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 Instructional Focus DocumentGeometry
 TITLE : Unit 05: Relationships of Right Triangles, including Trigonometry SUGGESTED DURATION : 22 days

#### Unit Overview

This unit bundles student expectations that address constructions of right triangles and right triangle relationships including the Pythagorean Theorem, Pythagorean triples, special right triangles, and trigonometric ratios. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this unit, in Grade 8, students were introduced to the Pythagorean Theorem and its converse and applications of these in problem situations. In Geometry Unit 04, students studied triangles, including triangle relationships, congruent triangles, similar triangles, and special segments of triangles.

During this unit, students concretely explore and prove the Pythagorean Theorem and its converse. Students examine patterns of given Pythagorean triples to discover other Pythagorean triples. Students use constructions of right triangles and their altitudes to investigate and analyze geometric relationships, including geometric mean. Students explore geometric relationships in special right triangles (30°-60°-90° and 45°-45°-90°) and similar triangles. Students develop right triangle trigonometry and trigonometric ratios to determine side lengths and angle measures in right triangles. Students apply all right triangle geometric relationships to solve both mathematical and real-world problem situations.

After this unit, in Geometry Units 06 and 08, students will continue to use the concepts of right triangles to investigate and solve problems involving properties and measurement of figures. In subsequent courses in mathematics, these concepts will continue to be applied to problem situations involving triangles.

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. According to the National Council of Teachers of Mathematics (2012), using diagrams and constructions to interpret and communicate geometric relationships is essential in geometry. Using definitions of figures to characterize figures in terms of their properties is another essential in geometry. In geometry, the “proving process involves working with diagrams, variation and invariance, conjectures, and definitions.” (p. 92)

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

#### OVERARCHING UNDERSTANDINGS and QUESTIONS

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?

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.

Numeric Reasoning

Algebraic Reasoning

• Equations
• Patterns/Rules
• Solve

Geometric Reasoning

• Congruence
• Geometric Attributes/Properties
• Geometric Relationships
• Logical Arguments
• Pythagorean Theorem
• Similarity
• Theorems/Axioms

Measurement Reasoning

• Angle Measures
• Length

Associated Mathematical Processes

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

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

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

Patterns can be investigated to make conjectures about geometric relationships.

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

Developing conjectures and theorems are important to the development of logical reasoning.

• How does the investigation of patterns lead to the development of conjectures and theorems?
• How is logical reasoning used to verify conjectures and postulates?

Right triangles exhibit various geometric relationships associated with their sides and angles that can be demonstrated and verified using a variety of methods including concrete models and coordinate geometry.

• What geometric relationship does the exploration of the squares of the lengths of the sides of a triangle reveal about the type of triangle?
• What geometric relationship can be verified by exploring the sum of the squares of the legs of any right triangle?
• What geometric theorem can be applied to solve application problems with right triangles?
• What geometric relationship describes dilations of right triangles?
• What relationship exists between sides and angles of triangles that are formed by Pythagorean triples?
• How can the Pythagorean Theorem and Pythagorean triples be used to solve problems?

Geometric tools such as Pythagorean Theorem are used to determine lengths in right triangles.

• For which particular geometric figures can Pythagorean Theorem be used?
• What information is needed in order to use the Pythagorean Theorem?
• How can the Pythagorean Theorem be used to calculate distances?
• How can the Pythagorean Theorem be used to determine if the triangle is a right triangle?

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.

• How does the dilation of Pythagorean Triples by a specified scale factor maintain the definition of similarity?
• What is meant by corresponding parts of similar triangles 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.

Numeric Reasoning

Algebraic Reasoning

• Equations
• Patterns/Rules
• Solve

Geometric Reasoning

• Congruence
• Constructions
• Geometric Attributes/Properties
• Geometric Relationships
• Logical Arguments
• Pythagorean Theorem
• Similarity
• Theorems/Axioms

Measurement Reasoning

• Angle Measures
• Length

Associated Mathematical Processes

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

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

• Why are diagrams necessary for visualizing the geometric relationships found in the problem situation?
• How are diagrams used to organize information from the problem situation?
• How do diagrams aid in calculations when solving problems?

Constructions provide insight into geometric relationships.

• How can constructions be used to validate the geometric relationships that occur when the altitude is constructed to the side of an equilateral triangle?
• How can constructions be used to validate the geometric relationships that occur when the altitude is constructed to the hypotenuse of an isosceles right triangle?
• 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 when the altitude is drawn to the hypotenuse to any right triangle?
• How can constructions be used to validate the geometric relationships that occur with perpendicular bisectors of a line segment?
• How can constructions be used to validate the geometric relationship that occurs with between the sides and angles of similar right triangles?

Right triangles exhibit various geometric relationships associated with their sides and angles that can be demonstrated and verified using a variety of methods including concrete models and coordinate geometry.

• What are the two types of special right triangles?
• How can the relationships in special right triangles be used to solve problems?
• What relationship occurs between the altitude of a right triangles and the segments created by the intersection of the altitude and the opposite side?
• What geometric relationships do explorations of the altitude of a right triangle reveal?

Geometric tools such as Pythagorean Theorem and special right triangle relationships are used to determine lengths and angles in right triangles.

• For which particular geometric figures can Pythagorean Theorem be used?
• What information is needed in order to use the Pythagorean Theorem?
• How can the Pythagorean Theorem be used to calculate distances?
• How can the Pythagorean Theorem be used to determine if the triangle is a right triangle?
• How can the special right triangle relationships be used to calculate distances?
• For which particular geometric figures can special right triangle relationships be used?
• What information is needed to use special right triangle relationships 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.

Numeric Reasoning

Algebraic Reasoning

• Equations
• Patterns/Rules
• Solve

Geometric Reasoning

• Congruence
• Geometric Attributes/Properties
• Geometric Relationships
• Logical Arguments
• Theorems/Axioms
• Trigonometric Ratios

Measurement Reasoning

• Angle Measures
• Length

Associated Mathematical Processes

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

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

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

Right triangles exhibit various geometric relationships associated with their sides and angles that can be demonstrated and verified using a variety of methods including concrete models and coordinate geometry.

• How do similarity, proportionality, and trigonometric ratios relate?
• What are the geometric relationships between the length of sides in any right triangle as defined by the trigonometric ratios sine, cosine, and tangent?
• How can the trigonometric ratios be used to solve real-world situations involving right triangles?
• What are the two types of special right triangles?
• How can the relationships in special right triangles be used to solve problems?

Geometric tools such as Pythagorean Theorem, special right triangle relationships, and trigonometric ratios are used to determine lengths and angles in right triangles.

• For which particular geometric figures can Pythagorean Theorem be used?
• What information is needed in order to use the Pythagorean Theorem?
• How can the Pythagorean Theorem be used to calculate distances?
• How can the Pythagorean Theorem be used to determine if the triangle is a right triangle?
• How can the special right triangle relationships be used to calculate distances?
• For which particular geometric figures can special right triangle relationships be used?
• What information is needed to use special right triangle relationships to solve problems?
• For what particular figures can trigonometric ratios be used to solve problems?
• What information is needed to use trigonometric ratios to solve problem situations?
• How can trigonometric ratios be used to calculate distances and angle measures in a right triangle?

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may not correctly identify which side lengths should be used for the legs (a and b) and the hypotenuse (c) in the Pythagorean Theorem.
• Some students may not use the correct order of operations when applying or solving an equation with the Pythagorean Theorem.
• Although some students may identify a Pythagorean Triple, such as 3, 4, and 5, the students may over-generalize its use to situations where the triple does not apply, such as where the legs of a right triangle are 3 and 5 (instead of 3 and 4).
• Some students may have trouble identifying the adjacent leg and opposite leg for a given acute angle in a right triangle.
• Some students may confuse which trigonometric function is appropriate for the given situation.
• Some students may confuse when to use a trigonometric function (such as sine) and its inverse (sin-1).
• Some students may not correctly identify which angle is the angle of elevation or angle of depression.
• Some students may not correctly apply the special right triangle relationships if the side length opposite the 60o angle or hypotenuse is given rather than the side length opposite the 30o angle.

Underdeveloped Concepts:

• Although some students may know how to work the problem, they may have trouble sketching and labeling the figure appropriately.
• Although some students may correctly use a calculator to evaluate trigonometric functions, the students may get strange or erroneous answers, if the MODE of the calculator is set in RADIANS instead of DEGREES.
• Although some students may know how to use the Pythagorean Theorem, they may have trouble simplifying any radicals produced when solving for a side length.

#### Unit Vocabulary

• 30°-60°-90° right triangle relationships – the hypotenuse is twice as long as the shorter leg, and the longer leg is  times as long as the shorter leg
• 45°-45°-90° right triangle relationships – the legs are equal in measure and the hypotenuse is  times as long as a leg
• Altitude of a triangle – a 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
• Congruent angles – angles whose angle measurements are equal
• Congruent segments – line segments whose lengths are equal
• Corresponding angles – angles in two figures whose relative position is the same
• Geometric construction – construction of accurate representations of lengths, angles, and geometric figures using only a straight edge and compass
• Geometric mean – a positive number x such that , so x2 = ab and x =
• Hypotenuse of a right triangle – the longest side of a right triangle, the side opposite the right angle
• Perpendicular bisector of a line segment – line, ray, or segment that divides a line segment into two congruent segments and forms a 90º angle at the point of intersection
• Perpendicular lines – lines that intersect at a 90º angle to form right angles
• Proportional sides – corresponding side lengths form equivalent ratios
• Segment bisector – point, line, ray, segment, or point that divides a line segment into two congruent segments
• Similar figures – shapes whose angles are congruent and side lengths are proportional (equal scale factor)
• Trigonometric ratios – a ratio of the measures of two sides of a right triangle based on their position in relation to an acute angle in the right triangle

Related Vocabulary:

 Angle of depression Angle of elevation Converse of Pythagorean Theorem Cosine Equilateral triangle Hypotenuse Isosceles right triangle Leg adjacent Leg of a right triangle Leg opposite Pythagorean Theorem Pythagorean triple Right triangle Sine Special right triangle Tangent
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.5 Logical argument and constructions. The student uses constructions to validate conjectures about geometric figures. The student is expected to:
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
• 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

Note(s):

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

Including, but not limited to:

• Concrete models and exploration activities
• Connections between models, pictures, and the symbolic formula
• Proof of the Pythagorean Theorem
• Dynamic geometry software

Apply

THE RELATIONSHIPS IN TRIANGLES, INCLUDING THE PYTHAGOREAN THEOREM, 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
• Pythagorean Theorem and the converse of the Pythagorean Theorem

Note(s):

• Previous grade levels investigated attributes of triangles.
• Grade 8 introduced and applied the Pythagorean Theorem and the converse of the Pythagorean Theorem to solve problems.
• Grade 8 used models and diagrams to explain the Pythagorean Theorem.
• Geometry proves the Pythagorean Theorem and uses the Pythagorean Theorem and the converse of the Pythagorean Theorem to solve problems.
• 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.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.8B Identify and apply the relationships that exist when an altitude is drawn to the hypotenuse of a right triangle, including the geometric mean, to solve problems.

Identify, Apply

THE RELATIONSHIPS THAT EXIST WHEN AN ALTITUDE IS DRAWN TO THE HYPOTENUSE OF A RIGHT TRIANGLE, INCLUDING THE GEOMETRIC MEAN, TO SOLVE PROBLEMS

Including, but not limited to:

• Altitude of a triangle – a line segment drawn from any vertex of a triangle perpendicular to the opposite side
• Hypotenuse of a right triangle – the longest side of a right triangle, the side opposite the right angle
• In a right triangle, the measure of the altitude from the vertex of the right angle to the hypotenuse is the geometric mean between the measures of the two segments formed where the altitude intersects the hypotenuse.
• If an altitude is drawn to the hypotenuse of a right triangle, then the length of either leg is the geometric mean of the length of the hypotenuse and the length of the segment of the hypotenuse adjacent to that leg.
• Geometric mean – a positive number x such that , so x2 = ab and x =
• Connection to similar triangles
• Concrete models and exploration activities
• Dynamic geometric software
• Real-world problem situations

Note(s):

• Geometry introduces the geometric mean.
• Previous grade levels solved problems involving similar figures.
• 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.
• 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.9 Similarity, proof, and trigonometry. The student uses the process skills to understand and apply relationships in right triangles. The student is expected to:
G.9A Determine the lengths of sides and measures of angles in a right triangle by applying the trigonometric ratios sine, cosine, and tangent to solve problems.

Determine

THE LENGTHS OF SIDES AND MEASURES OF ANGLES IN A RIGHT TRIANGLE BY APPLYING THE TRIGONOMETRIC RATIOS SINE, COSINE, AND TANGENT TO SOLVE PROBLEMS

Including, but not limited to:

• Trigonometric ratios – a ratio of the measures of two sides of a right triangle based on their position in relation to an acute angle in the right triangle
• Sine
• Cosine
• Tangent
• Right triangles
• Side lengths
• Angle measures
• Applications to real-world situations

Note(s):

• Middle School introduced ratios and unit rates when developing proportionality.
• Grade 8 uses the Pythagorean Theorem and its converse to solve problems.
• Geometry introduces trigonometric ratios.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• III. Geometric Reasoning
• A3 – Recognize and apply right triangle relationships including basic trigonometry.
• C1 – Make connections between geometry and algebra.
• C3 – Make connections between geometry and measurement.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
G.9B Apply the relationships in special right triangles 30°-60°-90° and 45°-45°-90° and the Pythagorean theorem, including Pythagorean triples, to solve problems.

Apply

THE RELATIONSHIPS IN SPECIAL RIGHT TRIANGLES 30°– 60°– 90° AND 45°– 45°– 90° AND THE PYTHAGOREAN THEOREM, INCLUDING PYTHAGOREAN TRIPLES, TO SOLVE PROBLEMS

Including, but not limited to:

• 30°– 60°– 90° right triangle relationships – the hypotenuse is twice as long as the shorter leg, and the longer leg is  times as long as the shorter leg.
• x represents the length of the short leg.
•  represents the length of the longer leg.
• 2x represents the length of the hypotenuse.
• 45°– 45°– 90° right triangle relationships – the legs are equal in measure and the hypotenuse is  times as long as a leg.
• x represents both congruent legs.
•  represents the hypotenuse.
• Pythagorean Theorem
• Pythagorean triples – three positive integers, a, b and c, such that a2 + b2 = c2, triples can be generated by multiplying a given Pythagorean triple by any positive integer
• Methods for finding Pythagorean triples
• Solutions with rounded decimal answers
• Applications to real-world situations

Note(s):

• Grade 8 used the Pythagorean Theorem and it's converse to solve problems.
• Geometry introduces special right triangles.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• III. Geometric Reasoning
• A3 – Recognize and apply right triangle relationships including basic trigonometry.
• C3 – Make connections between geometry and measurement.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections