Download Test #1 Review

Test #1 Review Sheet
M 333 L Spring 2017
READINGS: From the textbook Roads to Geometry :
Sections 1.1, 1.2, pp. 19-21
Sections 2.1, 2.2, 2.3, 2.6, 2.7
Sections 3.1, 3.2. 3.3, 3.4, 3.5, 3.6
Sections 4.1, 4.2.
HANDOUTS:
From the "First Handouts" Folder:
Review of High School (Euclidean) Geometry
Some Definitions from Euclidean Geometry -- Part I
Congruence between two triangles
Congruence Between Two Triangles Worksheet (Not the HW worksheet!)
Solutions to Congruence Between Two Triangles Worksheet (Not the HW worksheet!)
Triangle Congruence Conditions: S A S , A S A , A A S , S S S , and (for right triangles) H L
An Example Proof Using the Triangle Congruence Conditions
The Axiomatic Method
Consistency, Completeness, and Independence of a Set of Axioms
Two-Column Proofs of Fe-Fo Theorems 1, 2 and 3
Outlines of Three Types of Proof
Summary of Euclid's "Elements", Book I
The SMSG Postulates for Euclidean Geometry
Three Flaws in the Axiom System presented in Euclid's "Elements"
The Three Alternatives for a Parallel Postulate
From the "Neutral Geometry" Folder:
Introduction to Neutral Geometry
More on Neutral Geometry I
More on Neutral Geometry II
The "Common Perpendicular" Theorem
The Correct Statements of The "Alternate Interior Angles" Theorem
and of The Converse of the Alternate Interior Angles Theorem
Available Corollaries to the Alternate Interior Angles Theorem and its Converse
The Converse of AIA is False in Hyperbolic Geometry
The Saccheri-Legenre Proof (that Angle-sums ≤ 180o)
Proofs of Some EPP Equivalences (in Neutral Geometry)
Section 3.6 Annotated Summary
From the "Euclidean Geometry I" Folder:
Introduction to Euclidean Geometry (Section 4.2)
Available Corollaries to the Alternate Interior Angles Theorem and its Converse
Homework assignments: HW #01-HW #05
(Solutions are available online through the link to the homework assignments.)
Note: It is helpful to review the Chapter Summary on pp. 119 – 123 of the textbook.
Bring a Compass and Straight-edge to the test.
Questions on the test will be of various types:
Short answer or fill-in-the-blank, State certain definitions, theorems, postulates.
Compute various numbers, figures, correspondences, and equations using the concepts we have studied
(e.g., congruence of triangles, and the Poincare Half-plane model of Hyperbolic Geometry).
Understand the proofs presented in class, but, except for a proof that the EPP is equivalent to a particular
statement, the more complicated proofs will not need to be presented in full.
Presentation of complete proofs may be required for the simpler ones.
Proofs will be required on some problems.
Topics:
Historical Development:
Euclid’s Elements: Be able to state Euclid’s 5 postulates, especially the 5th postulate;
know three ways that the Elements is flawed as a formal axiom system;
Axiomatic Systems:
Terminology (e.g., consistency, completeness, model, etc.) and proofs;
know the four components of any axiom system.
Be able to: prove that a set of axioms is consistent, prove that a set of axioms is inconsistent,
prove that a particular axiom is independent from the rest of the axioms,
prove that particular axiom is dependent on the rest of the axioms,
prove simple theorems of the axiom system,
construct an abstract model of an axiom system
(thereby proving that the set of axioms is consistent)
prove (when possible) that a particular theorem statement is both true and false
(thereby proving that the set of axioms is inconsistent)
prove (when possible) that a particular axiom is true using only the other axioms
(thereby proving that the particular axiom is dependent on the rest)
Regarding the SMSG Postulates of Euclidean Geometry:
Know how to state:
The Plane Separation Postulate (Postulate #9).
The Angle Construction Postulate (Postulate #12).
The Supplement Postulate (Postulate #14).
The Euclidean Parallel Postulate (Postulate #16) .
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Be able to state:
the Euclidean Parallel Postulate,
the Hyperbolic Parallel Postulate,
the Elliptic Parallel Postulate,
the Definition of Euclidean Geometry
the Definition of Neutral Geometry,
the Definition of Hyperbolic Geometry
(No need to know the definition of Elliptic Geometry)
the Converse of any given IF-THEN statement
Is the Euclidean Parallel Postulate the same as Euclid’s 5 th Postulate? (Ans = NO)
Is the Euclidean Parallel Postulate the same as Playfair’s Postulate? (Ans = YES)
Some Topics From Neutral Geometry (Chapter 3)
Be able to state the following theorems, and be able to cite them by name when using them to justify a
step in a proof:
The Isosceles Triangle Theorem,
The Congruent Supplementary Angles Theorem,
The Vertical Angles Theorem,
Theorem 3.2.8: The "Perpendicular Bisector" Theorem
The Alternate Interior Angle Theorem
The Exterior Angle Theorem,
The Triangle Inequality
Know how to apply the following Triangle Congruence Conditions both for calculations and for use in a
proof:
SAS, ASA, AAS, SSS, and (for right triangles) HL.
Know all the statements which were proven to be equivalent to the Euclidean Parallel Postulate in
Neutral Geometry.
Be able to correctly identify and distinguish the following to theorems:
The Alternate Interior Angle Theorem (Theorem 3.4.1) and
The Converse of the Alternate Interior Angle Theorem (4.2.1).
Is the Converse of the Alternate Interior Angle Theorem true in Neutral Geometry? (Ans = NO)
Is the Alternate Interior Angle Theorem true in Neutral Geometry? (Ans = YES)
Some Topics From Euclidean Geometry (Chapter 4, Sections 4.1-4.2)
Understand Theorems 4.2.1, 4.2.2, 4.2.3 and 4.2.4.
You might be required to prove any one of Thms 4.2.11 – 4.2.11 and Thms 4.2.16 and 4.2.17.
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Regarding the Poincare half-plane model of Hyperbolic Geometry:
Know what “points” and “lines” are and how angle measure is defined in this model.
Be able to draw lines which provide counter-examples to theorems and statements which are
true in Euclidean Geometry, but which are false in Hyperbolic Geometry.
Compass and Straight-edge Constructions:
Be able to:
1) Given a line segment, construct the midpoint of that line segment.
2) Given a line segment, construct the perpendicular bisector of that line segment.
3) Given a line and a point on that line, construct the line which is perpendicular to the given line and
passes through the given point.
4) Given a line and a point NOT on that line, construct the line which is perpendicular to the given line
and passes through the given point.
5) Given two points, construct the line which is the set of all the centers of all the circles that pass
through the two given points, and construct 4 circles passing through the two given points.
Any problem in the homework can be adapted for and can appear as a test question.
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