
MA232A  Euclidean and NonEuclidean Geometry
Michaelmas Term 2015
Dr. David R. Wilkins

MA232A Michaelmas Term 2015
Syllabus for MA232A at the Annual Examination 2016, and
Sample Paper
 MA232A  Syllabus for Examination 2016
 MA232A  Sample Paper 2016
 Note that the final two pages contain a description of
Selected Propositions from Euclid. This is not
ideally suited to the questions that appear on the
sample paper, but is a verbatim reproduction of the
corresponding pages appended to the actual examination
paper. The Selected Propositions also gives
names to some of the propositions (SAS Congruence
Rule, Alternate Angles Theorem) that
match or resemble names attached to these results
in some Project Maths textbooks. Some propositions
whose statements are included in the Selected
Propositions are nevertheless included in the
formal list of examinable material, as candidates
are advised to be familiar with those propositions.
Recently noted Errata

Proof of Lemma 5.11, second paragraph:
“the angle between the vectors
n_{V} and n_{W}
is also a right angle”
changed to
“the angle between the vectors.
w and n_{W} is also a right angle”
(May 15, 2016)
 Proof of Proposition 5.14: corrections changing
r' to r'', and changing L'' to L',
at appropriate places (April 13, 2016)
 Statement of Corollary 8.3 corrected (May 16, 2016)
Note on Labelling of Congruence Rules and Other Theorems
The following table may assist in matching up labelled theorems
and congruence rules to propositions in Euclid.
Rule  Proposition 
SAS Congruence Rule  Euclid, Book I, Proposition 4 
SSS Congruence Rule  Euclid, Book I, Proposition 8 
ASA Congruence Rule  Euclid, Book I, Proposition 26 
SAA Congruence Rule  Euclid, Book I, Proposition 26 
Isosceles Triangle Theorem  Euclid, Book I, Proposition 5 
Vertically Opposite Angles Theorem  Euclid, Book I, Proposition 15 
Alternate Angles Theorem  Euclid, Book I, Proposition 27 and 29 
Corresponding Angles Theorem  Euclid, Book I, Proposition 29 
Pythagoras's Theorem  Euclid, Book I, Proposition 47 
Note that not all of the above are included in the
Selected Propositions for 2016.
To interpret congruence rule labellings: “SAS” stands for
“sideangleside” and encodes the fact that a first side
is incident on the angle which in turn is incident on the second side.
Note also that there is no valid “ASS” congruence rule!
Summary of Lectures
For an indication of what was been covered in lectures prior to
Study Week, consult the
Lecture Log.
Resource Pages
 Resources related to Euclid's Elements of Geometry
 Resources related to Gauss's General Investigations of Curved Surfaces
Curriculum
 Euclid's Elements of Geometry (6 weeks)
 We explored selected topics from the first six books
of Euclid's Elements of Geometry, including
in particular the following topics:
 in Book I, the Definitions, Postulates and Common Notions
 in Book I, the early propositions, including in particular
the Pons Asinorum taking note in particular
of places where proofs make unspecified assumptions
regarding existence of points of intersection and of
the homogeneity of the Euclidean plane;
 in Book I, rigidity of triangles, showing how triangles
are uniquely determined once appropriate lengths and
angles are given;
 in Book I, the theory of parallels (Proposition 27 to 33);
 in Book I, propositions (from Proposition 35 onwards)
concerning the areas of triangles
and parallelograms, and concern transformation of areas;
 in Book I, the proof of Pythagoras's Theorem (Proposition 47)
 the geometry of circles in Book III, up to Proposition 22;
 Eudoxus's Theory of Proportion, in Book V, covering the
basic definitions and Propositions 1, 2 and 8;
 Applications of the Theory of Proportion in
Propositions 1 and 2 of Book VI.
 Gauss's General Investigations of Curved Surfaces (4 weeks)
 This work by Gauss, originally published in Latin, gave
a considerable boost to the theory of curved surfaces, as not
much about them had been discovered before Gauss. Included
is the Theorema Egregium that ensures that
Gaussian Curvature (one of the measures of the curvature
of a surface at a point) is invariant under isometries
between surfaces. We aim to concentrate on sections 1 to 20.
 Geometry of the Hyperbolic Plane(1 week)
 We investigate 2dimensional hyperbolic geometry,
making use of the Poincaré
Disk model of the hyperbolic plane
Texts and other Resources on the Internet
Extensive use was made of online texts of mathematical
works that are freely available over the internet. Ideally
participants should be prepared to download selected texts themselves
for online viewing on desktop and laptop computers etc., maybe printing
off selected portions for further study. Note that Euclid's Elements of Geometry has a length of several hundred pages, so it may make
sense to restrict printing of texts to selected portions.
Photocopied portions of significant extracts may be distributed
from time to time.
Back to D.R. Wilkins: Lecture Notes
Dr. David R. Wilkins,
School of Mathematics,
Trinity College Dublin.