MATH 306-001 59458 College Geometry, Time MW 1600-1715 SMLC-124. 

Instructors: Lectures: MW 16:00-17:15 SMLC-124, Dimiter Vassilev, Associate Professor;   Office: SMLC, Office 326;   Email:

Office Hours:  Monday, Wednesday 2pm-3pm & Friday 10am-11am. Feel free to stop-by anytime if you have a quick question.

Textbook: Noronha H., Euclidean and Non-Euclidean Geometries

Other texts:

Catalog Description: An axiomatic approach to fundamentals of geometry, both Euclidean and non-Euclidean. Emphasis on historical development of geometry. Prerequisite: C (not C-) or better in 162 or 215.

Please note the following guidelines for the course:

GRADING: Your total course grade is based on your ranking and percentile in the class computed using the in-class exams, homework, and the final exam scaled as follows:

To get full credit on exams and quizzes you need to show your work, neatly, in clear and correct mathematical notation, annotated by English sentences where appropriate. You will be graded based on the work shown, not on the answer. All grades will be posted on UNMLearn.

Note on Ws: If you withdraw after the 3rd week of class, you will receive a W. If you do not withdraw, you will receive a letter grade of A,B,C,D or F (and not a W).

The Final Exam score will replace all midterm scores that are lower than the Final Exam score. All grades will be posted on UNMLearn. Although a small curve might be used, 90% , 80% or 70% of the possible maximum points guarantees at least an A, B or C, respectively.

CALCULATORS: We will not use any (graphing or non-graphing) calculators on the exams or quizzes.

EXAMS: The exam dates are given in the syllabus. No makeup exams will be given unless you contact your instructor ahead of time with a documented “university authorized absence” (illness, family emergency, active participation in scholarly or athletic events).

ATTENDANCE: Attendance at UNM and homework is mandatory. If you have missed more than 4 attendance+homework+quizzes in the first 3 weeks you will be dropped from the course. Similarly, students with absences and lack of work during the rest of the semester may be dropped. Tardiness or early departure may be regarded as absence. Please note that it is the students responsibility to drop the course if he/she stops attending. A failing grade of F may be assigned if the student stops attending and does not drop. You can find the precise statemenet of the University policy here.

STUDENT BEHAVIOUR: Be courteous and respectful towards the class: be on time for lectures, turn off cell-phones and refrain from talking in class, leaving the classroom in the middle of a lecture or doing any other activity that could be disruptive to the class. Cheating will not be tolerated.

ACCESSIBILITY STATEMENT: We will accommodate students with documented disabilities. During the first two weeks of the semester, those students should inform the instructor of their particular needs.

FINAL EXAM: Monday, December 11, 5:30pm - 7:30pm. Students having conflicts with this examination schedule must notify the appropriate instructor before Friday, November 3, 2017. Any student having more than three examinations scheduled in any one day may notify the instructor of the last examination listed. If notified before Friday, November 3, 2017, the instructor shall make arrangements to give a special examination. Conflicts arising as a result of scheduling out of normal hours-pattern or day sequences must be resolved by the instructor of the off-pattern courses. Changes in this examination schedule are not permitted except by formal approval of the instructor’s College Dean.

HOMEWORK: Homework related material including solutions of some problems will be posted on UNMLearn. The general rule is that homework assigned in one week is due the first class of the following week.

Class Week Topics Homework (due Monday following week)
1. Aug 21 The axioms of Incidence Geometries (IGs). HW1: p.14/ 1

2. Aug 28


Proofs of basic facts; common errors in proofs.

Examples and "non-examples" of IGs.

HW2: see UNMLearn (due Sep. 6)

3. Sep 4



Axiomatic systems - consistency, completeness, independence.

Parallel postulates - hyperbolic, elliptic and Euclidean axioms and their independence.

Hyperbolic, elliptic and Euclidean IGs and respective models (finite models,

Cartesian plane, models for Hyperbolic I.G - upper halpf-plane, Poincare and Klein disk models).

Dual IG. Affine plane, projective plane.

4. Sep 11


Axioms of betweennness - line separation property, plane

separation property.

HW3: p.14-15/2, 5.
5. Sep 18 Pasch's theorem. Problems HW4: UNMLearn
6. Sep 25 Angles and their interior. Betweennees of rays. The crossbar theorem. HW5: UNMLearn
7. Oct 2 Convex sets - intersection of convex sets. Convex polygons. HW6: p.21/ 2, 3, 5.

8. Oct 9



Characterization of convex polygons through semiparallel property,

diagonal splitting of a convex set. Jordan's theorem.

Measuring segments. Measuring angles.

HW7: p.21/4 p. 28/ 1, 2

9. Oct 16




Congruence of Triangles (SAS axiom) - ASA, SSS, AAS theorems,

isosceles triangles, angle bisector. Triangle related inequalities: exterior angle inequality,

sum of two angles in a triangle is less than 180, scalene inequality, triangle inequality.

Dropping a perpendicular. Parallel lines - alternate interior angles theorem.

HW8: UNMLearn
10. Oct 23 Neutral Geometries. Problems. Distance of a point to a line no due homework (see UNMLearn for practice problems)
11. Oct 30

Exam 1 (Monday)

Circles and lines tangency, inscribed circle.

HW9: p. 46/ 2, 3, 4

12. Nov 6


Hinge theorem. Tangency and intersection of two circles - principle of continuity.

Defect of a triangle, the Saccheri-Legendre theorem.

HW10: UNMLearn

13. Nov 13






Constructions using the Euclidean tools. Reflection across a line.

Existence pf parallel lines in N.G. - inconsistency of the Elliptic Axiom with the axioms of N.G.

Axioms equivalent to the EPP (Euclidean Parallel Postulate) - (1) Euclid's 5th, (2) Alternate

Interior Angels, (3) Proclus' Postulate, (4) Playfair's Postulate, (5) Common Perpendicular Postulate,

(6) Transitivity Postulate, (7) Triangle Defect Postulate. Proofs of the equivalences of EPP with

(1), (2), (3), (4), (5), (6); proof of EPP implies (7).

HW11: UNMLearn

14. Nov 20




Equivalence of the Triangle Defect Postulate (Angle Sum Postulate) and the EPP.

(8) The weak Angle Sum Postulate. Equivalence of the EPP and (8).

(9) Clairot's Postulate. Equivalence of the EPP and (9).

Similar Triangles. (10) Wallis' Postulate and its equivalence with the EPP - the AAA construction lemma.

(11) The equidistance Postulate and its equivalence to EPP.

The parallel projection theorem and side splitter theorem.

The fundamental theorem characterizing similar triangles in Euclidean Geometry.

HW12: UNMLearn

(see UNMLearn for practice problems)


15. Nov 27


The side-splitter lemma. The midsegment (of a triangle) property.

The Pythagorian theorem. Fundamental properties of N.G. - validity of either EPP or HPP; isometries,

categoricity, AAA congruence in H.G. Concurrence of the perpendicular bisectors, the altitudes and the medians.

Ceva and Minelaus' theorems.

Exam 2 (Friday)

16. Dec 4

Applications of similar triangles in E.G.: measure of arcs and inscribed angles; area of a triangle, complexes

and their areas. Trigonometric functions – the law of cosines, the law of sines.

11-Dec Final Exam Monday, December 11, 5:30pm - 7:30pm in SMLC-124 Final Exam Study Guide