Homework for Mat 304
March 18 – Collected problems due. Discussed Saccheri-Legendre Theorem – lemmas and corollaries. Homework:
Prove a second one of Section 3.4 – 1,2, or 4. Read the proof of Saccheri-Legendre Theorem and write a proof of Section
3.5 – 1 Test will be 26 or 27th.
March 16 – Proved #10 from 3.3 – HL. Discussed theorems equivalent to a parallel postulate. Homework: Exercises 3.4
- Write a “nice” proof of problems 1, 2, or 4 (as outlined in class). Earliest test date is the last of next week.
March 6 – Returned graded homework problem. Proved two homework problems. Homework: Read the proofs of the
hinge theorem, the triangle inequality and the statement of SSS. Assigned partners to write a proof due March 18.
Handed out “Mallory’s Theorem” for homework for everyone.
March 4 – Collected one homework problem of student’s choice. Wrote #11 on the board. Discussed theorems and
their proofs. Homework: Read Section 3.3. Exercise 3.2 – be sure that you have written proofs of both 12 and 13.
Section 3.3 – 1 and 2. Anna Kate is to polish a proof of 1 and Brooke is to polish a proof of 2 but all students are to work
on both of them.
March 2 – Wrote a proof for #5 page 89. Discussed the proof of the Exterior Angle Theorem; AAS; and SASAS.
Homework: Exercises 3.2 - 11, 12 or 13. Exercises 3.3 – 1.
February 27 – Started to write the proof of problem 5 in class and decided that there was probably a shorter way than
to deal with coordinates. Wrote the proof of “two angles complementary to the same angle are congruent”. Discussed
the Crossbar theorem, the Isosceles Triangle Theorem, and Th 3.2.8. Homework: Define – acute, obtuse and right
angles; perpendicular lines; scalene, isosceles and equilateral triangles. READ section 3.2 completely. Be prepared to
ask about any statement in the proof of any theorem that you do not understand. If there are homework questions
from Feb 16 – today, ask them on Monday. PROVE: Evan’s Theorem “All right angles are congruent.” And Shelby’s
Theorem “ Perpendicular lines form 4 right angles.” Then problem 5 (again) and 9 on page 89.
February 25 – No classes due to weather.
February 23 – Defined vertical angles and wrote three proofs. Homework - Define: Vertical angles. Four students are
assigned part of Theorem 3.2.2 or 3.2.3 to prove. All students are to work on proving the remaining parts of Theorems
3.2.2 and 3.2.3.
February 20 – Discussed models for elliptic and hyperbolic geometry. Began Chapter 3. Proved the reflexive and
symmetric parts of “congruence of angles is an equivalence relation”. Homework: Define – Congruent segments,
congruent angles, congruent polygons, midpoint, angle bisector and regular polygon. 2.7 – KNOW how to define lines
and points in the two Poincaré models and the Riemann model. problems 1 and 4. Section 3. 2 – Complete the proof
that congruence of angles is transitive and prove – “congruence of segments is an equivalence relation”.
February 18 – quiz on definitions. Completed discussion of SMSG postulates. Homework: Define – opposite rays and
linear pair. READ section 2.7. Be prepared to state the Hyperbolic Parallel Postulate and the Elliptic Parallel Postulate.
February 16 – Discussed SMSG Postulates through #15. Homework: Read Sections 2.6 and 2.7 Exercises 2.6 – 1-5, 10.
Define: polygon( triangle, quadrilateral, etc.), congruent, adjacent angles, supplementary angles, and complementary
angles.
February 13 – test
February 11 – General discussion.
February 9 – Reports on Beltrami and Poincare. Discussed Birkhoff’s axiomatic system. Homework: Exercises 2.4 – 3,
and exercises 2.5 – 2, 3, 6a, 7, and 8.
February 6 – Returned quiz. Completed Hilbert discussion. Test next Friday in MCC 105 or begin earlier by
appointment in my office. Homework: Read Section 2.5 Exercises 2.4 – 14; Exercises 2.5 – 4, 6, 8
February 4 – returned the last two quizzes. Had a quiz on Playfair’s Postulate. Began discussion of Hilbert’s Axioms and
defined several terms. Homework: Define – square, tangent to a circle, sector of a circle, segment of a circle. READ
Section 2.4 (at least). Exercises 2.4 – 7, 8, 9. Tentative test day is Friday February 13 .
February 2– Quiz on homework. Demonstrated the construction of a perpendicular to a line from a point on the line.
Discussed/defined several terms : Homework: Construct a perpendicular to a line from a point on it. Identify: halfplane, interior/exterior of an angle, minor arc, major arc and a central angle of a circle. Define: arc, radius, chord,
diameter, interior of an angle, and secant line. Construct the diagram on page 42 and determine what’s not correct in
the proof. Read the proof of the theorem on page 44-45 and determine what is implied by the proof. READ Section2.3
and 2.4. State Playfair’s Postulate. Section 2.3 – 3 - 7.
January 30 – quiz on construction. Demonstrated two constructions: construct a perpendicular to a line from a point
not on the line and bisect an angle. Discussed and created definitions. Homework: 1) practice new constructions. 2)
Handout to copy three segments and construct a triangle with sides equal to the lengths of the three segments. 3) READ
2.2 again. READ especially the “proof” on page 43 that any triangle is isosceles and find the “error”. 4) Each student
should know at least one statement that is equivalent to Euclid’s 5th Postulate. 5) State Euclid’s 5th. 6) Define:
“between” line segment (segment), ray, and angle.
January 28 – Began a discussion of Euclid’s axiomatic system. Constructed an equilateral triangle. Homework – State
the Euclid’s 5th Postulate and Read Section 2.1, and 2.2. Construct an equilateral triangle given the length of one side or
construct a triangle given all three sides. Exercises 2.2 – 1, 3 and 4 ( do write the definitions that you make in pencil, we
will agree in class which ones to use this semester.) and add a part vii to 4 with “angle”. You may use the definition of
“between” on page 63 as you create these definitions.
January 26 – Discussed #3 from homework. Defined circle, affine geometry and projective geometry. Constructed the
perpendicular bisector of a segment. Homework; Section 1.4 (finish/redo #3) , 5, 6, 11, 14 all parts except i. Define:
Circle, affine geometry and projective geometry.
January 23 – Returned the homework problem graded. Discussed form expected in a proof, incidence axioms, how to
construct parallel lines (copy a segment), and possible parallel relationships in an axiomatic system. Homework: Write
(when requested) the Axioms for the four point geometry, Axioms for Fano’s Geometry and Incidence Axioms. “redo”
homework proofs as needed – “catch up” if you need to. Section 1.3 – 19, Section 1.4 – #3.
January 21 – Collected the pairs problem and one homework proof of the student’s choice to grade. Discussed Fano’s
Geometry and created model(s) as well as the proof that the geometry has 7 lines and 7 points. Homework: Section
1.2 - # 25, Section 1.3 – 17 and 18.
January 16 – Returned the quiz from last class. Bryce showed us a proof. We proved #5 on the board. Discussion
involved form for proof writing and the four point geometry. Homework: Define (this means that you should write a
one sentence definition of the term any time I ask during the semester) – intersecting lines, concurrent lines, coplanar
lines, parallel lines (added a word to the textbook definition) and skew lines. Exercises 1.2 – 3 or 4, 6 or 9, 16 and 25.
READ pages 19 – 21 especially the proofs.
January 14 – Returned the quiz from last class. Quiz on homework reading. Discussed properties of an axiomatic system
and example 1.2.1 from the text. Homework: Exercises 1.2 – 1, 2, 5 and 6.
January 12 - Answered questions about the syllabus, then a quiz on the syllabus. Discussed parts of an axiomatic
system. Homework: Read Section 1.1 (at least), work #1, pairs problem is due 1-21, know the parts of an axiomatic
system.