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1. When is a line perpendicular to a plane? Show that a line at right angles to each of two intersecting lines is perpendicular to their plane.

2. How is the angle between two planes measured? Prove that the intersection of two planes perpendicular to a given plane is also perpendicular to that plane.

3. The volume of any prism is equal to the area of the base into the altitude. How is this applied to a similar theorem of the cylinder?

4. Define symmetrical spherical triangles and prove them equivalent.

5. The radii of the bases of a right circular cylinder and right circular cone are the same as a sphere's, say R. The altitudes of the cone and cylinder are 2R. Compare the volumes of the cylinder, cone, and sphere.


1. Prove the two propositions relating to the sum of the interior angles of a convex polygon, and the sum of the exterior angles formed by producing each side in one direction.

2. In a circle the greater chord subtends the greater arc, and conversely.

3. When is a line said to be divided harmonically? From the point P without a circle a secant through the center is drawn cutting the circle in A and B. Tangents are drawn from P and the points of contact connected by a line cutting AB in Q. Show that P and Q divide AB harmonically.

4. Derive an expression for the area of a regular polygon.

5. When two sides of a triangle are given at what angle must they intersect if the area shall be maximum? Prove your answer.


1. Enumerate and define the various sorts of symmetry. Show that two lines symmetrical with respect to a center are equal and parallel.

2. When two parallel lines are cut by any transversal, the alternate interior angles are equal.

3. The product of two sides of a triangle is equal to the perpendicular upon the third side into the diameter of the circumscribed circle. How may this be stated as a theorem in areas?

4. Given two similar polygons, construct a third similar to either and equivalent to their sum.

5. Define similar polygons. Triangles whose sides are proportional are similar.


I. Two angles not in the same plane with sides parallel and similarly directed are equal, and their planes are parallel.

2. Having cut a pyramid by a plane parallel to its base, show (a), the edges and altitude to be divided proportionally, and (b), the section made to be a polygon similar to the base. Apply the theorem to a cone.

3. Define symmetrical spherical triangles and show that they are equivalent.

4. Suppose a sphere of radius 5; upon a section of the sphere, at a distance 3 from the center, as a base, a cone is drawn whose elements touch the sphere; find the surface and volume of the cone.


1. (a) If a straight line is perpendicular to one of two parallel planes, it is perpendicular to the other. (b) All planes through a line perpendicular to a plane, are perpendicular to that plane.

2. Rectangular parallelopipeds having two dimensions equal are to each other as the third.

3. Two spherical surfaces intersect in a circumference whose plane is perpendicular to the line of centers, and whose center lies in that line.

4. Compute the volumes and surfaces of the regular tetrahedron and cube in terms of the edge E.

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