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6 7 Plane and sImPle your basic theorem toolbox The Elements of Euclid ca. 325 265 B.C.E. long ago set the standard for mathematical rigour and, being a popular textbook ever since, much of its contents have been absorbed into our common cultural heritage. Over the course of thirteen books, Euclid built up a complex network of theorems of ever increasing depth, connected by logical arguments and rooted in some intuitive facts, called axioms or postulates. To be prepared for the rest of this book, start with the four simple results on the right and, following the arrows, deduce the theorems on the left. You also need to be able to recognise in a flash the two main levels of sameness of triangles. Two triangles are similar if they have equal angles. Since two angles in a triangle determine the third, you know that two triangles are similar if you can show that they share two angles. Two triangles are congruent if they have equal sides. This is the case whenever one of the five configurations of three sides and angles drawn solid below are present in both triangles. For example, in the diagram on the right below, the two gray triangles share one such configuration consisting of the two sides r and m and one right angle and are therefore congruent. Hence the two tangent segments s and t to the circle from the point outside have the same length.
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