History of Mathematics 464 WI Sample Problems for Exam 1




  1. Carefully state any two ($2$) of the three famous ancient Greek construction problems, which allowed only an unmarked straightedge and a compass.


  2. Answer any two of the following questions:

    1. Prove in the manner of Euclid the following proposition (stated in modern language below):

      • I-6: If in a triangle two angles are congruent, then the sides opposite these angles are also congruent.


    2. Prove the following in the manner of Euclid. This was his first use of the $5^{th}$ postulate in a proof. Be sure to point out that use:

      • I-29: A straight line falling on parallel straight lines makes the alternate interior angles equal to one another.
        MATH

    3. Given the construction below in which $ACB$ is a semicircle with center $O$ and $AEC$ is a semicircle with center $D$, with everything else as marked. Prove the Great Theorem of Hippocrates of Chios as he did it:

      • Theorem: Lune $AECF$ is quadrable (squarable).
        MATH

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