Math 402, Assignment 8, Spring 2008

Due Wed. Mar 26 (at the beginning of class)

A polytope is convex if, given any edge AB, all the (other) vertices lie on the same side of the line AB. For this homework, you may use the following fact: Let X be a vertex of a convex polytope, and let Y be any point on the polytope. Then the ray XY lies inside the angle formed by the edges adjacent to X.
  1. A triangle (or polygon) is regular if each side has the same length, and each interior angle has the same measure. Consider triangles on a sphere with total surface area S. In terms of S, what is the area of each of the following three triangles? If the sphere were entirely covered with triangles of each of the following types, how many would fit? Show your calculations.
    1. A regular triangle with interior angle measure 120 degrees.
    2. A regular triangle with interior angle measure 90 degrees.
    3. A regular triangle with interior angle measure 72 degrees.

  2. Let s be the sum of the interior angles of a convex poltyope of area A. In class, we showed that
    s = pi + KA
    for triangles, where K = 0 for the Euclidean plane, K = 1/r2for a sphere of radius r, and K = -1/r2 for a hyperbolic plane of radius r.

    Prove that, for convex quadrilateral,

    s = 2 pi + KA.
    Hint: draw a diagonal.

  3. Let ABCD be a convex quadrilateral on any space which satisfies the five axioms. Assume that angles B and C are both perpendicular.
    Prove that if AB and CD are congruent, then angles A and D have the same measure.
    Hint: Draw the diagonals and look for congruent triangles, or look for a (reflection) symmetry.

  4. Let ABCD be a convex quadrilateral on any space which satisfies the five axioms. Assume that angles B and C are both perpendicular.
    1. Prove that if AB is longer than CD, then the measure of angle D is greater than the measure of angles A.
      Hint: follow the proof of the angle-side inequality in your notes.
    2. Prove that if the measure of angle D is greater than the measure of angles A, then AB is longer than CD.
      Hint: follow the proof of the side-angle inequality in your notes.