| Math 233 - Midterm | October 11, 2001 | |
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1. (20) Identify each of the following statements (there will be ten) in the language of Euclidean Geometry as either an axiom (A), a definition (D), or a theorem (T). 2. (8) What are the only two instruments (other than a writing implement) that are allowed to be used in classical geometric constructions? 3. (5) Why can't you have a triangle with sides of length 2, 4, and 8? (Hint: Look through the statements in question 1.) 4. (32) Prove the "Carpenter's Theorem": Let ABCD be a parallelogram (remember this means that opposite sides are not only parallel, but are congruent as well, that is AB = CD and AD = BC). The theorem then can be stated as follows: Theorem: If ABCD is a parallelogram and AC = BD, then ABCD is a rectangle. (Refer to the figure ABCD). 5. (25) Given that the sum of angles in a triangle is 180 degrees, find the sum of angles in a convex octagon (8-sided polygon), and prove your result! 6. (10) a) Use Heron's Theorem to find the area of a triangle with sides of length 4, 5, and 8. |