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adjacent altitude applied arranged base bisect called chord circle circumference coincide common construction contained Corollary corresponding curve demonstration described diagonals diameter diedral difference direction distance divided draw edges ends equal equal angles equally distant equivalent EXERCISES extend extreme faces figure formed four geometry given line given point gles greater half Hence homologous hypothesis included inscribed intercepted intersection isosceles Join length less limit means measured meet method opposite sides parallel parallel lines parallelogram pass perimeter perpendicular plane position principle prism problem produced proportional proposition proved pyramid quadrilateral radii radius ratio reason rectangle regular polygon respectively equal right angles secant sides similar solid sphere square straight line student Suppose surface tangent tetraedrons theorem third triangle triedral vertex vertices whole
Side 98 - If two triangles have two sides of the one respectively equal to two sides of the other, and the included angles unequal, the triangle which has the greater included angle has the greater third side.
Side 141 - The square described on the hypothenuse of a right-angled triangle is equivalent to the sum of the squares described on the other two sides.
Side 263 - The area of the surface of a sphere is equal to the area of the...
Side 258 - The sum of the angles of a spherical triangle is greater than two and less than six right angles ; that is, greater than 180° and less than 540°. (gr). If A'B'C' is the polar triangle of ABC...
Side 137 - The squa/re described on the difference of two straight lines is equivalent to the sum of the squares described on the two lines, diminished by twice the rectangle contained by the lines.
Side 227 - ... the two planes are equal polygons. Each side of one of the sections is parallel to the corresponding side of the other section, since they are the intersections of two parallel planes by a third. Hence, that portion of each side of the prism which is between the secant planes, is a parallelogram. Since the sections have their sides respectively equal and parallel, their angles are respectively equal. Therefore, the polygons are equal. 674. Corollary — The section of a prism made by a plane...
Side 237 - The volume of any prism is equal to the product of its base by its altitude. Let V denote the volume, B the base, and H the altitude of the prism DA'.
Side 191 - Theorem. — The intersections of two parallel planes by a third plane are parallel lines. Let AB and CD be the intersections of the two parallel planes M and N, with the plane P.