Geometry: A High School Course

Serge Lang needs no introduction as a mathematician or a writer of mathematics. Some of his books are excellent and some are less well written. This book is well written- it is clear, it is logical, and it tries to explain/motivate the important ideas of the book. I used it to teach my daughter geometry and found it well organized with an adequate number of exercises. What I liked best about it was that one can consider it as an introduction to mathematical thinking. By that I mean it tries to teach one not just some facts about geometry, but how mathematicians think about geometry, how mathematicians go about finding theorems in geometry and then how mathematicians go about proving those theorems. It is an excellent book for someone who wants to learn about what mathematics in a way so that they can understand and appreciate rigorous proofs.

If I didn't need this book immediately for the geometry class I'm currently teaching, I would exchange it. The content is fine, but, I didn't pay $45 for a "new" book only to receive a book with a big gash in the bottom. Amazon has been very good about delivering quality, undamaged stuff, but not this time. The box and packing material was undamaged so I can only assume it happened pre-shipping during handling. Not cool. It looks like it took some force to cut into the cover board and first few pages. I'm saying something here because I can't find anywhere else on Amazon to make my comment.

Is really very good product is just to expected to nongraduate course.

Perfect for up and coming high school students. Good exposition at an appropriate level. The flow of the mathematical logic is clear. The writing is clear. This book is filled with substance, unlike the standard high-school text books being used in the US with color pictures and all that.

The best high school geometry textbook on the market.Tired of math textbooks written by committee? Tired of textbooks full of distracting colors, pictures, and garbage written in the margins?This book was written by a math professor at Yale in collaboration with just a single other person, a high school math teacher. The content knowledge of Lang is second to none. The book is rigorous but approachable, not requiring previous knowledge of the subject. It has stood the test of time: released in 1983, it still has not been surpassed.Lang deftly incorporates algebra and the coordinate plane when appropriate without conceding any of Euclid. This prevents the "geometry sandwich" (a typical high school career also includes algebra I, algebra II, and pre-calculus that are heavy in algebra) from eroding algebra skills.

I found this text interesting because it does not follow the standard approach to high school geometry. In a standard high school geometry text, the material is developed from Euclid's postulates (axioms) and common notions in the manner that he did in the Elements, albeit in modern language. These books generally cover much the same material, although some include right triangle trigonometry and transformations. The authors of this text choose to introduce their own postulates, which leads to a quite different development of the material and alternative ways of proving standard theorems. Having previously read a standard development of geometry, I found reading it developed in an alternate way fascinating.The authors omit some topics in a standard geometry course, including inequalities, theorems about tangents, secants, and chords of circles, and concurrence theorems. The presentation of other topics is truncated. Instead, they include material on dilations, vectors, the dot product, transformations, and isometries. Right triangle trigonometry is not included. Another way in which this text differs from standard texts is that proofs are written in paragraph form, which is standard practice among mathematicians, rather than in the two column statement and reason format favored by geometry teachers for pedagogical reasons. I should caution you that some of the terminology and notation is also nonstandard, which could pose problems for you in your subsequent reading or examinations. Consequently, I recommend that you read a standard text before working through this one.The text begins with a discussion of lines and angles. Postulates are introduced. However, some statements initially stated as postulates about distance and parallel lines are later proved as theorems. Then coordinates are introduced, allowing the authors to use algebraic arguments throughout the text. From there, the authors cover area, the Pythagorean Theorem, the distance formula, circles, perpendicular bisectors, triangles, polygons, and triangle congruence. Dilations are used to explain similarity. Volume formulas are derived for some standard figures. The authors present fascinating geometric arguments that enable them to obviate the need to use calculus to find some of the limits involved in the derivations. The book concludes with nonstandard topics, including vectors, the dot product, transformations, and isometries. This material is the greatest strength of the text, which concludes with a proof that any isometry can be expressed as the composition of at most three reflections.The writing is generally clear, but there are errors. In one proof, there is a triangle whose three vertices are actually collinear. The authors do not distinguish between the Angle-Side-Angle congruence postulate for triangles and the Angle-Angle-Side Theorem for congruence of triangles, arguing that since the sum of the measures of a triangle is always 180 degrees, the measures of two angles of a triangle determine the third. Where this causes problems is that when they express that triangles are congruent, corresponding vertices do not necessarily match, which can be confusing.The problems in the text are both interesting and tractable. The problems in the final section of the text and those listed as Additional Exercises are more challenging. There are no answers to the problems in the text. A solution manual,  Solutions Manual for Geometry: A High School Course: by S. Lang and G. Murrow , written by Philip Carlson is available separately. Also, a problem involving similar triangles and another that hinges on the Side-Angle-Side congruence postulate for triangles are introduced before the relevant topics.I recommend using this text as a supplement to a standard course. That way you will be familiar with standard terminology and notation. You will also know what a standard course covers and the usual way in which the theorems are proved. That will help you appreciate the nonstandard material covered in this text and the alternative proofs that are presented.Alternatively, you could work through the text  Geometry  by Edwin E. Moise and Floyd L. Downs, Jr. That text, which is known for its challenging problems, is comprehensive enough to cover both the material in a standard course and much of the nonstandard material in this text.

I found this treatment to be an excellent introduction to geometry for students without much mathematical background. Students are taught how to think logically, but are not forced into the cookie cutter mold proof style that so many geometry courses use. I especially like that he introduces much of the interesting material as experiments and exercises so that the students get to discover geometry for themselves.

I teach Geometry with this book, and initially one of the reasons I wanted the Kindle was so that I could download this book. I downloaded the sample not longer after getting my Kindle, but I wasn't ready to pay the price for a number of reasons. At the time (about 3 months ago) it was about $17.00!!!! Now the Kindle version costs what I can get the hardcover for. This is ridiculous! What could have possibly happened in 3 months to nearly triple the Kindle version price?!? I am VERY disappointed.Needless to say, I am now NOT buying the Kindle version.By the way, the three stars is because I am giving Amazon 1 star for changing the price like this and I give the book itself 5 stars. That averages to 3.

Explanations clear and to the point

Los libros de Serge Lang son de los mejores que hay para aprender matemáticas. Estudiar con sus libros te hace disfrutar de las matemáticas, Lang (ya fallecido) era profesor y aplica su amplia experiencia didáctica en hacer fácil y ameno lo dificil. El libro está en inglés. 394 pags con algunas en blanco al final para tomar notas. Formato llevable cabe bien dentro de un bolso y no pesa demasiado. Este libro tiene otro complementario que son las soluciones.

Illuminante e non prolisso. Ottima collaborazione fra l'accademico e chi insegna

とても為になる、いい本です。内容も充実しています。直線とは何か？線分とは何かから始まります。