Solid Geometry

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Page 319 - A plane surface, or a plane, is a surface in which, if any two points are taken, the straight line joining these points lies wholly in the surface.
Page 269 - If two planes are perpendicular to each other, any line in one of them, perpendicular to their intersection, is perpendicular to the other.
Page 262 - Theorem. The acute angle which a line makes with its own projection on a plane is the least angle which it makes with any line in that plane. Given the line AB, cutting plane P at 0, A'B' the projection of AB on P, and XX' any other line in P, through 0.
Page 283 - Problem. Through a given point to pass a plane parallel to a given plane. Solution. Draw two intersecting lines in the given plane. Through the given point draw two lines parallel to these lines, thus determining the required plane. PROPOSITION XXXI. 380. Problem. Through a given point to draw a line perpendicular to a given plane : (1) the point being without the plane, (2) the point being in the plane.
Page 285 - If a polyhedron is such that no straight line can be drawn to cut its surface more than twice, it is said to be convex; otherwise it is said to be concave. Unless the contrary is stated, the word polyhedron means convex polyhedron.
Page 270 - Theorem. If each of two intersecting planes is perpendicular to a third plane, their line of intersection is also perpendicular to that plane. Given two planes, Q, R, intersecting in OP, and each perpendicular to plane M.
Page 311 - The lateral area of a regular pyramid equals half the product of its slant height and the perimeter of its base. For in the above theorem, let B' = 0; then s' and p
Page 254 - I, prop. XII 3. .'.by folding A ACP over AC as an axis, it can be brought to coincide with A ACP'. 57 339. Definitions. A line is said to be perpendicular to a plane when it is perpendicular to every line in that plane which passes through its foot, — ie the point where it meets the plane. The plane is then said to be perpendicular to the line. If a line meets a plane, and is not perpendicular to it, it is said to be oblique to the plane. COROLLARIES. 1. If a line is perpendicular to each of...
Page 315 - A polyhedron is called a prismatoid if it has for bases two polygons in parallel planes, and for lateral faces triangles or trapezoids with one side common with one base and the opposite vertex or side common with the other base.
Page 253 - O'A', OB = O'B', AB = A'B'. I, prop. XXIV 7. .'. A ^0 ^ A A'B'O', and Z ^fOB = Z ^'0'^'. I, prop. XII ' 2. THE RELATIVE POSITION OF A LINE AND A PLANE. PROPOSITION VI. 338. Theorem. If a line is perpendicular to each of two intersecting lines, it is perpendicular to every other line lying in their plane and passing through their point of intersection.

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