segments?
• triangle congruency?
• Accurate representations, models, or diagrams within a geometric system allows for visualizing, illustrating, and analyzing geometric relationships to aid in making and validating conjectures about those geometric relationships and is central to geometric thinking.
• What types of problem situations represent congruent triangles?
• How can representations and appropriate geometric language be used to effectively communicate and illustrate …
• side and angle relationships?
• triangle congruence criterion?
• What tools and processes can be used to construct …
• congruent segments?
• congruent angles?
• congruent triangles?
• How can constructions be used to make and validate conjectures about side and angle relationships in congruent geometric figures?
• What conjectures about congruency can be made and validated by exploring the patterns and properties of geometric figures?
• Attributes and quantifiable measures of geometric figures can be generalized to describe, determine, and represent algebraic and geometric relationships and be applied to solve problem situations.
• How can understanding congruent triangles be applied when solving problem situations?
• How can measurable attributes related to …
• midpoints
• side lengths
• angle measures of polygons
… be distinguished and described in order to generalize geometric relationships of congruent geometric figures?
• What processes can be used to determine the congruency of geometric figures?
• Logical Arguments and Constructions, Proof and Congruence
• Constructions
• Congruent segments
• Congruent angles
• Perpendicular lines
• Perpendicular bisector
• Deductive Reasoning
• Definitions
• Conjectures
• Theorems
• Proofs
• Geometric Relationships
• Congruence
• Equidistance
• Parallelism
• Perpendicularity
• Side relationships
• Angle relationships
• Triangle relationships
• Geometric Representations
• Angles
• Sides
• Lines
• Segments
• Triangles
• Polygons
• Transformations
• Geometries
• Euclidean
• Patterns, Operations, and Properties
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 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, spatial, and measurement reasoning are foundational to visualizing, analyzing, and applying relationships within and between scale, shapes, quantities, and spatial relations in everyday life. Why is developing geometric, spatial, and measurement reasoning essential? How does geometric, spatial, and measurement reasoning affect how one sees and works in the world? Logical reasoning can be used to make sense of claims, determine their validity, and construct and communicate arguments. Why is developing logical reasoning in mathematics important and how does this reasoning influence decision making in everyday life? What elements of logical reasoning influence the truth of a statement? How is logical reasoning used to uncover truths and/or make sense of, construct, and determine the validity of arguments and claims?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding and working with variance and invariance within geometry builds flexible algebraic and geometric reasoning and deepens understanding of intrinsic properties of geometric relationships.
• What invariant (unchanging) and variant (changing) relationships exist within side lengths and angle measures of triangles?
• How does examining variance and invariance lead to new conjectures and theorems about similarity of geometric figures?
• Deductive reasoning can be used to determine the validity of a conditional statement and its related statements and conjectures about geometric relationships in order to support or refute mathematical claims through the process of proving.
• How is deductive reasoning used to understand, prove, and apply geometric conjectures about …
• similarity relationships of geometric figures?
• the relationship of special triangle segments?
• triangle similarity relationships?
• What conjectures can be made about angle and side relationships of similar figures?
• Accurate representations, models, or diagrams within a geometric system allows for visualizing, illustrating, and analyzing geometric relationships to aid in making and validating conjectures about those geometric relationships and is central to geometric thinking.
• What types of problem situations represent similar triangles?
• How can representations and appropriate geometric language be used to effectively communicate and illustrate geometric relationships of similar geometric figures?
• What tools and processes can be used to construct …
• congruent angles?
• proportional side lengths?
• similar triangles?
• How can constructions be used to make and validate conjectures about geometric relationships in similar geometric figures?
• What conjectures about similarity can be made and validated by exploring the patterns and properties of geometric figures?
• Attributes and quantifiable measures of geometric figures can be generalized to describe, determine, and represent algebraic and geometric relationships and be applied to solve problem situations.
• How can understanding similar triangles be applied when solving problem situations?
• How can measurable attributes related to …
• side lengths
• angles
… be distinguished and described in order to generalize geometric relationships of similar geometric figures, including triangles?
• What processes can be used to determine the similarity of geometric figures, including triangles?
• Logical Arguments and Constructions; Similarity, Proof, and Trigonometry
• Constructions
• Congruent segments
• Congruent angles
• Segment bisector
• Angle bisector
• Perpendicular lines
• Perpendicular bisector
• Deductive Reasoning
• Definitions
• Conjectures
• Theorems
• Proofs
• Geometric Relationships
• Parallelism
• Perpendicularity
• Side relationships
• Angle relationships
• Triangle relationships
• Similarity
• Proportionality
• Geometric Representations
• Angles
• Sides
• Segments
• Triangles
• Polygons
• Transformations
• Geometries
• Euclidean
• Patterns, Operations, and Properties
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 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.

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think visual estimation provides sufficient evidence that triangles are congruent or similar rather than verifying this property using geometric relationships.
• Some students may think that the median goes from the midpoint of one side to the midpoint of another side rather than the vertex to the midpoint of the opposite side; students confuse the median with the midsegment.

Underdeveloped Concepts:

• Some students may confuse the different rules for congruence (e.g., SAS, ASA, SSS, etc.).
• Students may confuse the rules for congruence with the rules for similarity, which in many cases have the same names (SAS, SSS, etc.).

#### Unit Vocabulary

• Altitude of a triangle – line segment drawn from any vertex of a triangle perpendicular to the opposite side
• Angle bisector – line, ray, or segment that divides an angle into two congruent angles
• Angle-Angle criterion for triangle similarity – if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar
• Base angles of a triangle – the two angles that have one side in common with the base
• Centroid (point of concurrency) – the intersection point of the medians of a triangle
• Circumcenter (point of concurrency) – the intersection point of the perpendicular bisectors of a triangle
• Congruent angles – angles whose angle measurement are equal
• Congruent figures – figures that are the same size and same shape
• Congruent segments – line segments whose lengths are equal
• Conjecture – statement believed to be true but not yet proven
• Corresponding sides and angles – sides and angles in two figures whose relative position is the same
• Euidistant – equal distances
• Euclidean geometry – the study of plane and solid geometry based on definitions, undefined terms (point, line, plane) and the assumptions of mathematician Euclid
• 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
• Geometric construction – construction of accurate representations of lengths, angles, and geometric figures using only a straight edge and compass
• Incenter (point of concurrency) – the intersection point of the angle bisectors of a triangle
• Incircle – a circle inscribed in a triangle, using the incenter as the center of the circle
• Interior angle of a polygon – angle on the inside of a polygon formed by pairs of adjacent sides
• Median of a triangle – line segment drawn from any vertex of a triangle to the midpoint of the opposite side
• Midsegment of a triangle – ine segment drawn from the midpoints of two sides of the triangle
• Orthocenter (point of concurrency) – the intersection point of the altitudes of a triangle
• Perpendicular bisector of a line segment – line, ray, or segment that divides a line segment into two congruent segments and forms a 90° angle at the point of intersection
• Perpendicular bisector of a side of a triangle – line segment that is perpendicular to a side of a triangle at the midpoint of the side
• Perpendicular lines – lines that intersect at a 90° angle to form right angles
• Proportional sides – corresponding side lengths form equivalent ratios
• Rigid transformations (isometric transformations, congruent transformations) – transformations where size and shape are preserved
• Segment bisector – point, line, ray, or segment that divides a line segment into two congruent segments
• Similar figures – shapes whose angles are congruent and side lengths are proportional (equal scale factor)
• Spherical geometry – the study of figures on the two-dimensional curved surface of a sphere
• Triangle congruence – triangles whose corresponding side lengths and corresponding angle measures are equal

Related Vocabulary:

 Acute triangle Angle-Side-Angle (ASA) Angle-Angle-Side (AAS) Bisect Congruent triangle Constructions Corresponding parts, including CPCTC Equiangular triangle Dilation Equilateral triangle Hypotenuse-Leg (HL) Isosceles triangle Obtuse triangle Postulate Proofs Proportion Right triangle Side-Angle-Side (SAS) Similar triangle Side-Side-Side (SSS) Scale factor Scalene triangle Theorem Triangle Triangle Inequality theorem
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 Center 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

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

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE Legend:

• Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
• Student Expectations (TEKS) identified by TEA are in bolded, black text.
• Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a strike-through.

Specificity Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
• A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.
TEKS# SE# TEKS SPECIFICITY
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:
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world situation.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
• IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.
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:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.A. Statistical Reasoning – Design a study
• V.A.1. Formulate a statistical question, plan an investigation, and collect data.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A.2. Formulate a plan or strategy.
• VII.A.3. Determine a solution.
• VII.A.4. Justify the solution.
• VII.A.5. Evaluate the problem-solving process.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.2. Evaluate the problem-solving process.
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:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
• V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
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:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
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:
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
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:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
•  VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
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:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.4. Justify the solution.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII. C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
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.4D

Compare geometric relationships between Euclidean and spherical geometries, including parallel lines and the sum of the angles in a triangle.

Compare

GEOMETRIC RELATIONSHIPS BETWEEN EUCLIDEAN AND SPHERICAL GEOMETRIES, INCLUDING THE SUM OF THE ANGLES IN A TRIANGLE

Including, but not limited to:

• Euclidean geometry – the study of plane and solid geometry based on definitions, undefined terms (point, line, plane) and the assumptions of mathematician Euclid
• Spherical geometry – the study of figures on the two-dimensional curved surface of a sphere
• Definitions and undefined terms in Euclidean and spherical geometries
• Undefined terms • Postulates and theorems in Euclidean and spherical geometries
• Sum of the angles in a triangle

Note(s):

• Geometry introduces the concept of systems of geometry, including Euclidean geometry and spherical geometry.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VII.C.2. Understand attributes and relationships with inductive and deductive reasoning.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
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 CRITERIA REQUIRED FOR TRIANGLE CONGRUENCE, SPECIAL SEGMENTS OF TRIANGLES, AND 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)
• Criteria required for triangle congruence
• Side-Side-Side (SSS)
• If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.
• Side-Angle-Side (SAS)
• If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
• Angle-Side-Angle (ASA)
• If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
• Angle-Angle-Side (AAS)
• If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
• Hypotenuse-Leg (HL)
• If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent.
• Special segments of triangles
• Altitude of a triangle – line segment drawn from any vertex of a triangle perpendicular to the opposite side • Orthocenter (point of concurrency) – the intersection point of the altitudes of a triangle
• Orthocenter is not always in the interior of the triangle.
• Median of a triangle – line segment drawn from any vertex of a triangle to the midpoint of the opposite side • Centroid (point of concurrency) – the intersection point of the medians of a triangle
• Centroid of a triangle is twice as far from a given vertex than it is from the midpoint to which the median from that vertex is drawn.
• Angle bisector – line, ray, or segment that divides an angle into two congruent angles • Incenter (point of concurrency) – the intersection point of the angle bisectors of a triangle
• Incenter is equidistant from the three sides of the triangle (congruency of the radii of a circle).
• Incircle – a circle inscribed in a triangle, using the incenter as the center of the circle
• Perpendicular bisector of a side of a triangle – line segment that is perpendicular to a side of a triangle at the midpoint of the side • Circumcenter (point of concurrency) – the intersection point of the perpendicular bisectors of a triangle
• Circumcenter is the center of the circle circumscribed about the triangle and is equidistant from all vertices of the triangle.
• Circumcenter of a triangle does not always exist in the interior of the triangle.
• Midsegment of a triangle – line segment drawn from the midpoints of two sides of the triangle • Midsegment of a triangle is always parallel to the third side of the triangle whose midpoint is not included.
• Midsegment of a triangle is half as long as the third side of the triangle.
• 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

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.A. Geometric and Spatial Reasoning – Figures and their properties
• III.A.2. Form and validate conjectures about one-, two-, and three-dimensional figures and their properties.
• III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
• III.D.3. Determine indirect measurements of geometric figures using a variety of methods.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VII.C.2. Understand attributes and relationships with inductive and deductive reasoning.
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 SEGMENTS, CONGRUENT ANGLES, A SEGMENT BISECTOR, AN ANGLE BISECTOR, PERPENDICULAR LINES, AND THE PERPENDICULAR BISECTOR OF A LINE SEGMENT 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 segments – line segments whose lengths are equal
• Congruent angles – angles whose angle measurements are equal
• Segment bisector – point, line, ray, or segment that divides a line segment into two congruent segments
• Perpendicular bisector of a line segment – line, ray, or segment that divides a line segment into two congruent segments and forms a 90° angle at the point of intersection
• Angle bisector – line, ray, or segment that divides an angle into two congruent angles
• Perpendicular lines – lines that intersect at a 90° angle to form right angles

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
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
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 SEGMENTS, CONGRUENT ANGLES, ANGLE BISECTORS, AND PERPENDICULAR BISECTORS 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 segments
• Congruent angles
• Angle bisectors
• Perpendicular bisectors
• Perpendicular bisector of a segment
• 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
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
G.5D Verify the Triangle Inequality theorem using constructions and apply the theorem to solve problems.

Verify

THE TRIANGLE INEQUALITY THEOREM USING CONSTRUCTIONS

Including, but not limited to:

• Development of Triangle Inequality theorem using constructions of side lengths that form triangles and side lengths that do not form triangles
• Compass and straight edge
• Dynamic geometric software
• Patty paper
• Manipulatives
• Triangle Inequality theorem
• If a triangle has side of lengths a, b, and c, then a + b > c, a + c >b, and b + c >a.

Apply

THE TRIANGLE INEQUALITY THEOREM TO SOLVE PROBLEMS

Including, but not limited to:

• Application of Triangle Inequality theorem to solve mathematical and real-world problems

Note(s):

• Previous grade levels investigated attributes of triangles.
• Grade 6 laid the foundation for the Triangle Inequality Theorem when verifying that a given set of lengths could form a triangle.
• 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 verifies the Triangle Inequality Theorem using constructions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• III.A. Geometric and Spatial Reasoning – Figures and their properties
• III.A.2. Form and validate conjectures about one-, two-, and three-dimensional figures and their properties.
• III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
• III.D.3. Determine indirect measurements of geometric figures using a variety of methods.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A.3. Determine a solution.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VII.C.2. Understand attributes and relationships with inductive and deductive reasoning.
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.

Prove

EQUIDISTANCE BETWEEN THE ENDPOINTS OF A SEGMENT AND POINTS ON ITS PERPENDICULAR BISECTOR

Including, but not limited to:

• Methods of proof
• Coordinate geometry
• Two column proof
• Paragraph proof
• Flow chart
• Equidistant from endpoints of a segment and points on its perpendicular bisector

Apply

EQUIDISTANCE BETWEEN THE ENDPOINTS OF A SEGMENT AND POINTS ON ITS PERPENDICULAR BISECTOR TO SOLVE PROBLEMS

Including, but not limited to:

• Equidistant – equal distances
• Application of equidistant relationship between endpoints of a segment and points on its perpendicular bisector

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.A. Geometric and Spatial Reasoning – Figures and their properties
• III.A.2. Form and validate conjectures about one-, two-, and three-dimensional figures and their properties.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A.3. Determine a solution.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VII.C.2. Understand attributes and relationships with inductive and deductive reasoning.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
G.6B Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Side-Side-Side, Angle-Angle-Side, and Hypotenuse-Leg congruence conditions.

Prove

TWO TRIANGLES ARE CONGRUENT BY APPLYING THE SIDE-ANGLE-SIDE, ANGLE-SIDE-ANGLE, SIDE-SIDE-SIDE, ANGLE-ANGLE-SIDE, AND HYPOTENUSE-LEG CONGRUENCE CONDITIONS

Including, but not limited to:

• Triangle congruence – triangles whose corresponding side lengths and corresponding angle measures are equal
• Congruent figures and their corresponding parts
• Corresponding parts of congruent triangles are congruent (CPCTC)
• Congruence conditions
• Side-Angle-Side (SAS)
• Angle-Side-Angle (ASA)
• Side-Side-Side (SSS)
• Angle-Angle-Side (AAS)
• Hypotenuse-Leg (HL)

Note(s):

• Previous grade levels investigated attributes of triangles.
• Geometry introduces proving triangles congruent by triangle congruence relationships.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• III.A. Geometric and Spatial Reasoning – Figures and their properties
• III.A.2. Form and validate conjectures about one-, two-, and three-dimensional figures and their properties.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VII.C.2. Understand attributes and relationships with inductive and deductive reasoning.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
G.6C Apply the definition of congruence, in terms of rigid transformations, to identify congruent figures and their corresponding sides and angles.

Apply

THE DEFINITION OF CONGRUENCE, IN TERMS OF RIGID TRANSFORMATIONS, TO IDENTIFY CONGRUENT FIGURES AND THEIR CORRESPONDING SIDES AND ANGLES

Including, but not limited to:

• Congruent figures – figures that are the same size and same shape
• Rigid transformations (isometric transformations, congruent transformations) – transformations where size and shape are preserved
• Corresponding sides and angles – sides and angles in two figures whose relative position is the same
• Congruent figures and their corresponding parts
• Corresponding parts of congruent triangles are congruent (CPCTC)
• Types of rigid transformations
• Translation
• Reflection
• Rotation
• Combinations of transformations
• Transformations on the coordinate plane
• Image points from original figure
• Original points from image figure
• Verbal description of transformation
• Application of geometric properties to find missing points

Note(s):

• Previous grade levels investigated attributes of geometric figures.
• Geometry introduces proving figures congruent by congruence relationships.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• III.B. Geometric and Spatial Reasoning – Transformations and symmetry
• III.B.1. Identify transformations and symmetries of figures.
• III.B.2. Use transformations to investigate congruence, similarity, and symmetries of figures.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VII.C.2. Understand attributes and relationships with inductive and deductive reasoning.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
G.6D

Verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems.

Verify

THEOREMS ABOUT THE RELATIONSHIPS IN TRIANGLES, INCLUDING THE SUM OF INTERIOR ANGLES, BASE ANGLES OF ISOSCELES TRIANGLES, MIDSEGMENTS, AND MEDIANS

Including, but not limited to:

• Interior angle of a polygon – angle on the inside of a polygon formed by pairs of adjacent sides
• Base angles of a triangle – the two angles that have one side in common with the base
• Concrete models and exploration activities
• Connections between models, pictures, and the symbolic formula
• Sum of interior angles
• Base angles of isosceles triangles
• Midsegments
• Medians
• Dynamic geometry software

Apply

THE RELATIONSHIPS IN TRIANGLES, INCLUDING THE SUM OF INTERIOR ANGLES, BASE ANGLES OF ISOSCELES TRIANGLES, MIDSEGMENTS, AND MEDIANS TO SOLVE PROBLEMS

Including, but not limited to:

• Determination of length and angle measurements using relationships in triangles as needed to solve real-world problem situations
• Sum of interior angles
• Bases angles of isosceles triangles
• Midsegments
• Medians

Note(s):

• Previous grade levels investigated attributes of triangles.
• 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 conjectures about figures.
• Geometry introduces segments of a triangle.
• 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.7 Similarity, proof, and trigonometry. The student uses the process skills in applying similarity to solve problems. The student is expected to:
G.7A Apply the definition of similarity in terms of a dilation to identify similar figures and their proportional sides and the congruent corresponding angles.

Apply

THE DEFINITION OF SIMILARITY IN TERMS OF A DILATION TO IDENTIFY SIMILAR FIGURES AND THEIR PROPORTIONAL SIDES AND THE CONGRUENT CORRESPONDING ANGLES

Including, but not limited to:

• Similar figures – shapes whose angles are congruent and side lengths are proportional (equal scale factor)
• Proportional sides – corresponding side lengths form equivalent ratios
• Corresponding angles – angles in two figures whose relative position is the same
• Scale factor
• Ratios to show dilation relationships
• Identification of similar figures
• Properties of similar triangles
• Applications to real-world situations

Note(s):

• Previous grade levels defined similarity, applied similarity to solve problems, and used dilations to transform figures.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• III.A. Geometric and Spatial Reasoning – Figures and their properties
• III.A.2. Form and validate conjectures about one-, two-, and three-dimensional figures and their properties.
• III.A.3. Recognize and apply right triangle relationships including basic trigonometry.
•  III.B. Geometric and Spatial Reasoning – Transformations and symmetry
• III.B.1. Identify transformations and symmetries of figures.
• III.B.2. Use transformations to investigate congruence, similarity, and symmetries of figures.
• III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
• III.C.1. Make connections between geometry and algebraic equations.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
G.7B Apply the Angle-Angle criterion to verify similar triangles and apply the proportionality of the corresponding sides to solve problems.

Apply

THE ANGLE-ANGLE CRITERION TO VERIFY SIMILAR TRIANGLES

Including, but not limited to:

• Angle-Angle criterion for triangle similarity – if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar
• Angle congruency for similar triangles
• Hands-on exploration
• Dynamic geometry software

Apply

THE PROPORTIONALITY OF THE CORRESPONDING SIDES TO SOLVE PROBLEMS

Including, but not limited to:

• Formulation of equivalent ratios
• Proportional solutions for missing measures
• Applications to real-world situations

Note(s):

• Previous grade levels investigated similarity and proportionality.
• Previous grade levels solved problems involving similar 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 theorems of triangle similarity.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• III.A. Geometric and Spatial Reasoning – Figures and their properties
• III.A.2. Form and validate conjectures about one-, two-, and three-dimensional figures and their properties.
• III.A.3. Recognize and apply right triangle relationships including basic trigonometry.
• III.B. Geometric and Spatial Reasoning – Transformations and symmetry
• III.B.1. Identify transformations and symmetries of figures.
• III.B.2. Use transformations to investigate congruence, similarity, and symmetries of figures.
• III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
• III.C.1. Make connections between geometry and algebraic equations.
• III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
• III.D.3. Determine indirect measurements of geometric figures using a variety of methods.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A.3. Determine a solution.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
G.8 Similarity, proof, and trigonometry. 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.8A Prove theorems about similar triangles, including the Triangle Proportionality theorem, and apply these theorems to solve problems.

Prove

THEOREMS ABOUT SIMILAR TRIANGLES INCLUDING THE TRIANGLE PROPORTIONALITY THEOREM

Including, but not limited to:

• Angle-Angle Similarity theorem for triangles
• If two angles of one triangle are congruent to two corresponding angles of another triangle, the triangles are similar.
• Side-Side-Side Similarity theorem
• If the measures of the corresponding sides of two triangles are proportional, the triangles are similar.
• Side-Angle-Side Similarity theorem
• If the measures of two sides of a triangle are proportional to the two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar.
• Triangle Proportionality theorem
• If a line intersects two sides of a triangle and is parallel to other side, then the line divides the two sides proportionally.
• Corollary of Triangle Proportionality theorem
• The line that divides the two sides of the triangle and is parallel to the other side creates a smaller triangle similar to the original
• Incorporation of a variety of methods
• Coordinate
• Transformational
• Axiomatic
• Application of a variety of formats
• Two-column proof
• Paragraph proof
• Flow chart

Apply

THEOREMS ABOUT SIMILAR TRIANGLES TO SOLVE PROBLEMS

Including, but not limited to:

• Triangle Proportionality theorem
• Congruency of angles in similar triangles
• Proportionality of sides in similar triangles
• Applications to real-world situations

Note(s):

• Previous grade levels defined similarity and applied similarity to solve problems.
• Previous grade levels used direct variation to solve proportional problems.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• III.A. Geometric and Spatial Reasoning – Figures and their properties
• III.A.2. Form and validate conjectures about one-, two-, and three-dimensional figures and their properties.
• III.B. Geometric and Spatial Reasoning – Transformations and symmetry
• III.B.1. Identify transformations and symmetries of figures.
• III.B.2. Use transformations to investigate congruence, similarity, and symmetries of figures.
• III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
• III.D.3. Determine indirect measurements of geometric figures using a variety of methods.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A.3. Determine a solution.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VII.C.2. Understand attributes and relationships with inductive and deductive reasoning.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations. 