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I. 1. PROVE that two parallel lines are always in the same plane.
2. Prove that the sum of the plane angles, which form a solid angle, is always less than four right angles. (This theorem is sometimes stated thus : The sum of the face angles of a polyhedral angle is less than four right angles.)
3. Prove that parallel sections of a pyramid are similar polygons. What proposition relating to the volumes of pyramids is proved by aid of this proposition ? (State, but do not prove.)
4. Prove that the sum of the angles of a spherical triangle is greater than two right angles.
5. A spherical triangle has angles of 75°, 94°, and 91°; what is its area in degrees? How large a portion of the surface of the sphere does it cover ?
6. The surface of a sphere is 31.17 square feet; what is the surface of another sphere having three times the volume of the former ?
II. 1. Define a Plane, a Prism, a Great Circle. How many faces has a parallelopiped ? How many edges ? How is the angle between two planes measured ?
2. Prove that if two planes are perpendicular to a third plane, their line of intersection is also perpendicular to the third plane.
3. Prove that the section of a pyramid made by a plane parallel to the base is a polygon similar to the base.
4. Prove that a triangular pyramid is a third part of a triangular prism of the same base and altitude.
5. Prove that the sum of the angles of a spherical triangle is greater than two right angles. 6. Given the radius of a sphere = 2 inches.
= 2 inches. Compute the volume and convex surface.
1. If two planes are perpendicular to each other, the line drawn in one plane perpendicular to the common intersection is also perpendicular to the second plane.
2. The sum of all the plane angles which form a solid angle is always less than four right angles.
3. The solidity of a triangular prism is the product of its base by its altitude. Prove; and then show briefly how this theorem is made use of in finding the volume of a cylinder. Give the formula to express that volume.
4. Define similar polyhedrons. Prove that similar prisms, or pyramids, are to each other as the cubes of their altitudes.
5. Prove that if two spherical triangles on the same sphere, or on equal spheres, are equilateral with respect to each other, they are also equiangular with respect to each other.
6. The length of a perfectly round log of wood is 20 feet, and the diameter of each end is 12 feet. Find: (a.) Its convex surface. (6.) The surface of the greatest sphere which can be cut out of it. (c.) The volume of this sphere. IV. 1. Prove that the intersections of two parallel planes with any third plane are parallel lines. Define parallel planes.
2. Planes are passed through a pyramid parallel to its base; prove that the sections formed are similar polygons, and that these polygons are to each other as the squares of their distances from the vertex.
3. What are the regular polyhedrons ? How many faces has each ? how many vertices ? how many edges ? What are the faces in each case ?
4. A spherical triangle being given, to construct its polar. Prove the relations that exist between the sides and angles of a spherical triangle and those of the polar triangle.
5. The surface of a sphere is given, to find the surface of a sphere whose volume is five times as great.
6. A right cylinder and a right cone have the same circular base and the same altitude; compare their volumes. Compare with these the volume of a sphere having the same radius as the base of the cone.
1. Prove that oblique lines drawn from a point to a plane, at equal distances from the perpendicular, are equal; and that of two oblique lines unequally distant from the perpendicular the more remote is the greater. As a corollary to this theorem, show how a perpendicular may be drawn to a plane from a given point without the plane.
2. Prove that two straight lines, comprehended between three parallel planes, are divided into parts which are proportional to each other.
3. Prove that the sum of any two of the face angles of a triedral angle is greater than the third.
4. By what may a right cone be considered to be generated ? To what is the area of its convex surface equal ? To what is its solidity equal ? Compare the solidity of a right cone with that of a right cylinder, when both solids have the same altitude, and the radius of the base of the cylinder is double that of the base of the cone.
5. Prove that the sum of the sides of a spherical triangle is less than four right angles, and that the sum of the angles is greater than two right angles.
6. Prove that every triangular pyramid is one third of a triangular prism having the same base and altitude.
1. How do you find the co-ordinates of the point where two given lines intersect ?
2. Find the vertices of a triangle of which the sides are 2x + 4y +7= 0, 2x + y2 = 0, 220 = 0
2y + 1 3. Draw the lines just given and find the angles of the triangle they form.
4. What curve is represented by each of the following equations ? (i.) 22 + y2 + 4y = 0. (ii.) 9x2 + 25y2 400. (iii.) y -- 7x. (iv.) 16y2 — 9x2 + 36 = 0. Find the points at which each of these curves cuts the axes of co-ordinates.
5. Explain briefly how to construct a conic section when you have given the eccentricity (Boscovich's ratio), and the distance from the directrix to the focus. Take, for example, the eccentricity = 5, and the distance from the directrix to the focus 21
6. Find the equation of a conic section when the directrix is the axis of ordinates, and a perpendicular from the focus on the directrix is the axis of abscissas. Take, for example, the same data as are given in the preceding question.
Find what this equation becomes if transformed to a new set of axes parallel to the former and passing through the centre of the curve.
7. What is the locus of a point whose distance from a