C281 - College GeometryCourse of Study

This course supports the assessments for College Geometry. The course covers 4competencies and represents 4 competency units.

Introduction

OverviewThis course is designed for prospective secondary school mathematics teachers. It uses bothsynthetic and analytic approaches.

In this course, you will be introduced to formal proofs using geometric properties, and have theopportunity to explore basic concepts of transformational geometry. You will also becomefamiliar with the use of dynamic technologies and selected advanced topics in the study ofgeometry.

Watch the following video for an introduction to the course:

Note: To download this video, right-click the following link and choose "Save as...": downloadvideo.

CompetenciesThis course provides guidance to help you demonstrate the following 4 competencies:

Competency 218.1.1: Axiomatic SystemsThe graduate applies the axiomatic nature of geometry to analyze the fundamentalconcepts and principles of Euclidean and non-Euclidean geometries.Competency 218.1.2: Properties and RelationshipsThe graduate applies synthetic and analytic methods to construct proofs and solvesproblems involving the properties and relationships of two-dimensional objects.Competency 218.1.3: Congruence and SimilarityThe graduate proves theorems involving congruence and similarity of geometric objectsand applies them to solve problems.Competency 218.1.6: Geometric TransformationsThe graduate applies geometric transformations to explore and analyze objects andsolve problems.

Course Mentor AssistanceAs you prepare to successfully demonstrate competency in this subject, remember that coursementors stand ready to help you reach your educational goals. As subject matter experts,mentors enjoy and take pride in helping students become reflective learners, problem solvers,and critical thinkers. Course mentors are excited to hear from you and eager to work with you.

Successful students report that working with a course mentor is the key to their success. Coursementors are able to share tips on approaches, tools, and skills that can help you apply thecontent you're studying. They also provide guidance in assessment preparation strategies and

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troubleshoot areas of deficiency. Even if things don’t work out on your first try, course mentorsact as a support system to guide you through the revision process. You should expect to workwith course mentors for the duration of your coursework, so you are welcome to contact themas soon as you begin. Course mentors are fully committed to your success!

Preparing for Success

The information in this section is provided to detail the resources available for you to use as youcomplete this course.

Learning ResourcesThe learning resources listed in this section are required to complete the activities in this course.For many resources, WGU has provided automatic access through the course. However, youmay need to manually enroll in or independently acquire other resources. Read the fullinstructions provided to ensure that you have access to all of your resources in a timely manner.Enroll in Learning Resources

You will need to enroll in or subscribe to learning resources as a part of this course. You mayalready have enrolled in these resources for other courses. Please check the “LearningResources” tab and verify that you have access to the following learning resources. If you donot currently have access, please enroll or renew your enrollment at this time.

Note: For instructions on how to enroll in or subscribe to learning resources through the“Learning Resources” tab, please see the “Acquiring Your Learning Resources” page.

Geometer’s Sketchpad

Geometer’s Sketchpad is interactive geometry software that you will utilize throughout thiscourse of study. Geometer’s Sketchpad is used in many geometry classrooms throughout theUnited States, so becoming familiar with this resource now will also prepare you to teachgeometry in the future.Automatically Enrolled Learning Resources

You will be automatically enrolled at the activity level for the following learning resources.Simply click on the links provided in the activities to access the learning materials.

uCertifyYou will access a uCertify resource at the activity level within this course. The following will beyour primary learning resource throughout this course:

College Geometry 2Other Learning Resources

You will use the following learning resources for this course.

Online ResourcesYou will access content from several important mathematics education websites at the activity

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level throughout this course. You may also want to explore more of the resources available onthese sites:

Annenberg LearnerMath Open RefRegents Prep 

Pacing GuideThe pacing guide suggests a weekly structure to pace your completion of learning activities. It isprovided as a suggestion and does not represent a mandatory schedule. Follow the pacingguide carefully to complete the course in the suggested timeframe.

Pacing Guide: College Geometry

Note: This pacing guide does not replace the course. Please continue to refer to the course fora comprehensive list of the resources and activities.

Axiomatic Systems

Studying geometry presents an opportunity to develop and apply reasoning skills using aconcrete topic—you will be able to draw and see the ideas connected to your reasoning. Dr.Kenneth Ross of the Mathematical Association of America asserts: “One of the most importantgoals of mathematics courses is to teach students logical reasoning. . . . If reasoning ability isnot developed in the student, then mathematics simply becomes a matter of following a set ofprocedures and mimicking examples without thought as to why they make sense.”

Introductory ConceptsThe foundational introductory concepts related to geometry must first be understood before itcan be studied. You will first be introduced to the foundations of geometric thought and notation.

This topic addresses the following competency:

Competency 218.1.1: Axiomatic SystemsThe graduate applies the axiomatic nature of geometry to analyze the fundamentalconcepts and principles of Euclidean and non-Euclidean geometries.

This topic highlights the following key concepts:

the validity of a deductive argumentundefined terms, postulates, theorems, and definitionshow knowledge is built in an axiomatic system

Study Introductory Concepts

It is important for you to be aware of the symbols and markings used to identify geometricfigures and their properties as part of understanding the axiomatic system. As you learn aboutgeometric figures throughout the course, sketch these figures in your geometry notebook andinclude their notation.

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Read the following in College Geometry 2:

chapter 1 (“Introduction”)section 2.1 (“Reasoning”)

After you have read these sections of the resource, review all of the information on the followingpages from Regents Prep:

Undefined Terms: Point, Line and PlaneNotationTypes of Sentences (Logic)Negation (Logic)Conditional (Logic)Biconditional (Logic)

Geometer's Sketchpad Lab: Introduction to Geometer's Sketchpad

Complete the following lab to become familiar with the software that can be used to exploregeometry in an inductive manner:

Geometer’s Sketchpad (GSP) Lab 1: Introduction to Geometer’s Sketchpad

The links below from the GSP Learning Center can be used throughout the course whenworking with GSP, so you may want to bookmark them for your reference.  You will find anumber of activities, with video instruction, on how to complete various activities with GSP:

Getting Started TutorialsSketchpad Tips

You can also use the following document as a reference, which outlines the basic functions ofGSP for use as you familiarize yourself with the program:

Summary of Geometer’s Sketchpad Tools

You may want to save this to your computer or print it out so you can use it as you complete thelabs throughout this course and in the related performance tasks.

 

Axiomatic SystemsEuclid developed his geometry from five axioms (postulates) assumed to be true. Alternatives tothe fifth postulate suggest alternatives to Euclid’s geometry, including a geometry more suitedto a spherical planet as opposed to a flat one.  Prior to studying geometric concepts, you willlook at how to build knowledge deductively in an axiomatic system.

This topic addresses the following competency:

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Competency 218.1.1: Axiomatic SystemsThe graduate applies the axiomatic nature of geometry to analyze the fundamentalconcepts and principles of Euclidean and non-Euclidean geometries.

This topic highlights the following key concepts:

triangle properties given an alternative to the parallel postulategeometries and their unique characteristics

Explore and Study Alternatives to Euclidean Geometry

Become familiar with alternative geometries and their relationship to the parallel line postulate.Make note of the types of non-Euclidean geometries and their features.

Read the following in College Geometry 2:

section 2.5 (“Non-Euclidean Geometries”)

After you have read this section of the resource, review all of the information on the followingpages.  Take notes on the specific applications of these geometries:

Euclidean and Non-Euclidean GeometryEuclidean, Hyperbolic, and Elliptical Geometries

Explore Triangles in Non-Euclidean Geometries

Experiment with the following applets, which allow you to draw a triangle and observe its interiorangle sum for triangles and other properties in hyperbolic and spherical geometry:

Hyperbolic Geometry TriangleSpherical (Elliptic) Geometry Triangle (scroll down to the applet under the heading: TheArea of a Spherical Triangle Part 1)

For each applet, click and drag the vertices of the triangles given to change the triangle andobserve its angle sums. Draw an example of a triangle in each geometry (hyperbolic andspherical) in your geometry notebook, and record your observations about how triangles differ inEuclidean and non-Euclidean geometries. Share your observations in the College GeometryLearning Community.

Also, engage with the following applets to visually explore how objects in these geometriesappear and interact:

Hyperbolic GeometrySpherical Geometry

Axiomatic Systems

An axiomatic system begins with undefined terms and statements (axioms) that are assumed tobe true.  Read the information contained in the following site to gain an understanding of

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axiomatic systems:

Introduction to Axiomatic Systems

The following video also provides an introduction to axiomatic systems using an example thatwill be used again in the next activity.

Note: To download this video, right-click the following link and choose "Save as...": downloadvideo.Undefined Terms, Verifying Axiom Independence, and Proving Theorems

Once axioms have been established, theorems can be deduced, which are proven using theaxioms and logical reasoning. These theorems can be used to prove more theorems, and thesystem continues to grow. Euclid’s Elements represents a form of axiomatic system. A refinedaxiomatic system is at the heart of modern mathematics.

Consider the following set of axioms:

Axiom 1: Exactly 4 students are in each group.Axiom 2: Each student is in exactly 3 groups.Axiom 3: No 2 students are together in more than 1 group.

In these axioms, students, groups, and in are undefined terms.  Note that students and groupsare elements because they imply objects, and in is a relation because it denotes somerelationship between students and groups.

The three axioms are independent. That means each axiom cannot be proven using the otheraxioms, and cases exist where one axiom is not satisfied, given that the other two are satisfied. Independence of an axiom is shown by creating a model where the other axioms are true, whilethe axiom being shown independent is false.

For example, the following model would satisfy the conditions of axiom 1 and axiom 2: Studentsgrouped as A, B, C, D; A, B, E, F; A, B, G, H; C, D, E, F; C, D, G, H; E, F, G, H. Each group hasexactly four students, and each student is in exactly three groups. However, this does notsatisfy axiom 3, as some of these students are together in more than one group (e.g. student Ais with student B in three groups). So axiom 3 is independent of axiom 1 and axiom 2.  A similarargument would need to be made to show that axiom 1 is independent of the other two, andagain to show axiom 2 is independent of the other two.

Theorems can be deduced from these axioms. For example:

Theorem 1: There are at least ten students.

This theorem can be proven using the axioms, as follows:

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Since each group has exactly four students (axiom 1), one group looks like: A, B, C, D(where each letter represents a student).Each student is in exactly three groups (axiom 2), so there will be other groups withstudent A in them. But since no two students are together in more than one group(axiom 3), the other two groups with student A in them need to have the form: A, E, F, Gand A, H, I, J.Thus, under these conditions, there needs to be at least ten students (A through J).

Notice that this has not proven that ten students are sufficient to accommodate all students in allgroups, only that there needs to be at least ten to accommodate student A. Determining howmany students are sufficient is an entirely different theorem to be explored.Examples of Axiomatic Systems

Other examples of relatively simple axiomatic systems, theorems that can be deduced fromthem, and the idea of independence can be found at the following web page:

Example of Axiomatic Systems

Note: A video lecture of this example is contained here. 

Watch the following video sequence of a slight variation of the example above.

Note: To download this video, right-click the following link and choose "Save as...": downloadvideo.

Note: To download this video, right-click the following link and choose "Save as...": downloadvideo.

Note: To download this video, right-click the following link and choose "Save as...": downloadvideo.

Task 1: Axiomatic SystemsNow you have the competency necessary to complete Task 1.Complete Task 1

Complete the following task in TaskStream:

Task 1

For details about this performance assessment, see the "Assessment" tab in this course.

Lines, Angles, and Circles

There are multiple geometries possible depending on what is assumed to be true, but which

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geometry is used to model the Earth? Is the planet flat or round? If you look at the floor of theroom you are in, it looks flat, even though you know the planet is round. On a relatively smallscale, the “flat” Euclidean Geometry models the world well, so it makes sense to explore basicgeometric figures and how they relate to each other using Euclidean geometry.

Lines and AnglesLines and angles are the basic “building blocks” of geometric figures, and theirinterrelationships provide the student opportunity for exploration, deductive reasoning, andcreation of formal proofs.

This topic addresses the following competency:

Competency 218.1.2: Properties and RelationshipsThe graduate applies synthetic and analytic methods to construct proofs and solvesproblems involving the properties and relationships of two-dimensional objects.

This topic highlights the following key concepts:

analytic methods to calculate the midpoint and length of a line segmentanalytic methods to determine if two lines are parallel, perpendicular, or neitherangles and angle relationshipsproblems involving parallel lines, transversals, and their anglescommon line and angle constructions using a straight edge and compassproofs involving parallel lines, transversals, and their angles

Explore Angle Classification and Relationships in Euclidean Geometry

Complete the “Try This” activities from Math Open Ref, noting the terminology and notationused with the angles and how their angle measures relate. In your geometry notebook, sketchand label an example of the angles formed when two parallel lines are intersected by atransversal.

Angle (Review all the information on this page through the heading "Types of angle.")Transversal (Review all the information on this page through the heading "Properties ofa transversal of parallel lines.")

Explore Points and Lines from an Analytic Perspective

Complete the “Try This” activities from Math Open Ref. As you explore, be aware of how therelationships between the lines and points express themselves in coordinate geometry incontrast to what you know about these relationships in axiomatic geometry. Record importantformulas and the properties they express in your geometry notebook.

Perpendicular Lines (Coordinate Geometry)Parallel Lines (Coordinate Geometry)Midpoint of a Line Segment (Coordinate Geometry)Distance between two points (given their coordinates)Slope of a Line (Coordinate Geometry)

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Study and Review Angle Relationship

Read the following in College Geometry 2:

section 2.2 ("Building Blocks of Geometry")section 2.3 ("Starting Points")section 2.4 ("Early Constructions and Proofs")

View the embedded videos on constructions and parallel lines relationships. This portion of thetext includes deductive proofs of theorems as well as geometric constructions. Note importantproperties and theorems in your geometry notebook. As the compass and straightedgeconstructions are presented throughout the text, practice the technique by creating an examplein your geometry notebook and performing the construction.

If you need further help with compass and straightedge constructions related to the reading, youcan find them animated from Math Open Ref:

Perpendicular bisector of a line segment (i.e., construct the midpoint of a segment)Constructing a parallel through a point (angle copy method)Constructing a parallel through a point (rhombus method)

Synthesize and Practice with Angle Relationships and Coordinate Geometry

Complete the exercises at the end of chapter 2 ("Basics of Geometry") in College Geometry 2.Be sure to engage the setting "Learn Mode," while leaving all other settings as given. Then youcan "Start your test":

College Geometry 2

Access the following link from the Shodor website to practice with angle relationships bypracticing classifying angles and finding angle measures using the interrelationships amongangles made by transversals through parallel lines.

Angles

View the following video for explanation on how to use the link above:

Note: To download this video, right-click the following link and choose "Save as...": downloadvideo.

Access the following link from the Regents Prep website to apply the formulas and propertiesfrom coordinate geometry to problems; be sure to check your answers with the link at thebottom of the page.

Multiple Choice Practice: Coordinate GeometryGeometer?s Sketchpad Lab: Constructing Parallel and Perpendicular Lines

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Complete the following lab:

GSP Lab 2: Constructing Parallel and Perpendicular Lines

The links below from the GSP Learning Center can be used throughout the course whenworking with GSP, so you may want to bookmark them for your reference.  You will find anumber of activities, with video instruction, on how to complete various activities with GSP:

Getting Started TutorialsSketchpad Tips

You can also use the following document as a reference, which outlines the basic functions ofGSP for use as you familiarize yourself with the program:

Summary of Geometer’s Sketchpad Tools

You may want to save this to your computer or print it out so you can use it as you complete thelabs throughout this course and in the related performance tasks.

The CircleIn the previous topic, your explorations involved straight lines, how they intersect, and therelationships among the angles they create. You used a curved line, the circle, to construct linesand angles. Next you will look in depth at the circle itself.

This topic addresses the following competency:

Competency 218.1.2: Properties and RelationshipsThe graduate applies synthetic and analytic methods to construct proofs and solvesproblems involving the properties and relationships of two-dimensional objects.

This topic highlights the following key concepts:

common constructions involving circles using a straightedge and compasslines and angles associated with circlesanalytic methods to prove theorems involving circlessynthetic methods to prove theorems involving circlesmeasurements related to circlescoordinate geometry to solve problems involving circles

Explore Circles and Related Geometric Figures

Familiarize yourself with parts of circles and lines associated with them by sketching labeleddrawings in your geometry notebook from these examples.

Parts of a circle - pictorial index Explore Circles from an Analytic Perspective

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Review all of the information on the following site through the heading “Example.”  Then followthe steps on “Things to Try” near the bottom of this page:

Basic Equation of a Circle

Note the location of the center of the circle and how the radius of the circle is expressed in theequation. Sketch an example of a circle on the coordinate grid with its equation in yourgeometry notebook.Study Constructions and Circles

Read the following in College Geometry 2:

section 3.1 ("The Circle in Detail")section 3.2 ("Area of a Circle")section 3.3 ("Chords and Tangents")section 3.4 ("Theorems of Circles")section 3.5 ("Coordinate Geometry of Circles")

View the embedded videos on constructions and circle relationships. Make special note of theformulas for area and circumference and how they differ.

If you need further help with compass and straightedge constructions related to the reading, youcan find them animated by from Math Open Ref:

Finding the center of a circle or arcTangents to a circle through an external pointTangent to a circle through a point on the circle

Synthesize and Practice with Circle Properties

Complete and check your answers to the following multiple choice practice problems at MultipleChoice Practice: Circles:

1, 2, 3, 4, 5, 7, 8, 9, 10, 13, 15, 18, and 19Geometer?s Sketchpad Lab: Circles

Complete the following lab:

GSP Lab 3: Circles

The links below from the GSP Learning Center can be used throughout the course whenworking with GSP, so you may want to bookmark them for your reference.  You will find anumber of activities, with video instruction, on how to complete various activities with GSP:

Getting Started TutorialsSketchpad Tips

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You can also use the following document as a reference, which outlines the basic functions ofGSP for use as you familiarize yourself with the program:

Summary of Geometer’s Sketchpad Tools

You may want to save this to your computer or print it out so you can use it as you complete thelabs throughout this course and in the related performance tasks.

Triangles, Congruence, and Similarity

The most basic polygon is the triangle, yet studying it can allow us to learn much about otherpolygons.  Triangles may be considered a major building block of geometry.  Once a singletriangle is understood, can we compare two triangles?  What are the sufficient conditions toknow if two triangles are congruent? If they are not congruent, does that mean they are notrelated at all? Congruency provides opportunities for reasoning and proof, comparison andcontrast. Similarity has applications in scale models and map reading.

TrianglesYou have constructed line segments and angles. If you put these pieces together, you getpolygons. The simplest polygon is the triangle. Later on in this course, you will look at otherpolygons, which can be divided up or dissected into triangles, and thus the properties of atriangle can be used to find properties of other polygons.

This topic addresses the following competency:

Competency 218.1.2: Properties and RelationshipsThe graduate applies synthetic and analytic methods to construct proofs and solvesproblems involving the properties and relationships of two-dimensional objects.

This topic highlights the following key concepts:

classification of trianglesline segments and properties associated with trianglesusing a straightedge and compass to construct lines associated with trianglesproofs involving triangle properties

Explore Triangles and Triangle Properties

Study the different ways of classifying triangles, using angles and sides at the following link fromMath Open Ref:

Triangles 

Be sure to click on each triangle to link to more detailed properties of these triangles andinteractive applets that allow exploration of each different triangle type. Note the two types ofclassifications used for triangles (by sides or by angles) and properties associated with thesenames in your geometry notebook.  Review all of the information contained within each link ofeach type of triangle.

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Quiz yourself on your understanding:

Quiz: TrianglesStudy Triangle Relationships

Read the following in College Geometry 2:

section 3.6 ("Triangles")section 3.7 ("Types of Triangles")section 3.8 ("General Triangle Properties")section 3.9 ("Triangle Theorems")

View the embedded videos on constructions and triangle relationships. As a construction isdemonstrated, create a similar example repeating the construction in your geometry notebook.The Triangle Centers page has links to animations of the constructions for the triangle centers(also linked in the list below). Click into each of the four centers of a triangle from this site toexplore each center, its construction, and its properties. Note each center in your geometrynotebook.

If you need further help with compass and straightedge constructions related to the reading, youcan find them animated by from Math Open Ref:

Medians of a TriangleOrthocenter of a Triangle (i.e., construct an altitude of a triangle)Equilateral TriangleIsosceles Triangle (given base and one side)Isosceles Triangle (given base and altitude)Circumcenter of a Triangle (i.e., construct perpendicular bisectors of a triangle)Incenter of a Triangle (i.e., construct angle bisectors of a triangle)Centroid of a Triangle (i.e., construct medians of a triangle)

Synthesisze and Practice with Angles, Circles, and Triangles

Complete the exercises at the end of chapter 3 ("Simple Two Dimensional Shapes") in CollegeGeometry 2. Be sure to engage the setting "Learn Mode," while leaving all other settings asgiven. Then you can "Start your test": 

College Geometry 2

Also complete the practice on triangle inequalities from the Regents Prep website:

Practice: Triangle InequalitiesGeometer?s Sketchpad Lab: Constructing Triangles

Complete the following lab:

GSP Lab 4: Constructing Triangles

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The links below from the GSP Learning Center can be used throughout the course whenworking with GSP, so you may want to bookmark them for your reference.  You will find anumber of activities, with video instruction, on how to complete various activities with GSP:

Getting Started TutorialsSketchpad Tips

You can also use the following document as a reference, which outlines the basic functions ofGSP for use as you familiarize yourself with the program:

Summary of Geometer’s Sketchpad Tools

You may want to save this to your computer or print it out so you can use it as you complete thelabs throughout this course and in the related performance tasks.

Congruence and SimilarityRecall that higher order polygons can be divided into triangles. Knowing how to prove trianglescongruent allows you to prove various properties of other polygons using triangle congruency.

This topic addresses the following competency:

Competency 218.1.3: Congruence and SimilarityThe graduate proves theorems involving congruence and sSimilarity of geometric objects and applies them to solve problems.

This topic highlights the following key concepts:

why combinations of congruent corresponding parts do or do not prove trianglecongruence or similaritythe Pythagorean theorem using multiple methodsspecial triangles (30-60-90, 45-45-90)

 Explore Triangle Congruence

The interactive site Congruence Theorems allows for exploration of what congruent parts oftriangles are sufficient to guarantee two triangles are congruent. Read the instructions at the topof the page and complete the activity six times for each of the cases of triangles using thefollowing parts:

3 sides (SSS)3 angles (AAA)2 angles and a side (ASA, AAS)2 sides and an angle (SAS, SSA)

Note in your geometry notebook which combinations of corresponding triangle parts always

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create congruent triangles and which do not.

View the video below for explanation on how to use the activity in the link above:

Note: To download this video, right-click the following link and choose "Save as...": downloadvideo.Study Triangle Congruence and Similarity

Read the following in College Geometry 2:

section 4.1 ("Forms of Congruency")section 4.2 ("Forms of Similarity")

View the embedded videos, web links, and Connect the Idea problems. Compare the results ofyour exploration to the information in the Learning Resource: which corresponding congruentparts are sufficient to prove two triangles congruent? Which are similar?

If you need further help with compass and straightedge constructions related to the reading, youcan find them animated from Math Open Ref:

Triangle, given all 3 sides (SSS)Triangle, given two sides and included angle (SAS)Triangle, given one side and adjacent angles (ASA)

Geometer's Sketchpad Lab: Triangle Congruence and SSA Case

Complete the following lab:

GSP Lab 5: Triangle Congruence and SSA Case

The links below from the GSP Learning Center can be used throughout the course whenworking with GSP, so you may want to bookmark them for your reference.  You will find anumber of activities, with video instruction, on how to complete various activities with GSP:

Getting Started TutorialsSketchpad Tips

You can also use the following document as a reference, which outlines the basic functions ofGSP for use as you familiarize yourself with the program:

Summary of Geometer’s Sketchpad Tools

You may want to save this to your computer or print it out so you can use it as you complete thelabs throughout this course and in the related performance tasks.Explore the Pythagorean Theorem

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Read the following in College Geometry 2:

section 4.3 ("Pythagorean theorem")

Compare the proof of the Pythagorean Theorem presented in the reading to the two interactivepuzzles at the following site:

Pythagorean Theorem

Discuss in your College Geometry Learning Community how these puzzles visually andalgebraically demonstrate the Pythagorean Theorem. Sketch each puzzle’s solution in yourgeometry notebook, and include their algebraic equations that prove the Pythagorean Theorem.Study More Triangle Theorems and Special Triangles

Read the following in College Geometry 2:

section 4.4 ("Theorems")section 4.5 ("Particular Triangles Relationships")

Also review all of the information about these relationships on the Math Open Ref website:

30-60-90 Triangle45-45-90 Triangle

Include sketches and examples of right triangles and special triangles in your geometrynotebook.Synthesize and Practice Triangle Congruency, Similarity, and the Pythagorean Theorem

Complete the exercises at the end of chapter 4 ("Similarity and Congruency in triangles") inCollege Geometry 2.  Be sure to engage the setting “Learn Mode,” while leaving all othersettings as given, then you can “Start your test”:

College Geometry 2

Complete the extra practice with special right triangles by using the Khan Academy website:

Special right trianglesTangrams

The tangram is an example of a dissection puzzle. 

Explore the information about tangrams on the following page from Wolfram MathWorld:

Tangram 

Explore using Tangrams in the following interactive demonstration from the National Library of

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Virtual Manipulatives:

Tangram Demonstration

View the video below for explanation of using the link above.

Note: To download this video, right-click the following link and choose "Save as...": downloadvideo.

Quadrilaterals and Higher Order Polygons

After studying triangles, we will advance to working with polygons with more sides.  However,what was learned about triangles can assist with these higher order polygons because they canbe divided into triangles when we draw diagonals through them.

Quadrilaterals and Higher Order PolygonsYou will now move on from the triangle to other polygons by increasing the number of sides andangles within the figure. However, you can still use what you learned from triangles when youstudy these higher order polygons because they can be subdivided into triangles. The higherorder polygons increase in complexity and provide many opportunities for further reasoning andproof, applying what you have already learned about line, angle, and triangle relationships.

This topic addresses the following competency:

Competency 218.1.2: Properties and RelationshipsThe graduate applies synthetic and analytic methods to construct proofs and solvesproblems involving the properties and relationships of two-dimensional objects.

This topic highlights the following key concepts:

the hierarchy of quadrilateralsthe properties of quadrilateralstheorems involving properties of quadrilateralsproblems given a higher order polygon

Explore Quadrilaterals and Their Interrelationships

Explore the properties of the various quadrilaterals from this summary page by clicking on eachof the quadrilateral types (i.e., square, rectangle, parallelogram, trapezoid, rhombus, and kite)and reviewing all information on each page. Sketch an example of each quadrilateral given,noting its definition and properties, in your geometry notebook.

Quadrilateral Study Quadrilaterals and Their Interrelationships

Read the following in College Geometry 2:

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section 5.1 ("Quadrilaterals")section 5.2 ("Types of Quadrilaterals")

View the embedded videos and complete the Connect the Idea problems. Pay special attentionto the quadrilateral hierarchy and the interrelationships among the quadrilaterals. Choose atheorem about properties of a particular quadrilateral in the chapter and explain in your learningcommunity how you would prove that property.Review and Practice Concepts about Quadrilaterals

Complete all exercises in the first link and problems 1, 4, 5, 6, 8, 9, 10, and 11 in the secondlink, and check your answers:

Numerical Practice with QuadrilateralsProofs Involving Quadrilaterals

Verifying a Dissected Quadrilateral

Previously you learned about the tangram dissection puzzle. In the video below, you will see anon-tangram dissection of a quadrilateral. You are given the measurements of the sides andangles of each of the dissected pieces, and will use them to prove that the shape is a square.

Note: To download this video, right-click the following link and choose "Save as...": downloadvideo.Study Higher Order Polygons

Read the following in College Geometry 2:

section 5.3 ("Higher order polygons")section 5.4 ("Calculating Angles")

View the embedded videos. Note the methods for calculating interior and exterior angle sumsfor polygons and include an example in your geometry notebook. Notice how triangles are usedto calculate area of higher order polygons.Geometer?s Sketchpad Lab: Quadrilaterals

Complete the following lab:

GSP Lab 6: Quadrilaterals

The links below from the GSP Learning Center can be used throughout the course whenworking with GSP, so you may want to bookmark them for your reference.  You will find anumber of activities, with video instruction, on how to complete various activities with GSP:

Getting Started TutorialsSketchpad Tips

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You can also use the following document as a reference, which outlines the basic functions ofGSP for use as you familiarize yourself with the program:

Summary of Geometer’s Sketchpad Tools

You may want to save this to your computer or print it out so you can use it as you complete thelabs throughout this course and in the related performance tasks.Review Perimeter and Area of Polygons

Review the following in College Geometry 2:

section 5.5 ("Calculating perimeters")section 5.6 ("Ways of calculating area")

View the embedded videos. Notice how triangles are used to calculate area of higher orderpolygons. Include an illustration in your geometry notebook.Synthesize and Practice Concepts About Quadrilaterals and Higher Order Polygons

Complete the exercises at the end of chapter 5 ("Higher Order Polygons") in College Geometry2.  Be sure to engage the setting “Learn Mode,” while leaving all other settings as given, thenyou can “Start your test”: 

College Geometry 2

Check your answers against those provided.

Task 2: Properties and RelationshipsYou now have the competency necessary to complete Task 2.Complete Task 2

Complete the following task in TaskStream:

Task 2

Here is a guide for Task 2:

Guide to College Geometry Task 2

For details about this performance assessment, see the "Assessment Preparation" box in thiscourse.

Visualization

The NCTM standards advocate that students be able to problem solve using visualization andspatial reasoning, including representing three dimensional figures in two dimensions,visualizing three dimensional objectives from two dimensional representations, and analyzingcross sections of three dimensional figures. Visualization of three dimensional objectives in two

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dimensions allows for development of surface area and volume concepts.

VisualizationPaper and computer screens are flat and two dimensional. Objects in the world are threedimensional. Visualization techniques such as projections, views, cross sections, and netsrepresent these three dimensional objects in two dimensions. You can use these techniques todeepen your understanding of their properties.

This topic addresses the following competency:

Competency 218.1.2: Properties and RelationshipsThe graduate applies synthetic and analytic methods to construct proofs and solvesproblems involving the properties and relationships of two-dimensional objects.

This topic highlights the following key concepts:

how to represent three-dimensional shapes in two dimensionshow to construct three-dimensional shapes from two-dimensional representationsthe surface area and volume of three-dimensional objectsthe surface area and volume formula of three-dimensional objectsthe effect of scaling on perimeter, area, and volume

Explore and Study Two-Dimensional Representations of Three-Dimensional Shapes

From the following Annenberg Learner session, examine the shadows given in problems on thispage (problems C-5, C-6, and C-7), and predict the three-dimensional object that created thembefore viewing the provided solutions. Include a sketch of an example and its shadows in yourgeometry notebook.

Cross Sections

This exploration will be expanded upon in the performance assessment.

The link below from Math Open Ref will give you more practice on this topic.  At this site, clickon the option “Show cross-section” and observe the shape of the cross-section for theprism. Click and drag on the orange vertex of the cross-section and observe how thecross-section relates to the prism. Change the prism types in the drop-down menu and repeatthe exploration, and doing so again after checking the option “Allow oblique.” Record in yourgeometry notebook your observations on how a prism relates to its cross-sections.

PrismLine and Rotational Symmetry

From the following Annenberg Learner session, review the information on this page andcomplete the problems provided (Problems A1 and A2) before checking your solutions to theones provided:

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Line Symmetry

Note in your geometry notebook what lines of symmetry regular polygons have. Sketch severalexamples to help you remember.

From the following Annenberg Learner session, review the information at the top of the pageand then and then explore the “Interactive Activity” to investigate the symmetry of the shapes. Try to determine what angle of rotation each shape will have about its center in order to besymmetric.

Rotation Symmetry

Make a conjecture about a possible rule to determine the angle of rotational symmetry for aregular polygon and record your conjecture in your geometry notebookStudy and Review Visualization

Read the following in College Geometry 2:

section 6.3 (“Relations to 2D”)

Make note of examples of different visualization techniques in your geometry notebook.Representing Three-Dimensional Figures in Two Dimensions

A very practical problem that is particularly familiar to architects and builders is how to representthree-dimensional figures using two-dimensional media, such as paper or computer software.Read a brief introduction to various perspective drawings on the following website by reviewingall information on the page, but not accessing any of the provided links:

3D Drawing Solids of Revolution from Figures in Two Dimensions

Continue investigating shapes of revolution by using the following link to create solids ofrevolution. Examine the shapes you create and their appearance from different perspectives.

3D Transmographer

The following video will show how to use the 3D Transmographer application whileexperimenting with a few examples.

Note: To download this video, right-click the following link and choose "Save as...": downloadvideo.

Identify an object you see in your daily life that can be viewed as a solid of revolution and writeabout it in your geometry notebook.Study, Synthesize, and Practice Ideas Using Surface Area, Volume, and Formula

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Derivation

From the following Annenberg Learner session, familiarize yourself with three-dimensionalshapes, their names, and their parts, such as edges, faces, and vertices. Explore all informationon each of the pages, being sure to observe how a three dimensional object can be viewed“unfolded” as a net of its surfaces and how this relates to the surface area of the objects. Besure to assess your understanding of these concepts with the “Test Your Skills” Quiz at theend of the session.

Note: If any of the interactive pieces do not appear during your exploration, try refreshing thepage to have them load properly. 

Interactives: Geometry 3D Shapes

Sketch several prisms and pyramids with their associated nets in your geometry notebook.Share any questions you may still have about three dimensional figures with your learningcommunity.

Read the following in College Geometry 2:

section 6.1 ("Change from 2D")section 6.2 ("Common 3D shapes")section 6.4 ("Surface Areas")section 6.5 ("Volumes")

View the embedded videos. Note the differences between prisms and pyramids, as well as theirformula derivations, and note these formulas in your geometry notebook.

Watch the following video showing an animation of the method discussed in the embeddedvideo for deriving the volume of a sphere:

Interact with the very brief demonstration here:

Three Pyramids that Form a Cube

Discuss with your learning community how this demonstration relates the formulas for volume ofa prism and volume of a pyramid as given in section 6.5 of College Geometry 2.Synthesize Surface Area, Volume, and Formula Derivation

Complete the exercises at the end of chapter 6 in College Geometry 2.  Be sure to engage thesetting “Learn Mode,” while leaving all other settings as given, then you can “Start your test”:

College Geometry 2Explore Scaling

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If you have ever played with the enlargement or reduction features of a copy machine, you haveexperimented with scaling. Enlarging or reducing a figure makes it visually larger or smaller, andit has a predictable impact on its measurements, including length, area, and volume. Explorescaling using the following applet. In this applet, you will change the size of a rectangle throughscaling, and observe how the perimeters, areas, and ratios of perimeter and areas change.Drag the slider to change the scale factor on the rectangle and observe how thesemeasurements change. Start with a scale factor of 1, and then repeat your exploration byincreasing the scale factor multiple times. Note your observations in your geometry journal.

Scale FactorStudy and Review Scaling

Read the following in College Geometry 2:

section 7.4 ("Scaling Factors")

Note the scale factor theorems. Compare the theorem for scale factor and areas to what youobserved in the previous exploration, and add to your geometry notebook the theorem for scalefactor and volumes. Discuss in your Learning Community how you can use an example likethose given in section 7.4 to verify the scale factor theorems for area and volume.

Geometric Transformations

What happens to a figure if you slide it? Flip it? Turn it? Enlarge or reduce it? How is theresulting figure (i.e. image) related to the original figure (i.e. pre-image)? Your own reflectionlooking back at you in a mirror, the spinning of a wheel on sports car, the way a pattern repeatsin wallpaper or on quilt or Navajo rug, all represent geometric transformations. Geometrictransformations provide a means of exploring geometry from a different perspective and helpdevelop visualization skills.

Geometric TransformationsIn this topic you will explore slides (translations), flips (reflections), turns (rotations), and sizechanges (dilations). Furthermore, due to its concrete nature, geometry offers opportunities forexploration of shapes and their properties. Throughout the course you have used dynamicinteractives and software to explore geometric concepts and make conjectures about thepatterns you observed, and applied this reasoning to the student project tasks. In the final task,you will have the opportunity to conjecture using concrete models through paper folding,compass and straight edge constructions, and dynamic geometry.

This topic addresses the following competency:

Competency 218.1.6: Geometric TransformationsThe graduate applies geometric transformations to explore and analyze objects andsolve problems.

This topic highlights the following key concepts:

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the transformations that relate two objectsthe object and its properties that result from a set of transformationssymmetry in terms of transformationsif two-dimensional objects will tessellatethe center and magnitude of dilationexamples and counterexamples to explore and verify assertionsinductive reasoning and patterns to develop conjecturesconcrete models and dynamic technologies to draw conclusions about a conjecturethe process, reasoning, and results of geometric investigations

Explore Transformations

Review all of the information on the following links on transformations of polygons.  Be sure tomanipulate and explore the interactive figures on each link:

Translation of a polygonRotation of a polygonReflection of a polygonDilation of a polygon

Note in your geometry notebook which properties of a polygon change and which do not undereach transformation. Follow your observations with the “Try this” suggestion on each link andnote how the transformation changes when you change the parameters and locations of thefigures involved.Study Transformations

Read the following in College Geometry 2:

section 7.1 ("Preface")section 7.2 ("Reflection, Translation, and Glide reflections")section 7.3 ("Congruency Preserving Transformations")section 7.5 ("Transformations that do no preserve scale")section 7.6 ("Composite Transformations")

Reflect on how each transformation changes the figure and whether the resulting figures arecongruent or similar. Pay special attention to composite transformations: how an image can becreated from a pre-image using more than one transformation. Review your observations inyour geometry notebook and summarize the main concepts about transformations.Review and Investigate Fundamental Isometries

Read the summary of the four basic isometries.

Isometries of the PlaneExplore Tessellations

Read the simulated student/mentor discussion of tessellations.

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What Are Tessellations

Note the definition of tessellation in your geometry notebook.

Create your own tessellation using the Tessellate activity:

Tessellate

The video below will show you how to use the Tessellate activity:

Note: To download this video, right-click the following link and choose "Save as...": downloadvideo.Geometer's Sketchpad Lab: Transformations

Complete the following lab:

GSP Lab 7: Transformations

The links below from the GSP Learning Center can be used throughout the course whenworking with GSP, so you may want to bookmark them for your reference.  You will find anumber of activities, with video instruction, on how to complete various activities with GSP:

Getting Started TutorialsSketchpad Tips

You can also use the following document as a reference, which outlines the basic functions ofGSP for use as you familiarize yourself with the program:

Summary of Geometer’s Sketchpad Tools

You may want to save this to your computer or print it out so you can use it as you complete thelabs throughout this course and in the related performance tasks.Investigating Isometries with Geometer?s Sketchpad

Watch the following videos.

Note: To download this video, right-click the following link and choose "Save as...": downloadvideo. 

Note: To download this video, right-click the following link and choose "Save as...": downloadvideo. 

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Note: To download this video, right-click the following link and choose "Save as...": downloadvideo.Review, Synthesize, and Practice your Ideas

Complete the exercises at the end of chapter 7 in College Geometry 2.  Be sure to engage thesetting “Learn Mode,” while leaving all other settings as given, then you can “Start your test”:

College Geometry 2

Task 3: TransformationsYou now have the competency necessary to complete Task 3.Complete Task 3

Complete the following task in TaskStream:

Task 3

Here is a guide for Task 2:

Guide to College Geometry Task 3

For details about this performance assessment, see the "Assessment Preparation" box in thiscourse.

Final Steps

Congratulations on completing the activities in this course! This course has prepared you tocomplete the assessments associated with this course. If you have not already been directed tocomplete the assessments, schedule and complete them now.

The WGU Library

The WGU LibraryThe WGU Library is available online to WGU students 24 hours a day.

For more information about using the WGU Library, view the following videos on The WGUChannel:

Introducing the WGU library

Note: To download this video, right-click the following link and choose "Save as...": downloadvideo.

Searching the WGU library

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Note: To download this video, right-click the following link and choose "Save as...": downloadvideo.

Center for Writing Excellence: The WGU Writing Center 

If you need help with any part of the writing or revision process, contact the Center for WritingExcellence (CWE). Whatever your needs—writing anxiety, grammar, general college writingconcerns, or even ESL language-related writing issues—the CWE is available to help you. TheCWE offers personalized individual sessions and weekly group webinars. For an appointment,please e-mail writingcenter@wgu.edu.

Feedback

WGU values your input! If you have comments, concerns, or suggestions for improvement ofthis course, please submit your feedback using the following form:

Course Feedback

ADA Policy

Western Governors University recognizes and fulfills its obligations under the Americans withDisabilities Act of 1990 (ADA), the Rehabilitation Act of 1973 and similar state laws. WesternGovernors University is committed to provide reasonable accommodation(s) to qualifieddisabled learners in University programs and activities as is required by applicable law(s). ADASupport Services serves as the principal point of contact for students seeking accommodationsand can be contacted at ADASupport@wgu.edu. Further information on WGU?s ADA policyand process can be viewed in the student handbook at the following link:

Policies and Procedures for Students with Disabilities

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