MA1004: Geometry

School Cardiff School of Mathematics
Department Code MATHS
Module Code MA1004
External Subject Code 100405
Number of Credits 10
Level L4
Language of Delivery English
Module Leader Dr Timothy Logvinenko
Semester Autumn Semester
Academic Year 2014/5

Outline Description of Module

This module gives an introduction to elementary plane Euclidean geometry. We present this material in a way which emphasises axiomatic approach, logical thinking and rigorous proofs, as well as careful use of diagrams as an aid to understanding problems and finding solutions. In the latter half of the module we also introduce basic notions of spherical geometry, emphasising the differences between it and Euclidean geometry.

On completion of the module a student should be able to

  • Understand basic geometrical notions of lines, circles, triangles, distances, angles, tangents, bisects, isometries, similarities, etc.
  • Prove geometrical figures to be congruent
  • Being able to carry out classical geometrical constructions such as those of an inscribed and a circumscribed triangle
  • Visualise non-Euclidean geometries using the geometry on a sphere as an example
  • Compute areas and sums of internal angles of basic geometrical figures

How the module will be delivered

27 - 50 minute lectures

5 - 50 minute tutorial classes

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 50 hours private study including preparation of worked solutions for tutorial classes.

Skills that will be practised and developed

Constructing logical proofs based on a set of axioms and previously established results

Rigorous use of diagrams in the course of the proof

Developing geometrical intuition and visualisation in 2 and 3 dimensions

How the module will be assessed

Formative assessment is by means of regular tutorial exercises.  Feedback to students on their solutions and their progress towards learning outcomes is provided during lectures and tutorial classes.

The in-course element of summative assessment is based on selected problems on the tutorial sheets.

The major component of summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has two sections of equal weight.  Section A contains a number of compulsory questions of variable length but normally short.  Section B has a choice of two from three equally weighted questions.

Assessment Breakdown

Type % Title Duration(hrs)
Exam - Autumn Semester 85 Geometry 2
Written Assessment 15 Coursework N/A

Syllabus content

  • Introduction: Euclid's axioms. Distances and angles   
  • Isometries and congruences.                                              
  • Triangle congruences: SAS, ASA and SSS. Isosceles triangles.      
  • Perpendicular bisects. Distance from a point to a line.
  • Bisectors. Inscribed and circumscribed circles of a triangle.                                                    
  • Spherical geometry: intro. Lines and angles.                        
  • Parallel postulate. Sum of angles of a triangle / n-gon.                   
  • Area. Ceva's Theorem.                                                       
  • Spherical geometry: triangles on a sphere.                          
  • Similarity. Pythagoras theorem.                                         

Essential Reading and Resource List

Note on "Introduction to Geometry" – To be made available via Learning Central at the start of the module

Background Reading and Resource List

G.A. Jennings, Modern geometry with applications, Springer-Verlag

H.S.M. Coxeter, Introduction to Geometry, Wiley Classics Library, John Wiley and Sons

Euclid, Elements, Books I, III and IV, various publishers (the original treatise on the subject)


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