Elementary Geometry Volume 1 (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1868 Excerpt: ...Def. 31. A four-sided figure which has only one pair of opposite sides parallel is called a trapezium. The following exercises on parallelograms are important and should be thoroughly worked out by the student. Exercises On Parallelograms. 1. Prove that the diagonals of a rectangle are equal to one another. 2. The diagonals of any parallelogram bisect one another. 3. The diagonals of a rhombus bisect one another at right angles. 4. Shew that a trapezium may be divided into a parallelogram and a triangle. 5. Or into a rectangle and two right-angled triangles. 6. If two straight lines be drawn bisecting one another, and their extremities be joined, the figure so formed will be a parallelogram. 7. Given that a four-sided figure has its opposite sides equal, prove that it must be a parallelogram. 8. The straight lines which join the extremities of equal and parallel straight lines towards the same parts are themselves also equal and parallel. SECTION VI. THE EQUIVALENCE OF FIGURES. By equivalent figures are meant figures whose areas are equal although the figures may be of different shapes, and therefore not conceivable as superposed on one another. Thus a circular field may be as large as a square one, or a triangular piece of paper as large as a rectangular piece, and in such cases these figures would be called 'equivalent. The consideration of equivalent figures is an important part of Geometry. Def. 32. The altitude of a parallelogram is the perpendicular distance between one side which is called the base and the side opposite to it. Thus in the figure at the side the perpendiculars DE, EG, or Cff, which are equal (by Theorem 21) since DEGF, DEHC are parallelograms, are each of them the altitude of the parallelogram ABCD, AB being the base. Def. 33. The alt...