Math 402, Assignment 11, Spring 2008

Due Wed. April 23 (at the beginning of class)

      Note: These problems are for any space with absolute geometry. The goal of this homework is to prove the following famous and amazing theorem, due to Legendre. If there exists a single triangle whose angle sum is 180 degrees, then every triangle has angle sum 180 degrees. Note that the converse is obvious, if every triangle has angle sum 180 degrees, then there exists a triangle whose angle sum is 180 degrees.

    1. Let ABC be a triangle. Pick a point D on the interior of segment BC.
      1. Assume that the angle sum of triangle ABC is 180 degrees. Prove that the angle sum of triangle ABD is 180 degrees.
        Hint: LST.
      2. Assume that the angle sum of triangles ABD and ACD are each 180 degrees. Prove that the angle sum of triangle ABC is 180 degrees.


    2. Let ABC be a triangle. Let D be the unique point on line BC so that line AD is perpendicular to line BC.
      1. Prove that if BC is the longest side then D is in the interior of segment BC.
      2. Show that D may not be in the interior of segment BC otherwise.

      1. Let ABC be a triangle whose angle sum is 180 degrees. Prove that there exists a right triangle whose angle sum is 180 degrees.
        Hint: Use problem 1 and 2.

      2. Let ABC be a right triangle whose angle sum is 180 degrees. Prove that there exists arbitarily large right triangles whose angle sum is 180 degrees.
        Hint: To begin, put together four triangles to make a triangle twice as big.

      3. Assume that there exists arbitrarly large right triangles whose angle sum is 180 degrees. Prove that every right triangle has angle sum 180 degrees.
        Hint: Expand the edges adjacent to the 90 degree angle. Use problem 1.

      4. Assume that every right triangle has angle sum 180 degrees. Prove that every triangle has angle sum 180 degrees.
        Hint: Use problems 1 and 2.

      5. Assume that there exists a triangle with angle sum 180 degrees. Prove that every triangle has angle sum 180 degrees.
        Hint: This should be very easy!