Calendar for 2018 - 2019 Spring Semester MATH 3110/5110 Section 100 (Class Numbers 6354/6383), taught by Mark Barsamian

Week 1: Mon Jan 14 - Fri Jan 18

• Day #1: Axiom Systems: Intro to Axiom Systems Show/Hide details
  • Topics: Intro to Axiom Systems, Primitive Relations & Terms, Interpretations, Models
  • Book Section: 1.1 Introduction to Axiom Systems
• Day #2: Axiom Systems: Consistency, Independence Show/Hide details
  • Topics: Consistency, Independence
  • Book Section: 1.2 Properties of Axiom Systems I: Consistency, Independence
  • Class Presentations:
    • Carlsen CP01: What is the Contradiction Rule ? Explain. (The rule is presented in the book, and you may have encountered it in MATH 3030, Discrete Math.)
    • Fox CP01: Suppose that one suspects that an axiom system is inconsistent and wants to prove that it is inconsistent. How does one prove that an axiom system is inconsistent?
    • Goodfellow CP01: Consider Axiom System #4 that was introduced in Section 1.2.1. Is it consistent? Explain.
    • Huff CP01: Consider Axiom System #5 that was introduced in Section 1.2.2. Is it Independent? Explain.
• Day #3: Axiom Systems: Completeness (Quiz 1) Show/Hide details
  • Topics: Completeness
  • Book Section: 1.3 Properties of Axiom Systems II: Completeness
  • Class Presentations:
    • Miller CP01: What is a "complete" axiom system? Explain.
    • Strobel CP01: Consider Axiom System #6 that was introduced in Section 1.3.1. Is it Complete ? Explain.
    • Tobon CP01: Make up an axiom system that you think is not complete. Present it to the class and explain why you think it is not complete. (Do your work on paper, so that I can project it on the movie screen.)
    • Wait CP01: Is it okay to use a Model to prove a statement about an axiom system? Explain.
  • Quiz 1

Week 2: Mon Jan 21 - Fri Jan 25 (Monday is MLK Day Holiday)

• Monday is Martin Luther King Day Holiday: No Class
• Day #4: Axiomatic Geometries: Introduction and Basic Examples Show/Hide details
  • Topics: Introduction to Axiomatic Geometries, and Basic Examples
  • Book Section: 2.1 Introduction and Basic Examples
  • Class Presentations:
    • White CP01: What is an analytic geometry ? Explain.
    • Zink CP01: What is an axiomatic geometry ? Explain.
    • Steadman CP01: What does the book say in Section 2.1.2 about measuring distance in axiomatic geometry ?
    • Aryal CP01: Axiomatic Geometry is largely about very precise use of language. Some of the sentences below contain primitive relations, some contain defined relations, and some are not valid uses of terminology. Explain which is which.
      • Point P lies on line M .
      • Line M passes through point P .
      • Line M lies on point P .
      • Line M is parallel to line N .
      • Lines L,M,N are concurrent.
    • Wambua CP01: In axiomatic geometry, one is often interested in two particular questions about Parallel lines. Present those questions to the class.
• Day #5: Axiomatic Geometries: Fano's Geometry Show/Hide details
  • Topics: Fano's Geometry
  • Book Section: 2.2 Fano's Geometry and Young's Geometry
  • Class Drill: Drill for Section 2.2: Justifying and Illustrating a Proof of Fano's Theorem #3
  • Class Presentations:
    • Caitlin Carlsen CP02: At the end of Section 2.1, we learned about the Recurring Questions about Parallels :
      • (1) Do Parallel Lines Exist?
      • (2) Given a line L and a point P not on L , how many lines exist that pass through P and are parallel to L ?
      For Fano's Geometry, what are the answers to these questions?
    • Aja Fox CP02: Present a proof of Fano's Theorem #1: There exists at least one point. Explain why there is no choice about how the proof has to start. (Hint: Read the book's Remark about Proof Structure in Section 2.2.2 about the Proof of Fano's Theorem #3.)
    • Kayla Goodfellow CP02: Consider Fano's Theorem #2: For any two lines, there is exactly one point that lies on both lines. How would a proof of this theorem have to start? That is, what would have to be the first statement of the proof? (Hint: Read the book's Remark about Proof Structure in Section 2.2.3 about the Proof of Fano's Theorem #4.)
    • Samantha Huff CP02: Is Fano's Geometry Consistent? Explain
    • Rhett Miller CP02: (Refer to Caitlin Carlsen's question above.) For Young's Geometry, what are the answers to the Recurring Questions about Parallels ?
    • Nathan Steadman CP02: Is Young's Geometry Consistent? Explain.

Week 3: Mon Jan 28 - Fri Feb 1

• Day #6: Axiomatic Geometries: Incidence Geometry (Quiz 2) Show/Hide details
  • Topics: Incidence Geometry
  • Book Section: 2.3 Incidence Geometry
  • Class Presentations: Finished up Class Presentations from Friday.
  • Class Drill: Drill for Section 2.3: Two Theorems of Incidence Geometry
  • Quiz 2
• Wed Jan 30 Classes Cancelled
• Day #7: Neutral Geometry I: Neutral Geom Axioms and First 6 Theorems Show/Hide details
  • Topics: Neutral Geometry Axioms and the First 6 Theorems
  • Book Section: 3.1 Neutral Geometry Axioms and First 6 Theorems
  • Class Presentations:
    • Allyson Strobel CP02: Is the Axiom System for Neutral Geometry complete ?
    • Micah Tobon CP02: What do the Neutral Geometry axioms say about parallel lines?
    • Doug Wait CP02: The introduction to Chapter 3 says that we are seeking an axiom system that will be an abstract version of the straight line drawings that we are used to making. That is, we are seeking an abstract axiom system for which straight line drawings are a model. Another model would be ordinary analytic geometry of the ( x , y ) plane. In those models, there is an infinite set of points that lie on each line. Does axiom <N2> say that there are exactly two points on each line? Explain. What do axioms <N1> - <N5> say about how many points are on lines?
    • Baylie White CP02: The exercises for Section 3.1 (found on page 94 of the book) included exercises [6] - [10] that ask the reader to provide proofs for Neutral Geometry Theorems 1 - 5. But there is a clue in the reading about where to find proofs of those theorems in the book. Explain.

Week 4: Mon Feb 4 - Fri Feb 8

• Day #8: Neutral Geometry I: The Distance Function and Coordinate Functions Show/Hide details
  • Topics: The Distance Function and Coordinate Functions
  • Book Sections: 3.2, 3.3, 3.4, 3.5
  • Class Presentations:
    • Reilly Zink CP02: Topic: Coordinate Function and Distance in Drawings (Section 3.2) Using a meter stick, illustrate the process for measuring the distance between two drawn dots on the chalkboard. Present the process as a two-step process, involving a "coordinate function", as I do in Section 3.2 of the book.
    • Harman Aryal CP02: Topic: Coordinate Function and Distance in Analytic Geometry (Section 3.3) Give an example of a coordinate function for a line in Analytic Geometry. Don't use exactly one of my book examples, but feel free to make up something very similar.
    • Mitchelle Wambua CP02: Topic: Coordinate Function and Distance in Analytic Geometry (Section 3.3) Give an example of a function that fails to be a coordinate function for a line in Analytic Geometry. Don't use exactly one of my book examples, but feel free to make up something very similar.
    • Caitlin Carlsen CP03: Topic: Coordinate Functions and Distance in Axiomatic Geometry (Section 3.4) How many points are on lines in Neutral Geometry? Explain.
• Day #9: Neutral Geometry I: Basic Properties of Distance Function; Ruler Placement (Quiz 3) Show/Hide details
  • Topics: Basic Properties of Distance Function; Ruler Placement
  • Book Sections: 3.6, 3.7, 3.8
  • Class Presentations:
    • Aja Fox CP03: Topic: Basic Properties of the Distance Function in Neutral Geometry (Section 3.6) What does it mean to say that the Distance Function on the Set of Points is positive definite ?
    • Kayla Goodfellow CP03: Topic: Basic Properties of the Distance Function in Neutral Geometry (Section 3.6) What does it mean to say that the Distance Function on the Set of Points is symmetric ?
    • Samantha Huff CP03: Topic: Basic Properties of the Distance Function in Neutral Geometry (Section 3.6) Provide justifications for the proof of Theorem 9.
    • Rhett Miller CP03: Topic: Ruler Placement in Drawings (Section 3.7) Given points A and B on a drawn line L , we often want the ruler placed so that the resulting coordinate function has two properties:
      • the coordinate of point A is zero
      • the coordinate of point B is positive
      With an actual ruler and a drawing, we can simply put the ruler on the drawing in the right way. But the book presents a process for getting a ruler placed into a good position. Describe that process. (I'll bring a meterstick to class so you can illustrate at the board.)
    • Nathan Steadman CP03: Topic: Ruler Placement in Analytic Geometry (Section 3.8) In drawings , an actual physical ruler can be used to assign numbers (coordinates) to the points on a drawn line L . The book describes the processes of sliding and flipping a ruler in order to get it into a preferred positon. In Analytic Geometry , points are represented by ordered pairs of numbers, ( x , y ), and a mathematical function f can be used to assign assign numbers (coordinates) to the points on a line. Such a function f is called a coordinate function . What are the operations that one can do to a coordinate function f in analytic geometry that are analogous to sliding and flipping a ruler in a drawing? Explain.
  • Quiz 3
• Day #10: Neutral Geometry I: Ruler Placement Show/Hide details
  • Topics: Ruler Placement; Rulers in High School Geometry Books
  • Book Sections: 3.9, 3.10
  • Class Presentations:
    • Allyson Strobel CP03: (Topic: Ruler Placement in Neutral Geometry (Section 3.9)) To prove that some function \(g\) is a coordinate function for some line L , one needs to prove four things. What are the four things?
    • Micah Tobon CP03: (Topic: Ruler Placement in Neutral Geometry (Section 3.9)) What does it mean to say that a function \(g\) is one-to-one ?
    • Doug Wait CP03: (Topic: Ruler Placement in Neutral Geometry (Section 3.9)) What does it mean to say that a function \(g\) is onto ?
    • Baylie White CP03: (Topic: Ruler Placement in Neutral Geometry (Section 3.9)) Given an abstract line L and some function \(g:L\rightarrow\mathbb{R}\), we are often interested in whether \(g\) agrees with the distance function d . What does that mean?
    • Reilly Zink CP03: (Topic: Distance and Rulers in High School Geometry Books (Section 3.10)) In ordinary analytic geometry of the ( x , y ) plane, there is a formula for computing the distance d ( P , Q ) between points P and Q . Present the formula. Are points P and Q required to be distinct points in this formula? What happens if P and Q are the same point? Explain.
    • Harman Aryal CP03: (Topic: Distance and Rulers in High School Geometry Books (Section 3.10)) Ruler Placement behavior is guaranteed in both the SMSG geometry and in our Neutral Geometry, but in different ways. Explain.
    • Mitchelle Wambua CP03: (Topic: Distance and Rulers in High School Geometry Books (Section 3.10)) Are the SMSG Postulates independent ? Explain.

Week 5: Mon Feb 11 - Fri Feb 15

• Day #11: In-Class Exam 1 on Chapters 1,2,3
• Day #12: Neutral Geometry II: Betweenness, Segments, Rays, Angles, Triangles Show/Hide details
  • Topics: Betweenness, Segments, Rays, Angles, Triangles
  • Book Sections: 4.1, 4.2
  • Class Presentations:
    • Caitlin Carlsen CP04: How is betweenness of points defined? How is it related to betweenness of real numbers ? What is the link that brings betweenness of real numbers into a discussion of betweenness of points? (Vague enough?)
    • Aja Fox CP04: Why does Theorem 15 look so much like Theorem 12? Do we need them both? Explain.
    • Kayla Goodfellow CP04: Illustrate Theorems 13 & 16 with a yardstick.
    • Samantha Huff CP04: Illustrate Theorem 17 with a yardstick.
    • Rhett Miller CP04: Justify & illustrate Theorem 18
• Day #13: Neutral Geometry II: Segment Congruence; Segment Midpoints Show/Hide details
  • Topics: Segment Congruence; Segment Midpoints
  • Book Sections: 4.3, 4.4
  • Class Drills:
    • Drill for Section 4.4 The Congruent Segment Construction Theorem
  • Class Presentations:
    • Nathan Steadman CP04: Is segment congruence defined in Neutral geometry, or is it an undefined thing? If it is defined, how is it defined? Explain.
    • Allyson Strobel CP04: Suppose you have a drawn line segment, with endpoints A and B , and a ruler is sitting alongside the segment, but not necessarily placed in good position. (t\That is, it is not given that the ruler zero is at A .) How could you use this ruler to find the midpoint of the line segment? (Without moving the ruler.) Explain. Suppose the ruler is now placed differently on the same line (so it is effectively a new ruler) and your process is repeated. Would the location of the resulting midpoint change? Explain.
    • Micah Tobon CP04: Illustrate the statement of Theorem 24 on the chalkboard. (You don't have to prove it, just make a picture that illustrates what the theorem statement says.)

Week 6: Mon Feb 18 - Fri Feb 22

• Day #14: Neutral Geometry III: Separation Axiom, Theorems about Lines intersecting Triangles Show/Hide details
  • Topics: Separation Axiom, Theorems about Lines intersecting Triangles
  • Book Sections: 5.1, 5.2
  • Class Drills:
    • Drill for Section 5.1: Justifying and illustrating Steps in proof of Theorem 27
    • Drills for Section 5.2: Justifying and illustrating Steps in proof of Theorems 28 & 28
  • Class Presentations:
    • Doug Wait CP04: What does it mean to say that three sets B,C,D form a partition of some set A ?
    • Baylie White CP04: Does a half-plane include the edge ? (This is kind of subtle. See if you can figure it out and explain it.)
    • Reilly Zink CP04: What does it mean to say that some set of points is convex ?
    • Harman Aryal CP04: Axiom <N6> (ii) says that for every line L , there are two associated half-planes , and that those half-planes are convex . What is the restatement of Axiom <N6> (ii) as a conditional statement ?
    • Mitchelle Wambua CP04: Harman provided the restatement of Axiom <N6> (ii) as a conditional statement . What is the contrapositive of that conditional statement?
• Day #15: Neutral Geometry III: Angle and Triangle Interiors; Rays and lines intersecting them (Quiz 4) Show/Hide details
  • Topics: Angle and Triangle Interiors; Rays and lines intersecting them
  • Book Sections: 5.3, 5.4
  • Class Presentations: none
  • Quiz 4
• Day #16: Neutral Geometry III: Triangle can't enclose ray; Conv. Quads.; Plane Separation in H.S. books Show/Hide details
  • Topics: Triangle can't enclose ray; Convex Quadrilaterals; Plane Separation in H.S. books
  • Book Sections: 5.5, 5.6, 5.7
  • Class Presentations:

    Remember that to illustrate a Conditional Statement, it is helpful to make two drawings: one drawing that illustrates the situation described in the hypothesis, and another drawing that illustrates the situation described in the conclusion.

    • Caitlin Carlsen CP05: Make an illustration of the statement of Theorem 29 about a line intersecting two sides of a triangle between vertices.
    • Aja Fox CP05: Make an illustration of the statement of Theorem 30 about ray with endpoint on a line.
    • Kayla Goodfellow CP05: Make an illustration of the statement of Theorem 31 about a ray with endpoint on angle vertex
    • Rhett Miller CP05: Make an illustration of the statement of Theorem 33 points on a side of triangle are in interior of opposite angle.

Week 7: Mon Feb 25 - Fri Mar 1

• Day #17: Neutral Geom IV: Angle Measure, Construction, Addition Show/Hide details
  • Topics: Angle Measure, Angle Construction, Angle Addition
  • Book Sections: 6.1, 6.2, 6.3
  • Class Presentations:
    • Samantha Huff CP05: Axiom <N7> (The Angle Measurement Axiom) says that there exists a function m : \( A \rightarrow \) (0,180).
      The symbol (0,180) is used for the codomain. What does this mean?
      Why is this symbol used, and not [0,180]?
    • Nathan Steadman CP05: Alice, Bob, Charlie, Denise, and Emily are discussing the the diagram below.
      
      • Alice says that \(m( \angle ABC ) = 45 \)
      • Bob says that the measure of the smaller angle is 45, but the measure of the larger angle is 315.
      • Charlie says that \(m( \angle ABC ) = 315 \) because you're supposed to go counterclockwise from the first point listed to the third point listed.
      • Denise says that \(m( \angle ABC ) = -45 \) because you're supposed to go in the direction from the first point listed to the third point listed, and clockwise is negative.
      • Emily says that one can't say anything about the measures unless one knows whether one is speaking of the smaller angle or the larger angle, and whether one is supposed to go counterclockwise or clockwise.
      Comment on the validity or invalidity of each person's statement.
• Day #18: Neutral Geom IV: Angle Bisectors Show/Hide details
  • Topics: Angle Bisectors
  • Book Sections: 6.3
  • Class Drills: Drill for Section 6.3
  • Class Presentations:
    • Doug Wait CP05: Is the angle measurement function one-to-one? Is it onto? Discuss why you think it is or is not, using illustrations.
    • Baylie White CP05: Consider the diagram below.
      
      Can one use the Angle Measure Addition Axiom <N9> to say that the measure of angle \( \angle DBE \) is 180? Explain why or why not.
      And can one use the Angle Measure Addition Axiom <N9> to say that the measure of the large angle \( \angle ABC \) is 270? Explain why or why not.
    • Reilly Zink CP05: Illustrate the statement of Theorem 39 about points in the interior of angles. (Remark: This is the first time I have assigned an illustration for a theorem statement that is not in the form of a conditional statement. Think about what the statement of Theorem 39 is claiming, and consider ways of illustrating the statement. Come and see me to discuss if you want.) Perhaps illustrate the statement of the theorem in two cases: one case where both of the statements I and II are true, and another case where both of the statements I and II are false.
    • Allyson Strobel CP05: Alice, Bob, Charlie, Denise, and Emily are discussing the the diagram below.
      
      • Alice says that \( \overrightarrow{BD} \) and \( \overrightarrow{BE} \) are both bisectors of angle \( \angle ABC \), because they both create congruent smaller angles.
      • Bob says that \( \overrightarrow{BD} \) is the bisector of small angle \( \angle ABC \), but \( \overrightarrow{BE} \) is the bisector of big angle \( \angle ABC \), because E is on the outside.
      • Charlie says that both angles have two bisectors.
      • Denise says \( \overrightarrow{BE} \) is the bisector of big angle \( \angle ABC \), but since small angle \( \angle ABC \) is a clockwise angle, its bisector should be negative, so its bisector will be \( \overrightarrow{BE} \) as well.
      • Emily says that one can't say anything about the bisectors unless one knows whether one is speaking of the smaller angle or the larger angle, and whether one is supposed to go counterclockwise or clockwise.
      Comment on the validity or invalidity of each person's statement.
    • Micah Tobon CP05: Alice, Bob, Charlie, Denise, and Emily are discussing the the diagram below. The two lines cross at point B
      
      • Alice says that \( \overrightarrow{BD} \) and \( \overrightarrow{BE} \) are both bisectors of angle \( \angle ABC \), because they both create congruent smaller angles.
      • Bob says that \( \overrightarrow{BD} \) is the bisector of angle \( \angle ABC \), but \( \overrightarrow{BE} \) is the bisector of angle \( \angle CBA \), because \( \overrightarrow{BE} \) points down.
      • Charlie says that both angles have two bisectors.
      • Denise says \( \overrightarrow{BD} \) is the only bisector, and it bisects both angles, because the bisector is always the one that points up.
      • Emily says that \( \overrightarrow{BD} \) is not the bisector of angle \( \angle ABC \), because B is not the midpoint of segment \( \overline{AC} \). But \( \overrightarrow{BC} \) is the bisector of angle \( \angle DBE \), because B is the midpoint of segment \( \overline{DE} \).
      Comment on the validity or invalidity of each person's statement.
• Day #19: Neutral Geom IV: The Linear Pair Theorem (Quiz 5) Show/Hide details
  • Topics: The Linear Pair Theorem
  • Book Sections: 6.4
  • Class Drills: Drill for Section 6.4
  • Class Presentations:
  • Harman Aryal CP05: Illustrate the statements of Theorem 41 and 42.
  • Mitchelle Wambua CP05: Provide an illustration of the statement of Theorem 43 Vertical Pair Theorem, then present a proof of the theorem.
  • Quiz 5

Week 8: Mon Mar 4 - Fri Mar 8

• Day #20: Neutral Geom IV: Right Angles and Perpendicular Lines; Angle Congruence Show/Hide details
  • Topics: Right Angles and Perpendicular Lines; Angle Congruence
  • Book Sections: 6.6, 6.7
  • Class Presentations: None
• Day #21: In-Class Exam 2 on Chapters 4, 5, 6
• Day #22: Neutral Geom V: Axiom of Triangle Congruence, Theorems about Congruences in Triangles Show/Hide details
  • Topics: Axiom of Triangle Congruence, Theorems about Congruences in Triangles
  • Book Sections: 7.1, 7.2
  • Class Drills: Drill for Section 7.2
  • Class Presentations: None

Week 9: Mon Mar 11 - Fri Mar 15 is Spring Break: No Class

Week 10: Mon Mar 18 - Fri Mar 22

• Day #23: Neutral Geom V: Theorems about bigger and smaller parts Triangles Show/Hide details
  • Topics: Theorems about bigger and smaller parts Triangles
  • Book Sections: 7.3
  • Class Drills: Drill for Section 7.3
  • Class Presentations: None
• Day #24: Neutral Geom V: Perp Lines; Final Look at Triangle Congruence in Neutral Geom (Quiz 6) Show/Hide details
  • Topics: Perpendicular Lines; Final Look at Triangle Congruence in Neutral Geometry
  • Book Sections: 7.5, 7.6
  • Class Drills: Drill for Section 7.6
  • Class Presentations: None
  • Quiz 6 Due at the Start of Class
• Day #25: Neutral Geom V: Parallel Lines in Neutral Geom Show/Hide details
  • Topics: Parallel Lines in Neutral Geom
  • Book Sections: 7.7
  • Class Drills: Drill for Section 7.7
  • Class Presentations:
    • Caitlin Carlsen CP06: Make two drawings.
      • In the first drawing, show lines L and M and a transversal T that have the special angle property .
      • In the second drawing, show lines L and M and a transversal T that do not have the special angle property .
    • Aja Fox CP06: Your job is to do part of the proof of Theorem 73 . Prove (2) → (4)
    • Kayla Goodfellow CP06: Your job is to do part of the proof of Theorem 73 . Prove (7) → (1)
    • Samantha Huff CP06: Your job is to do part of the proof of Theorem 73 . Prove (5) → (8)
    • Rhett Miller CP06: Can the Alternate Interior Angle Theorem be used to prove that Alternate Interior Angles are congruent? Explain.
    • Nathan Steadman CP06: Bubba's proof of the Torpedo Theorem includes this step (13):
      (13) \( \angle \) ABC \( \cong \angle \) FEB (by Alternate Interior Angles).
      Could this step be valid? Or does it need some fixing to be a valid step? Or is it hopeless? Explain.

Week 11: Mon Mar 25 - Fri Mar 29

• Day #26: Neutral Geom VI: Lines Intersecting Circles Show/Hide details
  • Topics: Theorems about Lines Intersecting Circles
  • Book Sections: 8.1
  • Class Drills: Drill for Section 8.1
  • Class Presentations:
    • Allyson Strobel CP06: What is a line tangent to a circle? Explain and draw picture.
    • Micah Tobon CP06: Draw an illustration of the statement of Theorem 78. Make two pictures: One in which both statements (i) and (ii) are true, and one in which they are both false.
    • Doug Wait CP06: Can Theorem 78 be used to prove that a line L is perpendicular to a radial segment AB at a point B on a circle? Explain how, or why not.
    • Baylie White CP06: Can Theorem 78 be used to prove that a line L is NOT perpendicular to a radial segment AB at a point B on a circle? Explain how, or why not.
    • Reilly Zink CP06: Can Theorem 78 be used to prove that a line L is tangent to a circle a circle at some point B? Explain how, or why not.
    • Harman Aryal CP06: Can Theorem 78 be used to prove that a line L that intersects some circle at some point B is NOT tangent? Explain how, or why not.
    • Mitchelle Wambua CP06: Provide a proof of Theorem 79 with illustrations.
• Day #27: Neutral Geom VI: Theorems about Chords (Quiz 7) Show/Hide details
  • Topics: Theorems about Chords
  • Book Sections: 8.2, 8.3
  • Class Presentations:
    • Caitlin Carlsen CP07: Illustrate the statement of Theorem 87
    • Aja Fox CP07: Illustrate the statement of Theorem 88
    • Kayla Goodfellow CP07: Prove Theorem 88
    • Samantha Huff CP07: Illustrate the statement of Theorem 89
    • Rhett Miller CP07: Illustrate the statement of Theorem 90
    • Nathan Steadman CP07: In the proof of Theorem 90, there are two statements that say that certain triangles are congruent. What are the justifications for these two statements?
  • Quiz 7 due at the start of class
• Day #28: Neutral Geom VI: Tangent Lines, Angle Bisector Concurrence, Inscribed Circles Show/Hide details
  • Topics: Tangent Lines, Angle Bisector Concurrence, Inscribed Circles
  • Book Sections: 8.4, 8.5
  • Class Drills:
    • Drill for Section 8.4 about Points on Angle Bisectors
    • Drill for Section 8.4 about Concurrence of Angle Bisectors
  • Class Presentations:
    • Allyson Strobel CP07: Illustrate the statment of Theorem 91
    • Micah Tobon CP07: In the proof of Theorem 91, there are two statements that say that certain triangles are congruent. What are the justifications for these two statements?
    • Doug Wait CP07: Illustrate the statement of Theorem 92 about the concurrence of angle bisectors.
    • Baylie White CP07: Theorem 92 proves that there exists a point where the angle bisectors of the three angles of a triangle intersect. What is the name given to this point?
    • Reilly Zink CP07: Theorem 92, in book Section 8.4, proves that there exists a point where the angle bisectors of the three angles of a triangle intersect. That point turns out to be important in book Section 8.5. How so?
    • Harman Aryal CP07: Present a solution to Chapter 8 Exercise [18].
    • Mitchelle Wambua CP06: Present a solution to Chapter 8 Exercise [19].

Week 12: Mon Apr 1 - Fri Apr 5

• Day #29: Wrap up Neutral Geom VI: Tangent Lines, Angle Bisector Concurrence, Inscribed Circles Show/Hide details
  • Topics: Tangent Lines, Angle Bisector Concurrence, Inscribed Circles
  • Book Sections: 8.4, 8.5
  • Class Presentations:
    • Harman Aryal CP07: Present a solution to Chapter 8 Exercise [18].
    • Mitchelle Wambua CP06: Present a solution to Chapter 8 Exercise [19].
• Day #30: In-Class Exam 3 on Chapters 7, 8
• Day #31: Euclidean Geometry I: Parallel Lines and Triangles in Euclidean Geometry Show/Hide details
  • Topics: Parallel Lines and Triangles in Euclidean Geometry
  • Book Sections: 9.1, 9.2
  • Class Presentations:
    • Aja Fox CP08:
      • One Recurring Question in Geometry is : "Do parallel lines exist?" We discussed in class awhile back the fact that in Neutral Geometry, it can be proven that there exists lines J,K that are parallel. (This is also the subject of Chapter 8 Exercise 29) So parallel line do exist in Neutral Geometry. That means that Parallel lines do exist in Euclidean Geometry. So the answer to the first recurring question is "yes" in Neutral Geometry and "yes" in Euclidean Geometry.
      • The other recurring question is "Given a line L and a point P not on L, how many lines exist that pass through P and are parallel to L?"
        • What was the answer to this question in Neutral Geometry? (Explain.)
        • What is the answer to this question in Euclidean Geometry? (Explain.)
    • Kayla Goodfellow CP08: Illustrate the statement of Theorem 98
    • Samantha Huff CP08: Illustrate the statement of Theorem 100
    • Rhett Miller CP08: Abby, Bob, and Chris are arguing about how to justify a statement in a theorem. The statement says (17) The alternate interior angles \( \angle ABC \) and \( \angle DCB \) are congruent.
      • Abby says that the justification should be Theorem 74, the Alternate Interior Angle Theorem .
      • Bob says that the justification should be Theorem 100, the Converse of the Alternate Interior Angle Theorem .
      • Chris says that the justification is that Alternate Interior Angles are congruent.
      Comment on the validity or invalidity of each.

Week 13: Mon Apr 8 - Fri Apr 12

• Day #32: Euclidean Geometry I: Euclidean Triangles can be Circumscribed; Parallelograms Show/Hide details
  • Topics: Euclidean Triangles, Euclidean Triangles can be Circumscribed; Parallelograms
  • Book Sections: 9.3, 9.4, 9.5
  • Class Presentations:
    • Caitlin Carlsen CP08: Drawing for Theorem 103
      • Make a drawing to illustrate the statment of Theorem 103. (Note that we are used to the idea of making a pair of drawings for theorems that are presented as conditional statements . One drawing illustrates the hypothesis; the other, the conclusion. But Theorem 103 is a universal statement , not a conditional statement. Even so, it is also helpful to illustrate universal statement with two drawings. The universal statement says something about all objects of a certain type. A helpful pair of drawings would have a first drawing that just shows the generic object of that type, and a second drawing that illustrates what is being said about objects of that type.)
      • Also, make a drawing to illustrate statement (3) of the proof.
    • Nathan Steadman CP08: What justifies statement 3 in the proof of Theorem 103?
    • Allyson Strobel CP08: Justify statements 5,6,7 in the proof of Theorem 103.
    • Micah Tobon CP08: Justify statements 8,9 in the proof of Theorem 103.
    • Doug Wait CP08: Justify statement 10 in the proof of Theorem 103
    • Illustrations of the Statement of Theorem 108 This theorem presents six statements that can be made about a convex quadrilateral. The theorem says that the six statements are equivalent. That is, they are either all true, or they are all false. Of course, any convex quadrilateral where any of the six statements is true is going to look like a parallelogram: that's the point of the theorem. To illustrate the statment of the theorem, I would like to have on the chalkboard a big drawing that shows six convex quadrilaterals arranged in a circle, as shown below
      • Baylie White CP08: Make a drawing illustrating statement (i) at the 12:00 position on the circle.
      • Reilly Zink CP08: Make a drawing illustrating statement (ii) at the 2:00 position on the circle.
      • Harman Aryal CP08: Make a drawing illustrating statement (iii) at the 4:00 position on the circle and a drawing illustrating statement (iv) at the 6:00 position on the circle.
      • Mitchelle Wambua CP08: Make a drawing illustrating statement (vi) at the 8:00 position on the circle and a drawing illustrating statement (vi) at the 10:00 position on the circle.
      
• Day #33: Euclidean Geometry I: Parallelograms (Quiz 8) Show/Hide details
  • Topics: Theorem 108 about Parallelograms
  • Book Sections: 9.5
  • Class Presentations: None
  • Quiz 8 due at the start of class
• Day #34: Euclidean Geometry I: Triangle Midsegments, Altitude Concurrence, Median Concurrence Show/Hide details
  • Topics: Triangle Midsegments, Altitude Concurrence, Median Concurrence
  • Book Sections: 9.6, 9.7
  • Class Drills: Drill for Section 9.6
  • Class Presentations: None

Week 14: Mon Apr 15 - Fri Apr 19

• Day #35: Euclidean Geometry II: Parallel Projection, Similarity (Quiz 9) Show/Hide details
  • Topics: Parallel Projection, Similarity; Applications of Similarity
  • Book Sections: 10.1, 10.2, 10.3
  • Class Presentations about Theorem 122 The Angle Bisector Theorem
    Use an original triangle A,B,C of roughly this shape: A = (1,0) and B = (0,2) and C = (8,0)
    • Caitlin Carlsen CP09: Make a drawing to illustrate proof step (1)
    • Aja Fox CP09: Justify and make a drawing to illustrate proof step (2)
    • Kayla Goodfellow CP09: Justify and make a drawing to illustrate proof step (3)
    • Samantha Huff CP09: Justify and make a drawing to illustrate proof step (4)
    • Rhett Miller CP09: Justify and make a drawing to illustrate proof step (5)
    • Nathan Steadman CP09: Make a drawing to illustrate proof step (6)
    • Allyson Strobel CP09: Make a drawing to illustrate proof step (7)
    • Micah Tobon CP09: Justify and make a drawing to illustrate proof step (8)
    • Doug Wait CP09: Justify and make a drawing to illustrate proof step (9)
  • Quiz 9 Due at the start of class.
• Day #36: Euclidean Geometry II: Parallel Projection, Similarity; Applications of Similarity Show/Hide details
  • Topics: Applications of Similarity: Proving the Pythagorean Theorem using Similarity; Converse of the Pythagorean Theorem
  • Book Sections: 11.1, 11.2, 11.3, 11.4
  • Class Presentations: None
• Day #37: In-Class Exam 4 on Chapters 9, 10, 11

Week 15: Mon Apr 22 - Fri Apr 26

• Day #38: Euclidean Geometry III: Area; Using Area to Prove the Pythagorean Theorem, Area of Similar Polygons Show/Hide details
  • Topics: Area; Using Area to Prove the Pythagorean Theorem, Area of Similar Polygons
  • Book Sections: 11.1, 11.2, 11.3, 11.4
  • Class Presentations:
    • Baylie White CP09: Chapter 11 Exercise [8] Justify the steps in the proof of Theorem 135 (about the ratio of the areas of similar triangles) (found on page 253).
    • Reilly Zink CP09: Chapter 8 Exercise [12] If you want to triple the area of a square, by what factor should you multiply the lengths of the sides? Explain.
    • Harman Aryal CP09: Solve Chapter 11 Exercise [15].
    • Mitchelle Wambua CP09: Solve Chapter 11 Exercise [17].
• Day #39: Euclidean Geometry IV: Cyclic Quadrilaterals, Intersecting Secants (Quiz 10) Show/Hide details
  • Topics: Circular Arcs; Arc Angle Measure; Angles Intercepting Arcs
  • Book Sections: 12.1, 12.2, 12.3
  • Class Presentations are about Chapter 12 Exercises involving measures of the various types of angles intersecting circles: (Give brief presentations; avoid getting bogged down writing full statements on the board. Use figures and annotated arrows to explain the progress of proofs and the justification of the steps.)
    • Caitlin Carlsen CP10: Chapter 12 Exercise [4] about the proof of Theorem 141 about Type 2 angles. Just do case (i), which is the case where the center of the circle lies on one of the rays of the angle.
    • Aja Fox CP10: Chapter 12 Exercise [8] about the proof of Theorem 143 about Type 3 angles.
    • Kayla Goodfellow CP10: Chapter 12 Exercise [9] about the proof of Theorem 144 about Type 4 angles.
    • Samantha Huff CP10: Chapter 12 Exercise [10] about the proof of Theorem 145 about Type 5 angles. Only justify and illustrate steps (1),(2),(3) of the proof, which proves that equation (i) is true.
    • Rhett Miller CP10: Chapter 12 Exercise [11] about the proof of Theorem 146 about Type 6 angles. Only justify and illustrate Case (i) about right angles.
    • Nathan Steadman CP10: Chapter 12 Exercise [11] about the proof of Theorem 146 about Type 6 angles. Only justify and illustrate Case (ii) about obtuse angles.
    • Allyson Strobel CP10: Chapter 12 Exercise [11] about the proof of Theorem 146 about Type 6 angles. Make a drawing and fill in the missing proof steps, with justifications, for Case (iii) about acute angles. (Note that this may sound daunting. But in proving Case (iii), you are allowed to use the result of Case (ii), which has already been proven at this point. So in fact, you can weasel out of doing any heavy lifting.)
  • Quiz 10 due at the start of class.
• Day #40: Euclidean Geometry IV: Cyclic Quadrilaterals, Intersecting Secants Show/Hide details
  • Topics: Cyclic Quadrilaterals, Intersecting Secants
  • Book Sections: 12.4, 12.5
  • Class Presentations: TBA
    • Micah Tobon CP10: Can every quadrilateral be circumscribed? Explain using drawings. What is a cyclic quadrilateral ?
    • Doug Wait CP10: Chapter 12 Exercise [14] about proving Theorem 148 about Cyclic Quadrilaterals.
    • Baylie White CP10: Theorem 148 tells us that for any cyclic quadrilateral, the sum of the measures of each pair of opposite angles is 180. Is the converse statement true? That is, if it is known that a quadrilateral has the property that the sum of the measures of each pair of opposite angles is 180, then will the quadrilateral definitely be cyclic? Explain.
    • Reilly Zink CP10: The proof of Theorem 149 (on pages 277 - 278) includes two statements that are not justified. Justify them. (Hint: Start with the second statement to be justified, about the situation where point P lies in the exterior of the circle. Just consider the case where the angle formed is Type 4, as shown in the left-most figure. Imagine adding a line segment DC to the drawing. What is the measure of \( \angle ADC \)? Why? And what can you say about the relative sizes of \( \angle ADC \) and \( \angle APC \)? Why? That should take care of justifying the second statement to be justified. Once you have done that, apply the same kind of approach and reasoning to justify the first statement to be justified.)
    • Harman Aryal CP010: A problem just like Chapter 12 problem [17], but instead of the numbers 3,6,5, use the numbers 3,4,6.
    • Mitchelle Wambua CP10: Chapter 12 Exercise [18].

Week 16 (Finals Week): Mon Apr 29 - Fri Mar 3

• Day #42: Fri May 3 Final Exam 3:10pm - 5:10pm in Morton 126 Show/Hide details about the Final Exam
  • You may use your packet "Definitions and Theorems" for reference.
  • The Exam is 8 problems.
    
    
    1. Prove a Theorem from Chapter 5 Plane Separation. (I choose the theorem.)
       • Theorem 27 each half-plane contains three non-collinear points. (Ch. 4, p. 120)
       • Theorem 28 Pasch's Theorem about a line intersecting a side of a triangle between vertices. (Ch. 4, p. 122)
       • Theorem 29 about a line intersecting two sides of a triangle between vertices. (Ch. 4, p. 122)
       • Theorem 33 Points on a side of a triangle are in the interior of the opposite angle. (Ch. 4, p. 125)
       • Theorem 35 The Crossbar Theorem (Ch. 4, p. 126)
    2. Prove a Theorem involving concurrence of three lines associated to triangles. It will be one of these. (I choose the theorem.) (You might have to prove, or you might have to just justify and illustrate. And you might have to do the whole proof, or you might just have to do part of the proof.)
       • Theorem 92 about the concurrence of the three angle bisectors for triangles in Neutral Geometry. (Ch. 8, p. 202))
       • Theorem 95 about the existence of an inscribed circle for triangles in Neutral Geometry. (Ch. 8, p. 204)
       • Theorem 106 about the concurrence of perpendicular bisectors of the sides for triangles in Euclidean Geometry. (Ch. 9, p. 214)
       • Theorem 107 In Euclidean Geometry, every triangle can be circumscribed. (Ch. 9, p. 215)
       • Theorem 113 about the concurrence of the altitudes for triangles in Euclidean Geometry. (Ch. 9, p. 219)
    
    
    3. Solve a Geometric Problem Involving Triangles. (I choose one.)
       • Chapter 10 Exercises (p. 241-242) [15],[16],[17]
       • Chapter 11 Exercises (p. 258-261) [1],[13],[15],[16],[17],[18]
    
    
    4. Solve a Geometric Problem Involving Circles. (I choose one.)
       • Chapter 8 Exercises (p. 207-208) [17],[18],[19],[20],[21]
       • Chapter 12 Exercises (p. 282-284) [5],[6],[7],[12],[17],[18],[19] (on problem 17,instead of the numbers 3,6,5, use the numbers 3,4,6)
    
    
    5. Prove a Theorem that has a proof involving creating a triangle inside or adjacent to an existing triangle. (I choose the theorem.)
       • Theorem 59 The Neutral Geometry Exterior Angle Theorem. (Ch. 7, p. 170)
       • Theorem 61 BS ==> BA. (Ch. 7, p. 172)
       • Theorem 64 The Triangle Inequality for Neutral Geometry. (Ch. 7, p. 174)
       • Theorem 110 The Triangle Midsegment Theorem. (Ch. 9, p. 217)
       • Theorem 122 The Angle Bisector Theorem Theorem. (Ch. 10, p. 230)
    
    
    6. Prove a Theorem that I proved with an indirect proof in the book. (I choose one theorem.)
       • Theorem 62 BA ==> BS (Ch. 7, p. 172)
       • Theorem 74 Alternate Interior Angle Theorem (Ch. 7, p. 188)
       • Theorem 99 about parallel lines, transversals, and the special angle property in Euclidean Geometry. (Ch. 9, p. 211)
    
    
    7. Prove a Theorem that is stated as an Equivalence Theorem. (I choose the theorem, and which part you will prove.) (You won't have to prove all the parts.)
       • Theorem 73 about angles formed by two lines and a transversal in Neutral Geometry (Chapter 7, p. 186)
       • Theorem 85 about special rays in isosceles triangles (Ch. 8, p. 199)
       • Theorem 86 about points equidistant from the endpoints of a line segment. (Ch. 8, p. 200)
       • Theorem 108 about convex quadrilaterals. (Ch. 9, p. 216)
    
    
    8. Prove a Theorem involving an Application of Similarity or involving area of Similar Triangles. (I choose the theorem.)
       • Theorem 130 The Pythagorean Theorem. (Ch. 10, p. 237)
       • Theorem 132 The product of base*height does not depend on the choice of base. (I would have you prove just one of the cases.) (Ch. 10, p. 238)
       • Theorem 135 about the ratio of the areas of similar triangles. (Ch. 11, p. 253)
    
    

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