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1. Define a dihedral angle, the plane angle of a dihedral angle, a polyhedral angle, symmetrical polyhedral angles, a spherical angle, a spherical polygon, a polar triangle.

2. If two angles, not in the same plane, have their sides respectively parallel and lying in the same direction, they are equal and their planes are paraliel.

3. Sections of a prism made by parallel planes are equal polygons.

4. In two polar triangles each angle of the one is the supplement of the opposite side in the other. What is the spherical excess of the polar triangle of the spherical triangle whose sides are 50°, 90°, and 120°?

5. An equilateral triangle revolves about one of its altitudes. If its altitude is 3h, find the volumes and total surfaces of the solids generated by the triangle, the inscribed circle, and the circumscribed circle.


1 Define a polyhedron, a tetrahedron, a prism, a parallelopiped, a pyramid, a cone, a cylinder, a sphere.

2. If a straight line is perpendicular to each of two straight lines at their point of intersection, it is perpendicular to the plane of those lines.

3. If a pyramid is cut by a plane parallel to its base, (a) the edges and altitude are divided proportionally; (b) the section is a polygon similar to the base.

4. A sphere may be circumscribed about any tetrahedron. 5. Compare the volume of a right prism 6a ft. long, the base of which is a square with a side 2a ft. long, with the volume of the largest cylinder, sphere, pyramid, and cone which can be made from it.


I. Two angles whose sides are parallel each to each are either equal or supplementary. When will they be equal, and when supplementary?

2. An angle formed by two chords intersecting within the circumference of a circle is measured by one-half the sum of the intercepted arcs.

3. A triangle having a base of 8 inches is cut by a line. parallel to the base and 6 inches from it. If the base of the smaller triangle thus formed is 5 inches, find the area of the larger triangle.

4. Construct a parallelogram equivalent to a given square, having given the sum of its base and altitude. Give proof.

5. What are regular polygons? A circle may be circumscribed about, and a circle may be inscribed in, any regular polygon.


I Define homologous points, homologous lines, homologous angles, equal figures, equivalent figures, symmetrical figures, isoperimetrical figures.

2. Deduce expressions for the sum of the interior angles and the sum of the exterior angles of a polygon of n sides.

3. If through each of the vertices of a given triangle a line be drawn parallel to the opposite side, a new triangle will be formed equal to four times the given triangle.

4. The square described on the hypothenuse of a right triangle is equivalent to the sum of the squares described on the other two sides.

5. Of all triangles having the same base and equal perimeters, the isosceles triangle is the maximum.


I. When is a line perpendicular to a plane? What is meant by the angle a line makes with a plane? When is one plane perpendicular to another plane? What is meant by the angle between two curves passing through the same point?

2. If two straight lines are intersected by three parallel planes, their corresponding segments are proportional.

3. A triangular pyramid is equal to one-third of the prism having the same base and altitude.

4. A sphere may be inscribed in any given tetrahedron. 5. Find the area and volume of a sphere of radius IO inches; also find the area of a spherical triangle upon it whose angles are 120°, 40°, and 110°, respectively.

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