By A. P. Kiselev
This is often an English translation of a classical Russian grade school-level textual content in airplane Euclidean geometry. (The reliable geometry half is released as Kiselev's Geometry / ebook II. Stereometry ISBN 0977985210.) The booklet ruled in Russian math schooling for numerous a long time, was once reprinted in dozens of thousands of copies, prompted geometry schooling in jap Europe and China, and remains to be lively as a textbook for 7-9 grades. The ebook is customized to the trendy US curricula through a professor of arithmetic from UC Berkeley.
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Extra resources for Kiselev's Geometry, Book 1: Planimetry
The congruence of the triangles implies: ZABF = ZCBF. C / A \F A ¶ C Figure 67 Figure B 68 65. Problem 4. From a given point C on the line AB, to erect a perpendicular to this line (Figure 68). On both sides of the point C on the line AB, mark congruent segments CD and CE (of any length). Describe two arcs centered at D and E of the same radius (greater than CD) so that the arcs intersect at a point F. The line passing through the points C and F will be the required perpendicular. Indeed, as it is evident from the construction, the point F will have the same distance from the points D and F; therefore it will lie on the perpendicular to the segment AB passing through its midpoint Since the midpoint is C, and there is only one line passing through C and F, then FC I DE.
For example, in the theorem "if central angles are congruent, then the corresponding arcs are congruent" the hypothesis is the first part of the theorem: "if central angles are congruent," and the conclusion is the second part: "then the corresponding arcs are congruent;" in other words, it is given (known to us) that the central angles are congruent, and it is required to prove that under this hypothesis the corresponding arcs are congruent. " 30. The converse theorem. The theorem converse to a given theorem is obtained by replacing the hypothesis of the given theorem with the conclusion (or some part of -the conclusion), and the conclusion with the hypothesis (or some part of the hypothesis) of the given theorem.
Why? 47. Define a circle. 48. " Is the converse proposition true? Why? 49. In the proposition from §10: "Two arcs of the same circle are congruent if they can be aligned so that their endpoints coincide," separate the hypothesis from the conclusion, and state the converse proposition. Is the converse proposition true? Why? 50. In the theorem: "Bisectors of supplementary angles are perpendicular," separate the hypothesis from the conclusion, and formulate the converse proposition. Is the converse proposition true?