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

#### Unit Overview

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).Student 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 Geometry 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): III. Geometric Reasoning A1, A2, A3, B1, B2, B3, D1, D2; IV. Measurement Reasoning C3; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

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.

#### OVERARCHING UNDERSTANDINGS and QUESTIONS

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

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

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

• What roles do undefined terms, defined terms, postulates, and theorems serve in an axiomatic system?
• How does the investigation of geometric patterns lead to the development of conjectures and postulates?
• How is the two-dimensional coordinate system 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 can constructions be used to validate conjectures about geometric figures?

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?
• How do geometric and spatial reasoning allow for the understanding of different geometric systems as models for the world?

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

• Equations
• Inequalities
• Solve

Geometric Reasoning

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

Measurement Reasoning

• Angle Measures
• Length

Associated Mathematical Processes

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

Triangles exhibit various geometric relationships associated with their sides and angles that can be demonstrated and verified using a variety of methods, including concrete activities.

• What geometric relationship does exploration of the sum of the interior angles of a triangle reveal?
• What geometric relationship does exploration of the side lengths of a triangle reveal?
• What geometric relationship does exploration of the base angles of an isosceles triangle reveal?
• What geometric relationship does exploration of the interior and exterior angles of a triangle reveal?

Geometric structures behave differently in Euclidean and spherical geometries.

• What is the Euclidean Triangle Sum theorem?
• How does the sum of the angles in a triangle in spherical geometry compare to the sum of the angles in a triangle in Euclidean geometry?
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic Reasoning

• Patterns/Rules
• Rate of Change/Slope

Geometric Reasoning

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

Measurement Reasoning

• Angle Measures
• Formulas
• Length

Associated Mathematical Processes

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

Patterns can be investigated to make conjectures about geometric relationships.

• What conjectures can be made about the properties of special segments (altitude, median, angle bisector, perpendicular bisector, and midsegment) in triangles through investigation of patterns?

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

• How does the investigation of patterns lead to the development of conjectures, postulates, and theorems?

Constructions provide insight into geometric relationships.

• How can constructions be used to validate the geometric relationships that occur with segment bisectors?
• How can constructions be used to validate the geometric relationships that occur with perpendicular bisectors of a line segment?

Triangles exhibit various geometric relationships associated with special segments that can be demonstrated and proved using a variety of methods, including concrete activities and coordinate geometry.

• What geometric relationships do explorations of the special segments (altitude, median, angle bisector, perpendicular bisector, and midsegment) of a triangle reveal?
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic Reasoning

• Patterns/Rules

Geometric Reasoning

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

Measurement Reasoning

• Angle Measures
• Length

Associated Mathematical Processes

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

Patterns can be investigated to make conjectures about geometric relationships.

• What conjectures can be made about sides and angles in congruent triangles through investigation of patterns?

Constructions provide insight into geometric relationships.

• How can constructions be used to validate the geometric relationships that occur with congruent segments?
• How can constructions be used to validate the geometric relationships that occur with congruent angles?

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

• How is logical reasoning used to verify conjectures, theorems, and postulates?
• How are proofs used to verify conjectures, theorems, and postulates?
• What types of proofs can be used to verify geometric relationships?

Congruent triangles exhibit various geometric relationships associated with congruency of corresponding sides and angles that can be demonstrated and proved using a variety of methods.

• What is the definition of congruence in terms of rigid transformations?
• What role do corresponding sides and angles have in determining criteria for congruency?
• What relationships must occur in order to prove triangles are congruent?
• What is meant by corresponding parts of congruent triangles are congruent (CPCTC), and how is this used to solve problems?
 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
• Ratios
• Solve

Relations

• Proportional Relationships

Geometric Reasoning

• Constructions
• Geometric Attributes/Properties
• Geometric Relationships
• Logical Arguments
• Proofs
• Scale Factors
• Similarity
• Theorems/Postulates/Axioms
• Transformations

Measurement Reasoning

• Angle Measures
• Length

Associated Mathematical Processes

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

Constructions provide insight into geometric relationships.

• How can constructions be used to validate the geometric relationships that occur with congruent segments?
• How can constructions be used to validate the geometric relationships that occur with congruent angles?
• How can constructions be used to validate the geometric relationships that occur with segment bisectors?
• How can constructions be used to validate the geometric relationships that occur with perpendicular bisectors of a line segment?

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

• How is logical reasoning used to verify conjectures, theorems, and postulates?
• How are proofs used to verify conjectures, theorems, and postulates?
• What types of proofs can be used to verify geometric relationships?

Similar triangles exhibit various geometric relationships associated with proportionality of corresponding sides and congruency of corresponding angles that can be demonstrated and proved using a variety of methods.

• What is the definition of similarity in terms of a dilation?
• What role do corresponding sides and angles have in determining criteria for similarity?
• What relationships must occur in order to prove triangles are similar?
• What is meant by corresponding parts of similar triangles and how is this used to solve problems?

#### 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-Angle criterion for triangle similarity – if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar
• Angle bisector – line, ray, or segment that divides an angle into two congruent angles
• Base angles of a triangle – angles that have one side in common with the base
• Congruent angles – angles whose angle measurement is 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
• 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
• 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 – a line segment drawn from the midpoints of two sides of the triangle
• Parallel line – line parallel to a given line through a point not on a line
• 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 the triangle at the midpoint of the side
• Perpendicular lines – lines in the same plane that intersect at 90° angles whose slopes are opposite reciprocals,
• 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
• Spherical geometry – the study of figures on the two-dimensional curved surface of a sphere
• Similar figures – shapes whose angles are congruent and side lengths are proportional (equal scale factor)
• 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 Centroid Circumcenter Congruent triangle Conjecture Constructions Corresponding parts,   including CPCTC Equiangular   triangle Dilation Equilateral triangle Hypotenuse-Leg (HL) Incenter Isosceles triangle Obtuse triangle Orthocenter Parallel lines Postulate Proofs Proportion Right triangle Rigid transformation 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 Creator if your district has granted access to that tool.

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

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

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

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

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

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

Texas Instruments – Graphing Calculator Tutorials

TEKS# SE# TEKS Unit Level Specificity

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

Apply

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

Including, but not limited to:

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

Note(s):

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

Use

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

Including, but not limited to:

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

Note(s):

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

Select

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

Including, but not limited to:

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

Note(s):

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

Communicate

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

Including, but not limited to:

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

Note(s):

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

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

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

Note(s):

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

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

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

Note(s):

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

Display, Explain, Justify

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

Including, but not limited to:

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

Note(s):

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

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

Compare

GEOMETRICRELATIONSHIPS 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
• III. Geometric Reasoning
• D1 – Make and validate geometric conjectures.
• D2 – Understand that Euclidean geometry is an axiomatic system.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
G.5 Logical argument and constructions. The student uses constructions to validate conjectures about geometric figures. The student is expected to:
G.5A

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

Investigate

PATTERNS TO MAKE CONJECTURES ABOUT GEOMETRIC RELATIONSHIPS, INCLUDING CRITERIA REQUIRED FOR TRIANGLE CONGRUENCE, SPECIAL SEGMENTS OF TRIANGLES INTERIOR AND EXTERIOR ANGLES OF POLYGONS OF CIRCLES 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
• Special segments of triangles
• Altitude of a triangle – line segment drawn from any vertex of a triangle perpendicular to the opposite side
• Median of a triangle – line segment drawn from any vertex of a triangle to the midpoint of the opposite side
• Angle bisector – line, ray, or segment that divides an angle into two congruent angles
• Perpendicular bisector of a side of a triangle – line segment that is perpendicular to a side of the triangle at the midpoint of the side
• Midsegment of a triangle – a line segment drawn from the midpoints of two sides 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. Geometric Reasoning
• A1 – Identify and represent the features of plane and space figures.
• A2 – Make, test, and use conjectures about one-, two-, and three-dimensional figures and their properties.
• B2 – Identify the symmetries in a plane figure.
• D1 – Make and validate geometric conjectures.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
G.5B

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

Construct

CONGRUENT 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. Slopes of perpendicular lines are opposite reciprocals, my2.
• Parallel line – line parallel to a given line through a point not on a line

Note(s):

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

Use

THE CONSTRUCTIONS OF CONGRUENT 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
• 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.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. 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.
• 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.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. 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.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. 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.
• B1 – Identify and apply transformations to figures.
• B2 – Identify the symmetries in a plane figure.
• B3 – Use congruence transformations and dilations to investigate congruence, similarity, and asymmetries of plane figures.
• D1 – Make and validate geometric conjectures.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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. Geometric Reasoning
• B1 – Identify and apply transformations to figures
• B3 – Use congruence transformations and dilations to investigate congruence, similarity, and symmetries of plane figures.
• D1 – Make and validate geometric conjectures.
• IV. Measurement Reasoning
• C3 – Determine indirect measurements of figures using scale drawings, similar figures, Pythagorean Theorem, and basic trigonometry.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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. 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.
• B3 – Use congruence transformations and dilations to investigate congruence, similarity, and asymmetries of plane figures.
• D1 – Make and validate geometric conjectures.
• IV. Measurement Reasoning
• C3 – Determine indirect measurements of figures using scale drawings, similar figures, Pythagorean Theorem, and basic trigonometry.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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. Geometric Reasoning
• B1 – Identify and apply transformations to figures
• B2 – Identify the symmetries in a plane figure.
• B3 – Use congruence transformations and dilations to investigate congruence, similarity, and symmetries of plane figures.
• D1 – Make and validate geometric conjectures.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections