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K12MATH010: Geometry

Unit 1: Congruence, Proof, and Constructions   Geometry is all about recognizing relationships and patterns within a single figure and among two or more figures. This begins with the concept of rigid transformations (translations, reflections, and rotations) to explain congruence. Congruence is one of the prominent concepts that will allow you to explain figures and their relationships. With a strong understanding of congruence, you will investigate congruency shortcuts and will be introduced to formal proofs using this foundational knowledge. You will then continue to develop your comfort with proofs, in a variety of formats, by using them to prove geometric theorems, or relationships, introduced in previous years and solving problems involving lines, angles, triangles, and quadrilaterals. Finally, formal constructions with compass and straightedge are introduced, and you will use your knowledge of geometric properties of figures to complete a variety of constructions and explain why they work.[[1]](#_ftn1)


[1]National Governors Association Center for Best Practices and Council of Chief State School Officers, “Common Core State Standards for Mathematics,” National Governors Association Center for Best Practices, Council of Chief State School Officers, Washington D.C. (2010): HSF.IF.C.7A, accessed September 21, 2012, http://www.corestandards.org/Math/Content/HSF/IF/C/7/a. “© Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.”

Unit 1 Time Advisory
Completing this unit should take approximately 12 hours.

☐    Subunit 1.1: 3 hours, 30 minutes

☐    Subunit 1.2: 3 hours

☐    Subunit 1.3: 1 hour

☐    Subunit 1.4: 1 hour

☐    Subunit 1.5: 1 hour

☐    Subunit 1.6: 2 hours, 30 minutes

Unit1 Learning Outcomes
Upon successful completion of this unit, you will be able to: - Explain congruence in terms of rigid motions. - Prove congruence using formal proofs. - Prove geometric theorems, using a variety of formats. - Apply geometric theorems to the solution of problems. - Make geometric constructions.

Standards Addressed (Common Core): - CCSS.Math.Content.HSG-CO.A.1 - CCSS.Math.Content.HSG-CO.A.2 - CCSS.Math.Content.HSG-CO.A.3 - CCSS.Math.Content.HSG-CO.A.4 - CCSS.Math.Content.HSG-CO.A.5 - CCSS.Math.Content.HSG-CO.B.6 - CCSS.Math.Content.HSG-CO.B.7 - CCSS.Math.Content.HSG-CO.B.8 - CCSS.Math.Content.HSG-CO.C.9 - CCSS.Math.Content.HSG-CO.C.10 - CCSS.Math.Content.HSG-CO.C.11 - CCSS.Math.Content.HSG-CO.D.12 - CCSS.Math.Content.HSG-CO.D.13 - CCSS.ELA-Literacy.RST.9-10.3 - CCSS.ELA-Literacy.RST.9-10.4 - CCSS.ELA-Literacy.RST.9-10.5 - CCSS.ELA-Literacy.WHST.9-10.1d - CCSS.ELA-Literacy.WHST.9-10.1e

1.1 Review: Rigid Transformations and Congruence   This section reviews the rigid transformations (translations, rotations, and reflections) studied in previous mathematics courses to further examine congruence. The structure of things in both nature and the man-made world depends on congruence: beehives, architecture, railroads, and more! The concept of congruence is critical in analyzing attributes of one figure and between figures. The resources in this section provide explanatory content as well as opportunities to investigate the transformations and how they relate to congruence and opportunities to apply this knowledge in problem-solving situations.

1.1.1 Rigid Transformations   - Explanation: CK-12: Geometry: “Translations and Vectors” Link: CK-12: Geometry: “Translations and Vectors” (HTML)
 
Instructions: Please click on the link above, and scroll down to read and take notes on the sections titled “Transformations” and “Translations.” Complete examples 1 and 2 of the Translations section as well. Answers follow each example. Upon reading this section, you should be able to explain what a translation is and draw a translation.
 
Reading these sections and taking notes should take approximately 15 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-CO.A.2](http://www.corestandards.org/Math/Content/HSG/CO/A/2)  
 ●     
[CCSS.Math.Content.HSG-CO.A.4](http://www.corestandards.org/Math/Content/HSG/CO/A/4)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)  

 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
  • Explanation: CK-12: Geometry: “Reflections” Link: CK-12: Geometry: “Reflections” (HTML)
     
    Instructions: Please click on the link above, and read and take notes on the information provided. Complete the example problems. Answers follow each example. Upon reading this section, you should be able to explain what a reflection is and draw a reflection over a variety of lines of symmetry.
     
    Reading this section and taking notes should take approximately 30 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-CO.A.2
    ●      CCSS.Math.Content.HSG-CO.A.4
    ●      CCSS.ELA-Literacy.RST.9-10.3
    ●      CCSS.ELA-Literacy.WHST.9-10.1e
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Explanation: CK-12: Geometry: “Rotations” Link: CK-12: Geometry: “Rotations” (HTML)

    Instructions: Please click on the link above, and read and take notes on the information provided. Complete the example problems. Answers follow each example. Upon completion of this activity, you should be able to explain rotations of 90, 120, 180, and 270 degrees in both a clockwise and counterclockwise direction. Additional questions to consider include the following: In which situations can a rotation or a reflection be used to create the same transformation? Which example did you find most difficult? Why?
     
    Reading this section and taking notes should take approximately 30 minutes.

    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-CO.A.2
    ●      CCSS.Math.Content.HSG-CO.A.4
    ●      CCSS.ELA-Literacy.RST.9-10.3
    ●      CCSS.ELA-Literacy.WHST.9-10.1e

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Did I Get This? Activity: Wikispaces: Justin Allen’s “U3—Lesson 1 hw—Translations” Link: Wikispaces: Justin Allen’s “U3—Lesson 1 hw—Translations” (DOCX)
     
    Instructions: Please click on the link above, download the lesson entitled “U3—Lesson 1 hw—Translations,” and complete the problems. When you finish, download the answers to the activity to check your work.
     
    Completing this activity should take approximately 30 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-CO.A.2
    ●      CCSS.Math.Content.HSG-CO.A.4
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Did I Get This? Activity: Wikispaces: Justin Allen’s “U3—Lesson 2 hw—Reflections” Link: Wikispaces: Justin Allen’s “U3—Lesson 2 hw—Reflections” (DOCX)
     
    Instructions: Please click on the link above, download the lesson entitled “U3—Lesson 2 hw—Reflections,” and complete the problems. When you finish, download the answers to the activity to check your work.
     
    Completing this activity should take approximately 30 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-CO.A.2
    ●      CCSS.Math.Content.HSG-CO.A.4
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Did I Get This? Activity: Wikispaces: Justin Allen’s “U3—Lesson 4 hw—Rotations” Link: Wikispaces: Justin Allen’s “U3—Lesson 4 hw—Rotations” (DOCX)
     
    Instructions: Please click on the link above, download the lesson entitled “U3—Lesson 4 hw—Rotations,” and complete the problems. When you finish, download the answers to the activity to check your work.
     
    Completing this activity should take approximately 30 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-CO.A.2
    ●      CCSS.Math.Content.HSG-CO.A.4
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

1.1.2 Definition of Congruent   - Explanation: CK-12: Geometry: “Congruent Figures” Link: CK-12: Geometry: “Congruent Figures” (HTML)
 
Instructions: Please click on the link above, and read and take notes. Then complete the example problems. Answers follow each example. As you read this section, consider how you would explain or define congruence to another person. What information must you know in order to demonstrate that two figures are congruent?
 
Reading this section and answering the questions should take approximately 30 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.8.G.A.2](http://www.corestandards.org/Math/Content/8/G/A/2)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

1.1.3 Composition of Transformations   - Explanation: CK-12: Geometry: “Composition of Transformations” Link: CK-12: Geometry: “Composition of Transformations” (HTML)
 
Instructions: Please click on the link above, and read and take notes on the information provided. Then complete the “Review Queue Questions.” You can check your answers at the bottom of the page. Upon reading this section, you should be able to identify compositions of transformations, as well as draw them.
 
Reading this section and answering the questions should take approximately 30 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-CO.A.2](http://www.corestandards.org/Math/Content/HSG/CO/A/2)  
 ●     
[CCSS.Math.Content.HSG-CO.A.4](http://www.corestandards.org/Math/Content/HSG/CO/A/4)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
  • Interactive Lab: Shodor: “Transmographer 2” Link: Shodor: “Transmographer 2” (Java)
     
    Instructions: Please click on the link above. You will find a page with four tabs. Click on the tab labeled “Learner.” Here you will need to download the worksheet titled “Transmographer2 Exploration Questions.” Next, click on the tab labeled “Activity” and use the instructions from the worksheet to guide you.
     
    Completing this activity should take approximately 30 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-CO.A.2
    ●      CCSS.Math.Content.HSG-CO.A.4
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

1.2 Reasoning and Proof: Congruence   This section introduces the idea of the proof, using reasoning and logic to explain geometric relationships. After introducing some of the important concepts of reasoning and logic, the proof is introduced by building on the student’s solid understanding of congruence. The resources in this section provide explanatory content, practice problems, some investigatory activities, and a real-world application of the concepts covered.

1.2.1 Inductive Reasoning   - Explanation: CK-12: Geometry: “Inductive Reasoning” Link: CK-12: Geometry: “Inductive Reasoning” (HTML)
 
Instructions: Please click on the link above, and take notes and complete the example exercises. Answers to examples are provided after each example. As you read, consider what additional examples of inductive reasoning you have seen or can identify in your own life.
 
Reading this section and taking notes should take approximately 15 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-CO.C.9](http://www.corestandards.org/Math/Content/HSG/CO/C/9)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1d](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/d)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

1.2.2 Conditional Statements   - Explanation: CK-12: Geometry: “Conditional Statements” Link: CK-12: Geometry: “Conditional Statements” (HTML)
 
Instructions: Please click on the link above, and take notes and complete the example exercises. Answers are provided after each example. How would you explain conditional statements to another person? Create three of your own examples of conditional statements.
 
Reading this section and taking notes should take approximately 30 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-CO.C.9](http://www.corestandards.org/Math/Content/HSG/CO/C/9)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1d](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/d)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
  • Did I Get This? Activity: MoodleShare: “Conditional Statements: Guided Practice” Link: MoodleShare: “Conditional Statements: Guided Practice” (HTML)
     
    Instructions: Please click on the link above. This page provides a series of practice problems that you can answer and check online. Each question has solutions worked out step-by-step if you need hints along the way.
     
    Completing this activity should take approximately 15 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSF-IF.A.1
    ●      CCSS.ELA-Literacy.WHST.9-10.1d
    ●      CCSS.ELA-Literacy.WHST.9-10.1e
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

1.2.3 Deductive Reasoning   - Explanation: CK-12: Geometry: “Deductive Reasoning” Link: CK-12: Geometry: “Deductive Reasoning” (HTML)
 
Instructions: Please click on the link above, and take notes and complete the example exercises. Answers are provided after each example. What is the difference between deductive and inductive reasoning? Provide an example of deductive reasoning.
 
Watching this lecture and taking notes should take approximately 30 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-CO.C.9](http://www.corestandards.org/Math/Content/HSG/CO/C/9)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1d](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/d)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

1.2.4 Algebraic and Congruence Properties   - Explanation: CK-12: Geometry: “Algebraic and Congruence Properties” Link: CK-12: Geometry: “Algebraic and Congruence Properties”  (HTML)
 
Instructions: Please click on the link above, and take notes and complete the example exercises. Answers are provided after each example. Did you already know these properties from previous mathematics courses? Which ones were new to you? Which properties are the most difficult to understand or remember? Why? Be sure to review these properties a second time.
 
Watching this lecture and taking notes should take approximately 30 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-CO.C.9](http://www.corestandards.org/Math/Content/HSG/CO/C/9)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

1.2.5 Proving Triangle Congruence Using SSS and SAS   - Explanation: CK-12: Geometry: “Triangle Congruence Using SSS and SAS” Link: CK-12: Geometry: “Triangle Congruence Using SSS and SAS” (HTML)
 
Instructions: Please click on the link above, and take notes and complete the example exercises. Answers are provided after each example. What are the congruency shortcuts explained in this section? Why are they called shortcuts?
 
Reading this section and taking notes should take approximately 30 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-CO.B.7](http://www.corestandards.org/Math/Content/HSG/CO/B/7)  
 ●     
[CCSS.Math.Content.HSG-CO.B.8](http://www.corestandards.org/Math/Content/HSG/CO/B/8)  
 ●     
[CCSS.Math.Content.HSG-CO.C.9](http://www.corestandards.org/Math/Content/HSG/CO/C/9)  
 ●     
[CCSS.Math.Content.HSG-CO.C.10](http://www.corestandards.org/Math/Content/HSG/CO/C/10)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.5](http://www.corestandards.org/ELA-Literacy/RST/9-10/5)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1d](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/d)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
  • Explanation: YouTube: Khan Academy’s “Finding Congruent Triangles” Link: YouTube: Khan Academy’s “Finding Congruent Triangles” (YouTube)
     
    Instructions: Please listen to the lesson. As you listen, take notes on the various congruence postulates that the speaker describes. What are the congruence postulates? Why are they useful? What combinations of three pairs of congruent corresponding parts do not make up a congruence postulate?
     
    Watchin this lecture and taking notes should take approximately 15 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-CO.B.7
    ●      CCSS.Math.Content.HSG-CO.B.8
    ●      CCSS.Math.Content.HSG-CO.C.9
    ●      CCSS.Math.Content.HSG-CO.C.10
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

1.2.6 Triangle Congruence Using ASA, AAS, and HL   - Explanation: CK-12: Geometry: “Triangle Congruence Using ASA, AAS, and HL” Link: CK-12: Geometry: “Triangle Congruence Using ASA, AAS, and HL” (HTML)
 
Instructions: Please click on the link above, and take notes and complete the example exercises. Answers are provided after each example. What are the congruency shortcuts explained in this section? What combination of corresponding angles or sides do not prove congruency? Why?
 
Completing this activity should take approximately 30 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-CO.B.7](http://www.corestandards.org/Math/Content/HSG/CO/B/7)  
 ●     
[CCSS.Math.Content.HSG-CO.B.8](http://www.corestandards.org/Math/Content/HSG/CO/B/8)  
 ●     
[CCSS.Math.Content.HSG-CO.C.9](http://www.corestandards.org/Math/Content/HSG/CO/C/9)  
 ●     
[CCSS.Math.Content.HSG-CO.C.10](http://www.corestandards.org/Math/Content/HSG/CO/C/10)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.5](http://www.corestandards.org/ELA-Literacy/RST/9-10/5)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1d](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/d)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

1.3 Reasoning and Proof: Lines and Angles   This is the first of several sections that uses proofs to revisit geometric properties of geometric figures introduced in previous mathematics courses. Specifically, this section examines properties of lines and angles. The resources in this section provide explanatory content and practice problems with immediate feedback.

1.3.1 Vertical Angles   - Explanation: CK-12: Geometry: “Angle Pairs” Link: CK-12: Geometry: “Angle Pairs” (HTML)
 
Instructions: Please click on the link above, and scroll down to read and take notes on the section titled “Vertical Angles.” The other sections may prove useful, if you need a review of angle pairs. Upon completion of the reading, be sure you can explain the relationship between vertical angles and how to identify a pair of vertical angles.
 
Reading this section and taking notes should take approximately 30 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-CO.C.9](http://www.corestandards.org/Math/Content/HSG/CO/C/9)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)  
    
 Terms of Use: Please respect the copyright and terms of use  
 displayed on the webpage above.
  • Did I Get This? Activity: Khan Academy’s “Vertical Angles” Link: Khan Academy’s “Vertical Angles” (HTML)
     
    Instructions: Please click on the link above. This page provides a series of practice problems that you can answer and check online. Each question has explanations for the example, worked out step-by-step if you need hints along the way. The program monitors your success, and when you have correctly completed six questions consecutively, it will tell you that you are ready to move on to a new skill.
     
    Completing this activity should take approximately 15 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-CO.B.7
    ●      CCSS.Math.Content.HSG-CO.B.8
    ●      CCSS.Math.Content.HSG-CO.C.9
    ●      CCSS.Math.Content.HSG-CO.C.10
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

1.3.2 Parallel Lines and a Transversal   - Explanation: CK-12: Geometry: “Properties of Parallel Lines” Link: CK-12: Geometry: “Properties of Parallel Lines” (HTML)
 
Instructions: Please click on the link above, and take notes and complete the example exercises. Answers are provided after each example. Upon completion of this reading, be sure you can define parallel. Also, what angle pairs do parallel lines and a transversal create? How can these angle pairs be identified and what is their relationship?
 
Reading this section and taking notes should take approximately 30 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-CO.C.9](http://www.corestandards.org/Math/Content/HSG/CO/C/9)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
  • Explanation: YouTube: Khan Academy’s “Angles of Parallel Lines 2” Link: YouTube: Khan Academy’s “Angles of Parallel Lines 2” (YouTube)
     
    Instructions: Please click on the link above, and take notes as you watch the 5-minute video. Note any steps or terms that are unclear.
     
    Watching this lecture and taking notes should take approximately 15 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-CO.C.9
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

1.3.3 Perpendicular Bisector   - Explanation: CK-12: Geometry: “Perpendicular Bisectors in Triangles” Link: CK-12: Geometry“Perpendicular Bisectors in Triangles” (HTML)
 
Instructions: Please click on the link above, and take notes and complete the example exercises. Answers are provided after each example. Be sure you can define perpendicular bisector and its properties or implications for defining special triangles.
 
Reading this section and taking notes should take approximately 15 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-CO.C.9](http://www.corestandards.org/Math/Content/HSG/CO/C/9)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

1.4 Reasoning and Proof: Triangle Properties   This is the second of several sections that uses proofs to revisit geometric properties of geometric figures introduced in previous mathematics courses. Specifically, this section examines properties of triangles, beyond congruency, such as relationships of the angles and sides of special triangles and properties of special lines that exist in relation to triangles. The resources in this section provide explanatory content with embedded examples and solutions.

1.4.1 Triangle Angle Sum   - Explanation: CK-12: Geometry: “Triangle Sums” Link: CK-12: Geometry: “Triangle Sums” (HTML)
 
Instructions: Please click on the link above, and take notes and complete the example exercises. Answers are provided after each example. Upon reading this section, you should be able to summarize the triangle sum theorem and apply it when solving for unknown angles in a triangle.
 
Reading this section and taking notes should take approximately 30 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-CO.C.9](http://www.corestandards.org/Math/Content/HSG/CO/C/9)  
 ●     
[CCSS.Math.Content.HSG-CO.C.10](http://www.corestandards.org/Math/Content/HSG/CO/C/10)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

1.4.2 Isosceles Triangles   - Explanation: CK-12: Geometry: “Isosceles and Equilateral Triangles” Link: CK-12: Geometry: “Isosceles and Equilateral Triangles” (HTML)
 
Instructions: Please click on the link above, and take notes and complete the example exercises. Answers are provided after each example. Upon completion of this reading, you should be able to define isosceles and equilateral triangles and identify some of their key properties or characteristics.
 
Reading this section and taking notes should take approximately 20 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-CO.C.9](http://www.corestandards.org/Math/Content/HSG/CO/C/9)  
 ●     
[CCSS.Math.Content.HSG-CO.C.10](http://www.corestandards.org/Math/Content/HSG/CO/C/10)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
  • Did I Get This? Activity: Khan Academy’s “Angles 1” Link: Khan Academy’s “Angles 1” (HTML)
     
    Instructions: Please click on the link above. This page provides a series of practice problems that you can answer and check online. Each question has explanations for the example, worked out step-by-step if you need hints along the way. The program monitors your success, and when you have correctly completed six questions consecutively, it will tell you that you are ready to move on to a new skill.
     
    Completing this activity should take approximately 15 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-CO.C.9
    ●      CCSS.Math.Content.HSG-CO.C.10
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

1.5 Reasoning and Proof: Parallelograms   This is the last of several sections that uses proofs to revisit geometric properties of geometric figures introduced in previous mathematics courses. Specifically, this section examines properties of parallelograms. The resources in this section provide explanatory content with embedded examples and solutions.

1.5.1 Opposite Sides   - Explanation: CK-12: Geometry: “Properties of Parallelograms” Link: CK-12: Geometry: “Properties of Parallelograms” (HTML)
 
Instructions: Please click on the link above, and read the sections entitled “What is a Parallelogram?” and “Proof of Opposite Sides Theorem.” Take notes and consider this question: What is true about the opposite sides of parallelograms?
 
Reading this section and taking notes should take approximately 15 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-CO.C.11](http://www.corestandards.org/Math/Content/HSG/CO/C/11)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

1.5.2 Opposite Angles   - Explanation: CK-12: Geometry: “Properties of Parallelograms” Link: CK-12: Geometry: “Properties of Parallelograms” (HTML)
 
Instructions: Please click on the link above. In the previous section, you read about the definition of a parallelogram and the proof of the opposite sides theorem. The reading explains that the opposite angles theorem is almost identical. What does the opposite angles theorem say? Review the opposite sides theorem and then create a two-column proof for the opposite angles theorem.
 
Reviewing this section and writing the proof should take approximately 15 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-CO.C.11](http://www.corestandards.org/Math/Content/HSG/CO/C/11)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

1.5.3 Diagonals   - Explanation: CK-12: Geometry: “Properties of Parallelograms” Link: CK-12: Geometry“Properties of Parallelograms” (HTML)
 
Instructions: Please click on the link above, and read the section entitled “Diagonals in a Parallelogram.” Take notes and complete the two example exercises. Answers are provided after each example.
 
Reading this section and taking notes should take approximately 15 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-CO.C.11](http://www.corestandards.org/Math/Content/HSG/CO/C/11)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
  • Explanation: CK-12: Geometry: “Proving Quadrilaterals are Parallelograms” Link: CK-12: Geometry: “Proving Quadrilaterals are Parallelograms” (HTML)
     
    Instructions: Please click on the link above, and take notes and complete the example exercises. Answers are provided after each example. While reading, consider these questions: What properties help to differentiate one quadrilateral from another? What are the different quadrilaterals? How are they similar? How are they different?
     
    Reading this section and taking notes should take approximately 30 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-CO.C.11
    ●      CCSS.ELA-Literacy.RST.9-10.3
    ●      CCSS.ELA-Literacy.WHST.9-10.1e
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above

  • Did I Get This? Activity: MoodleShare: “Guided Practice” Link: MoodleShare: “Guided Practice” (HTML)
     
    Instructions: Please click on the link above, and then click on the link on the page that appears. This page provides a series of practice problems that you can answer and check online.
     
    Completing this activity should take approximately 15 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-CO.C.11
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

1.6 Geometric Constructions   This section introduces the skill of constructions, using a straightedge and compass to create geometric figures. The resources in this section provide explanatory content, video demonstrations of the constructions, and additional practice problems in order to build proficiency in this skill.

1.6.1 Definition of Construction   - Explanation: Math Open Reference: “Introduction to Constructions” Link: Math Open Reference: “Introduction to Constructions” (HTML)

 Instructions: Please click on the link above, and read the overview
to gain a context of why constructions are used and how they came to
exist.  
    
 Reading this material should take approximately 15 minutes.  
    
 Standards Addressed (*Common Core*):  

 ●     
[CCSS.Math.Content.HSG-CO.D.12](http://www.corestandards.org/Math/Content/HSG/CO/D/12)  
 ●     
[CCSS.Math.Content.HSG-CO.D.13](http://www.corestandards.org/Math/Content/HSG/CO/D/13)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

1.6.2 Congruent Segments   - Web Media: Math Open Reference: “Copying a Line Segment” Link: Math Open Reference: “Copying a Line Segment” (Flash)
 
Instructions: Please click on the link above, and watch the short demonstration of how to construct a congruent segment. Then scroll down to where it says “Try It Yourself.” Here you will find a printable worksheet that provides two opportunities to try out this skill.
 
Completing this activity should take approximately 15 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-CO.D.12](http://www.corestandards.org/Math/Content/HSG/CO/D/12)  
 ●     
[CCSS.Math.Content.HSG-CO.D.13](http://www.corestandards.org/Math/Content/HSG/CO/D/13)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

1.6.3 Congruent Angles   - Web Media: Math Open Reference: “Copying an Angle” Link: Math Open Reference: “Copying an Angle” (Flash)
 
Instructions: Please click on the link above, and watch the short demonstration of how to construct a congruent angle. Then scroll down to where it says “Try It Yourself.” Here you will find a printable worksheet that provides two opportunities to try out this skill.
 
Completing this activity should take approximately 15 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-CO.D.12](http://www.corestandards.org/Math/Content/HSG/CO/D/12)  
 ●     
[CCSS.Math.Content.HSG-CO.D.13](http://www.corestandards.org/Math/Content/HSG/CO/D/13)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

1.6.4 Segment Bisector   - Web Media: Math Open Reference: “Perpendicular Bisector of a Line Segment” Link: Math Open Reference: “Perpendicular Bisector of a Line Segment” (Flash)
 
Instructions: Please click on the link above, and watch the short demonstration of how to construct a perpendicular bisector of a segment. Then scroll down to where it says “Try It Yourself.” Here you will find a printable worksheet that provides three opportunities to try out this skill.
 
Completing this activity should take approximately 15 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-CO.D.12](http://www.corestandards.org/Math/Content/HSG/CO/D/12)  
 ●     
[CCSS.Math.Content.HSG-CO.D.13](http://www.corestandards.org/Math/Content/HSG/CO/D/13)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

1.6.5 Angle Bisector   - Web Media: All Math Words Encyclopedia: “How to Bisect an Angle” Link: All Math Words Encyclopedia: “How to Bisect an Angle” (HTML)

 Instructions: Please click on the link above, and read the
step-by-step instructions for constructing an angle bisector. Watch
the short demonstration as well. Next, draw an angle and follow the
steps for constructing its bisector. Repeat this activity three
times to ensure you are comfortable with this skill.  
    
 Completing this activity should take approximately 15 minutes.  
    
 Standards Addressed (*Common Core*):  

 ●     
[CCSS.Math.Content.HSG-CO.D.12](http://www.corestandards.org/Math/Content/HSG/CO/D/12)  
 ●     
[CCSS.Math.Content.HSG-CO.D.13](http://www.corestandards.org/Math/Content/HSG/CO/D/13)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

1.6.6 Perpendicular Lines   - Web Media: Math Open Reference: “Perpendicular to a Line from an External Point” Link: Math Open Reference: “Perpendicular to a Line from an External Point” (Flash)
 
Instructions: Please click on the link above, and watch the short demonstration of how to construct a line perpendicular to another from an external point. Then scroll down to where it says “Try It Yourself.” Here you will find a printable worksheet that provides two opportunities to try out this skill.
 
Completing this activity should take approximately 15 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-CO.D.12](http://www.corestandards.org/Math/Content/HSG/CO/D/12)  
 ●     
[CCSS.Math.Content.HSG-CO.D.13](http://www.corestandards.org/Math/Content/HSG/CO/D/13)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

1.6.7 Parallel Lines   - Web Media: Math Open Reference: “Constructing a Parallel through a Point” Link: Math Open Reference: “Constructing a Parallel through a Point” (Flash)
 
Instructions: Please click on the link above, and watch the short demonstration of how to construct a line parallel to another through a given point. Then scroll down to where it says “Try It Yourself.” Here you will find a printable worksheet that provides two opportunities to try out this skill.
 
Completing this activity should take approximately 15 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-CO.D.12](http://www.corestandards.org/Math/Content/HSG/CO/D/12)  
 ●     
[CCSS.Math.Content.HSG-CO.D.13](http://www.corestandards.org/Math/Content/HSG/CO/D/13)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

1.6.8 Equilateral Triangles, Squares, and Regular Polygons   - Explanation: Utah Electronic High School: “Constructing an Equilateral Triangle” Link: Utah Electronic High School: “Constructing an Equilateral Triangle” (HTML)
 
Instructions: Please click on the link above, and read the steps for constructing an equilateral triangle. Use your own straightedge and ruler to repeat the steps. Construct three equilateral triangles of different sizes to develop a comfort with this skill.
 
Completing this activity should take approximately 15 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-CO.D.12](http://www.corestandards.org/Math/Content/HSG/CO/D/12)  
 ●     
[CCSS.Math.Content.HSG-CO.D.13](http://www.corestandards.org/Math/Content/HSG/CO/D/13)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
  • Explanation: Utah Electronic High School: “Constructing Squares and Rectangles” Link: Utah Electronic High School: “Constructing Squares and Rectangles” (HTML)
     
    Instructions: Please click on the link above, and read the steps for constructing squares and rectangles. Use your own straightedge and ruler to repeat the steps. Construct three squares of different sizes to develop a comfort with this skill.
     
    Completing this activity should take approximately 15 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-CO.D.12
    ●      CCSS.Math.Content.HSG-CO.D.13
    ●      CCSS.ELA-Literacy.RST.9-10.3
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Web Media: Math Open Reference: “Regular Pentagon Inscribed in a Circle” Link: Math Open Reference: “Regular Pentagon Inscribed in a Circle” (Flash)
     
    Instructions: Please click on the link above, and watch the short demonstration of how to construct a regular pentagon. Then scroll down to where it says “Try It Yourself.” Here you will find a printable worksheet that provides two opportunities to try out this skill. To challenge yourself, take some time to try and develop a construction for another regular polygon such as a hexagon. How is the process similar? How is it different?
     
    Completing this activity should take approximately 30 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-CO.D.12
    ●      CCSS.Math.Content.HSG-CO.D.13
    ●      CCSS.ELA-Literacy.RST.9-10.3
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Activity: Mathematics Vision Project: “Secondary One Mathematics: An Integrated Approach, Module 5 Congruence, Construction and Proof” Link: Mathematics Vision Project: “Secondary One Mathematics: An Integrated Approach, Module 5 Congruence, Construction and Proof” (PDF)
     
    Instructions: Please click on the link above, and scroll down to page 70. Here you will find a series of activities that provide you the opportunity to practice your construction skills. The problems are on pages 70 through 79. No answer key is provided; however, with an understanding of the different geometric figures, along with the use of your notes, you should be able to determine if you are constructing the figure well or not.
     
    Completing this activity should take approximately 1 hour.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-CO.D.12
    ●      CCSS.Math.Content.HSG-CO.D.13
    ●      CCSS.ELA-Literacy.RST.9-10.3
    ●      CCSS.ELA-Literacy.RST.9-10.4
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.