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To prepare oneself to teach high school geometry, one should do two things: (1) learn considerably more elementary geometry than is covered in the high school course itself, (2) become acquainted with modern notions of geometric structure.
The necessity of the first requirement follows from the well-known pedagogical fact that one cannot give a strong, interesting, and inspiring course in a subject without knowing a good deal more about the subject than that which one teaches. The necessity of the second requirement follows from the fact that one cannot attain a true appreciation of present-day geometry without knowing something of the evolution of geometric structure from the
days of the ancient Greeks up to modern times.
The above observations led to the planning of this text, wherein Part I is devoted to modern elementary geometry and Part II to the evolution of geometric structure. It was felt that Part I should be a sequel to the basic geometry of the high school course and introduce the reader to some of the important modern extensions of elementary geometryextensions that have largely entered into the mainstream of mathematics. On the other hand, Part II should treat notions of geometric structure via the non-Euclidean revolution that arose in the first half of the nineteenth century. It is hoped that such a two-part text can respectively furnish suitable introductions to (so-called) college geometry on the one hand and to non-Euclidean geometry on the other. It is through these two avenues that teachers should better be able to prepare themselves to rescue high school geometry from the doldrums of its present plight, and to rebreathe into it some of its inherent romance, beauty, and excitement.
The text is adaptable to either a two-semester or a one-semester course. For a two-semester course, which is much the better option if time permits, essentially the whole text can be covered, with the exception of most or all of the optional sections. Greater discrimination will be required for a one-semester course