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Euclidean and Transformational Geometry: A Deductive Inquiry

Shlomo Libeskind
Jones and Bartlett
Publication Date: 
Number of Pages: 
[Reviewed by
Barbara E. Reynolds
, on

This is an engaging textbook for a college geometry course. It would be an appropriate text to use in a course for prospective middle and high school teachers. It covers, as the title suggests, both Euclidean and transformational geometries in some depth. For instructors requiring an introduction to axiom systems, the Appendix is a solid introduction to Basic Notions, including a set of axioms that are equivalent to Euclid’s Five Postulates. Using these Basic Notions, students can begin immediately to write careful synthetic proofs. For instructors who prefer a more intuitive approach, these Basic Notions can be woven into the course at later times as their students are prepared for more rigorous detail.

The book opens with a Prolog, which gives an overview of five intriguing problems. Throughout the text, the author refers back to these problems in various ways — and presents a number of solutions to each of these problems in the context of various geometric tools or strategies. For example, the Treasure Island Problem is first presented in the endpapers of the book. In the Prolog, students are invited to think about the problem and to look for a solution experimentally, possibly using The Geometer’s Sketchpad (GSP). By the end of Chapter 1, “Congruence, Constructions, and the Parallel Postulate,” the students are able to make and prove their first conjectures about the location of the treasure using ideas about perpendicular bisectors and right triangles. This problem appears again in the Exercises at the end of Section 5.1, “Reflections, Translations, and Rotations,” where the students are invited to develop a proof of the same conjecture using tools of transformational geometry. This problem is solved again using a composition of transformations in Chapter 6, and one last time in Chapter 7, using vectors in the complex plane.

A strength of this text is the careful attention the author gives to developing methods for writing proofs. The author encourages students to restate each theorem or conjecture making it clearer what is given and what needs to be proved. He gives repeated examples of proofs in which he lays out the plan of the proof, and then carries out the proof. There are many examples of well written proofs, followed by carefully structured Exercises where the students can practice writing their own proofs. It is easy to see how students using this text would become excited about developing proofs as a form of problem solving. The problems posed both in the body of the chapters and in the exercises are interesting and enticing.

This text is written at a level which makes it accessible — even engaging — to students who have a good background in high school algebra, but who may not remember much from their high school geometry course. Students who complete a course based on this text would be prepared to teach a high school geometry course that requires some writing of proofs. A student would also be well prepared to teach in a middle school mathematics program where children are asked to explain their reasoning.

Sr. Barbara E. Reynolds, SDS, is Professor of Mathematics & Computer Science at Cardinal Stritch University, Milwaukee, WI.


0     Prologue
       1.  The Treasure Island Problem
       2.  The Nine-Point Circle
       3.  Morley's Theorem
       4.  The Hiker's Path
       5.  The Shortest Highway
       6.  Steiner's Minimum Distance Problem
       7.  The Pythagorean Theorem

1.     Congruence, Constructions, and the Parallel Postulate
       1-1   Angles and Their Measurement
       1-2   Congruences of Triangles
       1-3   The Parallel Postulate and Its Consequences
       1-4   More on Construction

2.     Circles
       2-1   Basic Properties of Arcs, Central and Inscribed Angles
       2-2   Circles Inscribed in Polygons
       2-3   More on Constructions

3.     Area and the Pythagorean Theorem
       3-1   Areas of Polygons
       3-2   The Pythagorean Theorem
       3-3   The Distance Formula

4.     Similarity
       4-1   Ratio, Proportion and Similar Polygons
       4-2   Further Applications of the Side Splitting Theorem and Similarity
       4-3   Areas of Similar Figures
       4-4   The Golden Ratio and the Construction of a Regular Pentagon
       4-5   Circumference and Area of a Circle
       4-6   Other Recursive Formulas for Evaluating
       4-7   Trigonometric Functions

5.     Isometries
       5-1   Reflections, Translations, and Rotations
       5-2   Congruence and Euclidean Constructions
       5-3   More on Extremal Problems
       5-4   Similarity Transformation with Applications to Constructions

6.     Composition of Transformations and Transformation Groups
       6-1   In Search for New Isometries
       6-2   Composition of Rotations, The Treasure Island Problems and Other Treasures

7.     More Recent Discoveries
       7-1   The Nine-Point Circle and Other Results
       7-2   Complex Numbers and Geometry