Miscellaneous Review Exercises

\begin{document} last edited 29apr15 \maketitle \begin{itemize} \item Recall \textit{Euclid's Parallel Postulate} (E5) says that if two lines are intersected by a transversal so that two interior angles on one side of the transveral add upt to less than $\pi$ then they meet in that halfplane of the transversal. \item[Problem 1:] Prove directly, using only absolute geometry that Proclus' Axiom is equivalent to Euclid's Parallel Postulate. {Proclus' Axiom says that \textit{ If a line intersects one of a pair of parallel lines then it also intersects the oher.} is equivalent to Euclid's Parallel Postulate. Do not use an other equivalent axioms, such as Playfair. \item[Problem 2:] Prove by any means in absolute geometry (including Playfair and models) that another E5equivalent axiom is \textit{A entire line cannot lie entirely inside of an angle} (i.e. not cross either ray.) \item[Problem 3:] Prove by any means in absolute geometry that another E5equivalent axiom is \textit{Opposite sides of a parallelogram are congruent.} \item[Problem 4:] Calculate the area of a Lambert quadrilateral in non-Euclidean geometry as a function of its angles alone. \item[Problem 5:] Given $0 \le x \le 1$ and $y = \frac{e-1}{e+1}$, consider the Lambert with base the segment $0\, x$ as on base, and $0\, iy$ ( on either side of the middle right angle of the quadilateral). Calculate the hyperbolic lengths of its heigh and base. \item[Problem5b] Show that if the Lambert's non-Euclidean area were also base$\times$height, then it would acquire infinitely large area as $x \rightarrow 1$. Why can this not be? \item[Problem 6:] Give two proofs of Pythagoras' Theorem, one using the analytic method of proof, the other using the synthetic method of proof. \item Using only your memory of having used Geogebra and GEX, and whatever you have in your Journal, give brief answers to the following questions. \item[Problem 7:] How do the buttons on GGB and GEX (choose and say which) correspond to axioms and theorems in Geometry? \item[Problem 8:] What is the chief difference in how GGB and GEX are designed to explore non-Euclidean geometry? \end{itemize} \end{document}