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# Basic Geometry - Manual for Teachers by George David Birkhoff, Ralph Beatley

By George David Birkhoff, Ralph Beatley

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You would need to know that two triangles that have their corresponding sides proportional are similar. That is, you would need Princi- ple 8, Case 3 of Similarity. The teacher should consider all these exercises together and note that they establish certain familiar and very important ideas concerning proportion. numerical. The content of these exercises is essentially Hence the treatment is numerical throughout and purposely avoids all reference to "taking a proportion by alternation," or "by composition," or the like.

If books on geometry always took pains to write their rropoeitions so that a literal interchange of this sort would yield the converse, there would still be the problem of training pupils to frame the converses of non-mathematical propositions, where the separation of hypothesis and conclusion calls for considerable discrimination. The pupils might as well face this problem in geometry, particularly since a studied effort to avoid it would result in very stilted statements of many theorems. " For while it is possible to regard the wording of certain propositions in such a way that the hypothesis, or the conclusion, or both, shall seem to have more than one part; and while it is possible then to devise all the partial converses that can result from Interchanging one or more of these partial hypotheses and conclusions, It is neither necessary nor desirable to do this.

38 - plied by Principle 3, the Principle of Angle Measure, as stated on page 47. Consequently, BASIC GZOMWTRY Is not troubled by the question of "hypothetical constructions" that plagues other geometries. When we set out to prove the first theorem that involves the bisector of an angle, we do not need to puzzle over the problem of how to demonstrate the constructibility of the bisector without making use of the theorem we wish to prove, or - foiled in that - to satisfy our consciences that it will be all right to prove the theorem first and demonstrate the existence of the bisector later.

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