Homework Mat 304 – Spring 2011
January 11 – Discussed: the syllabus for the course, parts of an axiomatic system, reviewed formulas needed for
homework problems, properties of an axiomatic system (consistent and independent especially). Homework: Section
1.1 - #1 is due by Thursday, special problems were selected and are due 1-18, (students are to work one more from the
selected problems (2, 4, 5, 8, 9, 10) than their particular problem). Read Section 1.1 and know something about each
man of geometry. Know the parts of an axiomatic system and what makes a system independent and/or consistent.
January 13 – Collected problem 1 from section 1.1. Discussed finite axiomatic systems including: the fe-fo example, the
x-y problem, and a four point geometry. Students worked in small group to prove one theorem, #1 in Ex. 1.2 then write
it on the board for discussion. Homework: Define – collinear, coplanar, intersecting lines, concurrent lines, and parallel
lines. Exercise 1.2 – on problem from 2 - 4, select one from 6-8, 16 and 25. These are due for the next class meeting.
January 18 – Collected the problem assigned to each individual last week. Summer and James wrote a proof on the
board. Discussion included discussing what a plane dual is (#5) in the third exercise, making a model for FANO’s
geometry, and how Young’s geometry differed from Fano’s Geometry. Homework: due next time - 1.3 - #4, and select
two problems from 14-18.
January 20 – Deryk and Dennis wrote problems on the board. Discussion included Incidence Axioms, and the three
possibilities for parallels. Homework: Section 1.3 – 24-27 select at least one. Section 1.4 - #3 (except v and vi the parts
worked in class) 6 and 11. Define affine geometry and projective geometry.
January 25 – Returned problems assigned from Section 1.1. The class discussed #3 from Section 1.4. Discussion began
on Euclid’s geometry from the Elements, his definitions, postulates, axioms, and propositions. Appendix A is a good
summary. Specifics that were pointed out include weaknesses in his system of geometry and the unique statement of
his Postulate 5, his parallel postulate. Homework: Read the section about Euclid’s geometry. Find the flaw in the proof
on page 42. State Postulate 5. Define: circle, concentric circles, sphere, radius, chord, arc, segment of a circle, and
annulus (of concentric circles.) BRING YOUR COMPASS NEXT TIME. No Class February 3.
January 27 – Discussed Science and Mathematics tournament. Constructed 3 kinds of perpendiculars, defined line
segment, ray, “between”, perpendicular lines, parallel lines (including the “coplanar”),diameter of a circle and sector of
a circle. Homework: Read sections 2.3 and 2.4 - Practice constructions, Know definitions, State Playfair’s Postulate, and
write a definition of a polygon.
February 1 – Quiz on Construction. Discussed Hilbert’s axioms, defined: polygon (sides and angles), half-plane, interior
of a polygon, exterior of a polygon, special polygons (triangle, quadrilateral, pentagon, hexagon, heptagon, octagon,
nonagon, decagon, dodecagon, and n-gon), angle, (acute, obtuse, and right angles). Homework; Read Section 2.5,
Section 2.4 – 7 (other than the first part), 8, 9, 11 (use ∠𝐴𝐵𝐶) and Find the content of the 13 books of Euclid’s Elements.
NO Class on February 4th – Science and Mathematics Tournament.
February 8 – Returned quiz. Reviewed the content of Euclid’s 13 books that comprised the Elements. Discussed
questions/problems with Hilbert’s axiomatic system and Birkhoff’s. Introduced SMSG postulate system. Homework:
Define: congruent segments, congruent angles, triangle names for sides (scalene, isosceles, and equilateral) and angles
(acute, obtuse, right, equiangular), and regular polygon. Section 2.4 – 2, Section 2.5 – 2 and 6i. Read Section 2.6 Test
next week.
February 10 – Students scheduled the day(Feb 17 or 18) and time to take the test on Chapter 1 – 2.6. Two proofs were
discussed #2 and 6i. The Ruler Postulate and the Supplement Postulate were illustrated. Defined convex set, adjacent
angles, opposite rays, linear pair, supplementary angles, and complementary angles. Homework: Select some unit
length then construct length of √2 𝑎𝑛𝑑 √5. Place the unit on a line after you select an arbitrary point to be 0, then
construct points have coordinates of √2 𝑎𝑛𝑑 √5. Section 2.6 - #10. After questions on Tuesday discussion will begin of
Section 2.7.
February 15 – Discussed possible items for the test. Constructed an angle bisector and parallel lines. Discussed Section
2.7 Discussed the Poincare half-plane model of hyperbolic geometry, and discussed the spherical model of elliptical
geometry. Homework: 2.7 – State the Elliptical parallel Axiom and the Hyperbolic parallel axiom. Bisect an angle,
construct a Euclidean parallel. Read Section 2.7, be able to describe a model for each non-Euclidean geometry .
Problems 1 & 4 in the section.
February 17 – Began the discussion of neutral geometry with emphasis on writing correct symbols (say what you mean).
Defined – midpoint of a segment, congruent angles, congruent segments, congruent polygons, vertical angles, midpoint
of a segment, bisector of an angle, median of a triangle, centroid of a triangle, altitude of a triangle, and orthocenter of a
triangle. Homework: Section 3.2 - #4 part of the theorem - either segments of angles, define the words above, construct
the medians and altitudes of three different kinds of triangles (scalene right, scalene obtuse, and scalene acute) and
paraphrase Theorem 3.2.3 into four conditional statement.
February 22 – Answered questions about orthocenter. Went to the lab and constructed the orthocenter and centroid
with Geometer’s Sketchpad. Answered questions about paraphrase of Theorem 3.2.3. Discussed the proof of Paush’s
“Axiom”, read the crossbar theorem, and discussed the proof of the isosceles triangle theorem. Students were
reminded to read the text. Homework: Parts of theorem 3.2.3 and two more were assigned to specific students.
Students were to prove their assigned part and one other of the six things to prove. Define: incenter and circumcenter.
Students are to practice the construction of those two points.
February 24 – Proved two theorems in class. Outlined the proofs of two more theorems. Quiz on constructions.
Homework: READ THE CHAPTER THROUGH SECTION 3.2. READ POSTULATES OF SMSG 1-15. Prove: If two congruent
angles form a linear pair, then the sides of one of the angles are perpendicular. (Deryk’s Theorem) Section 3.2 problem 9.
Polish any proofs that you worked for today that need a rewrite.
March 1 - Collected one proof from the last week of problems (not the ones that were discussed in detail on the board
last time). Discussion included how to prove POLYGONS congruent: ASA, AAS,SASAS, and SSS. Proofs were read and
discussed or partially written on the board. Demonstrated the construction “how to copy an angle”. Basic constructions
that have been demonstrated to date include: construct perpendiculars (3 kinds), parallel to a line through a point not
on the line, angle bisector, copy a segment, and copy an angle. Homework: Select and write a proof for 12 or 13 in
section 3.2 , outline a proof for #1 in Section 3.3, and complete the handout on construction. Deryk, yours is outside my
office.
March 3 – Discussed theorems left in Section 3.3. Student drew for a problem to solve in pairs and write the proof on
the board. Third one will come next class period. Homework: Three problems were assigned by pairs. . . 6, 7, and 10
from section 3.3. Students should attempt problems other than the one due from the pair. Test on Chapter 3 the week
after spring break.
March 8 – Collected the pairs proofs. Discussed Saccheri and Lambert quadrilaterals and theorems associated with
them. Homework: Define Saccheri quadrilateral and Lambert quadrilateral. READ THE PROOF OF THE SACCHERILENGENDRE THEOREM AND TH. 3.6.6. Read the last few sections of the chapter again. Prove: Summer’s Theorem: “ If
two angles are congruent and “ corresponding angles” (that are part of the larger angle) are congruent, remainding
angles (that are part of the larger angle) are congruent.
Brittany’s Theorem: All right angles are congruent.
Kelsey’s Theorem: Perpendicular lines form four right angles.
James’ Theorem: If point P is equidistant from two points, A and B, then P is on the perpendicular bisector of ̅̅̅̅
𝐴𝐵.
March 10 - Wrote two proofs during class (#9 and # 12). Then students wrote proofs from 3.6 in pairs. Homework:
̅̅̅̅ ≅ ̅̅̅̅
Dennis’ Theorem: If A-B-C , X-Y-Z, ̅̅̅̅
𝐴𝐶 ≅ ̅̅̅̅
𝑋𝑍, 𝑎𝑛𝑑𝐴𝐵
𝑋𝑌 then ̅̅̅̅
𝐵𝐶 ≅ ̅̅̅̅
𝑌𝑍. Section 3.6 - 11 and 15. Test week after
spring break.
March 22 – Two students wrote proofs on the board and they were critiqued. Class discussion was about the test this
week and chapter 4 – Euclidean Geometry - defined terms. Homework: Define – trapezoid, isosceles trapezoid,
rhombus, parallelogram, rectangle, square.
March 24 – Continued in Chapter 4 – Outlined a synthetic proof of “opposite sides of a parallelogram are congruent”
and then wrote an analytic proof. Discussed placement of shapes on a grid to best use the properties of the grid.
Homework: Section 4.2 - #2 , #6 by synthetic methods and #9 by analytic methods.