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1. Enumerate and define the different kinds of quadrilaterals. Show that if the middle points of the non-parallel sides of a trapezoid be joined by a straight line, this line is parallel to the other sides and equal to one-half their sum.

2. Give constructions for the inscribed, escribed, and circumscribed circles of any triangle.

3. (a) Two polygons are similar when composed of the same number of triangles, similar each to each and similarly placed. (b) When the areas of two similar polygons are in the ratio of m to n, in what ratio are the homologous sides?

4. In the triangle whose sides are a, b, and c, determine the segments of each side made by the bisector of the opposite angle.

5. A circle may be inscribed in and circumscribed about a regular polygon.


1. What are equal angles, complementary angles, supplementary angles? Show that angles whose sides are parallel or perpendicular are equal or supplementary.

2. If through a point without a circle a secant be drawn, the product of the whole secant and the part without the circle is constant, in whatever direction the secant is drawn. State the corresponding theorem when the point is inside the circle.

3. What is it to measure a quantity? When are two magnitudes commensurable; or incommensurable? When a line is drawn parallel to any side of a triangle, show that it divides the other two sides proportionally.

4. Two triangles have sides a, b, and c, and a', b', and c'; the angles opposite a and a' are equal; find the ratio of their areas.

5. Show that in a series of regular polygons of equal areas, the perimeters decrease as the number of sides increase.


1; (a) What is the projection of a line upon a plane? Show that the projection of a straight line is a straight line. (b) What is the angle a line makes with a plane? Show it is the least angle the line makes with any line drawn in the plane through its foot.

2. A regular triangular pyramid has a for its altitude and each side of its base. Find the area of a section parallel to the base and distant a from the vertex, and get the volume of the original pyramid.

3. Define similar cylinders of revolution. Show in such cylinders that the lateral and total areas are as the squares of homologous dimensions, the volumes as the cubes.

4. A sphere may be inscribed in, and circumscribed about any tetrahedron.


1. Define a plane, dihedral angle, polyhedral angle, polyhedron, regular polyhedron, pyramid, cylindrical surface, frustum of a cone, spherical polygon, lune, and zone.

2. (a) Two planes perpendicular to the same line are parallel. (b) Two lines perpendicular to the same plane are parallel.

3. Two prisms are equal when three faces including a trihedral angle of one are equal and similarly placed to three faces including a trihedral angle of the other.

4. What is the spherical excess of a spherical triangle? Prove the area of a spherical triangle is to the area of the sphere as its spherical excess is to eight right angles.


1. If two lines are parallel, the angles made with them by a transversal are equal in pairs. State and prove the converse proposition.

2. If a tangent and a secant are drawn from a point to a circle, the tangent is a mean proportional between the whole secant and the external segment.

3. The four tangents drawn to two circles from any point in their common chord are equal.

4. Construct a square whose area is 3 times that of a given square.

5. Circumferences of circles are as the radii, areas as the squares of the radii.

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