A Second Book in Geometry

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Brewer and Tileston, 1863 - Geometry - 136 pages
 

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Page 18 - After remarking that the mathematician positively knows that the sum of the three angles of a triangle is equal to two right angles...
Page 17 - Thus the proposition, that the sum of the three angles of a triangle is equal to two right angles, (Euc. 32. 1.) may be demonstrated, either in common language, or by means of the signs used in algebra. Let the side AB, of the triangle ABC, (Fig. 1.) be continued to D ; let the line BE be parallel to AC; and let GHI be a right angle.
Page 7 - С,) which is impossible, and therefore the angles cannot be altered. 95. Corollary. If the three sides of a triangle are respectively equal to the three sides of another triangle, the angles of one must be equal to those of the other, and the equal angles are enclosed in the equal sides. 96. Theorem. If the opposite sides of a quadrangle are equal, the quadrangle is a parallelogram. — Proof. If in the quadrangle А В С D (Fig.
Page 17 - ... that the maximum of polygons formed of given sides may be inscribed in a circle ; secondly, that the maximum of isoperimetrical polygons having a given number of sides has its sides equal ; and thirdly, that such a regular polygon is of smaller area than a circle isoperimetrical with it. 134. Theorem. The area of a triangle is found by multiplying the base by half the altitude. This theorem has been already proved (Art. 111). 135. We shall need the Pythagorean proposition, which implies all the...
Page 22 - ... these are self-evident truths. 51. By a self-evident step in reasoning, I mean the statement of the relation of one truth to another, or of the dependence of one truth upon another, when that dependence or that relation is itself a self-evident truth.
Page 9 - ... to make the remainder small enough to be neglected. 101. Definitions. The right angle, right triangle, legs, and hypotenuse, are defined in articles 14 and 17. 102. Theorem. The sum of the three angles of a triangle is equivalent to two right angles. This proposition has been proved in articles 26-31, 34-36, and 57-62. 103. Corollary. The sum of the two angles opposite to the legs of a right triangle is equivalent to one right angle. 104. Corollary. If an angle opposite a leg in one right triangle...
Page 4 - N, and this will give us MXQ : NXQ = NXP : NX Q. Thus from the self-evident truth of article 78 we find that the product of the means bears the same ratio to the product NXQ that is borne to it by the product of the extremes. And as it is self-evident that two quantities, bearing the same ratio to a third, must be equal to each other, we have proved that the product of the means is equal to that of the extremes. 82. Definition. When both the means are the same quantity, that quantity is called a...
Page 1 - PROPOSITION. 65. The Pythagorean proposition or theorem might be suggested in different ways. But in whatever way we were led to suspect that the square on the hypothenuse is equivalent to the sum of the squares on the legs, we should, in reflecting upon it, probably begin by drawing a right triangle with a square built upon each side. 66. We should inquire whether. the square on the hypothenuse could be divided into two parts that should be respectively equal to the other two squares.
Page 88 - If two triangles on the same sphere, or on equal spheres, are mutually equiangular, they will also be mutually equilateral. Let A and B be the two given triangles; P and Q their polar triangles. Since the angles are equal in the triangles A and B, the sides will be equal in.
Page 4 - Proof. For as the straight line has but one direction, and each of the parallel lines may always be considered as going in the same direction as the other, the difference of that direction from the direction of the third straight line must be the same for each of the parallel lines. 88. Corollary. If a straight line is parallel to one of two parallel lines, it is parallel to the other ; if at right angles to one of the two, it is at right angles to the other. 89. Theorem. If a straight line make...

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