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\centerline{MATH 392 -- Geometry Through History}
\centerline{Problem Set 2}
\centerline{Due: Friday, February 12}
\vskip 10pt
\noindent
I.  Proposition 30 in Book I of Euclid's \emph{Elements} says:  \emph{Two lines that
are parallel to the same line are parallel to each other}.  
\vskip 10pt
Here's the proof Euclid provides:  Let it be given that the lines $AB$ and
and $EF$ are parallel and the lines $CD$ and $EF$ are parallel.  Pick any two points
$G$ on the line $AB$ and $K$ on the line $CD$, and draw the line segment $GK$ (and extend it -- 
Postulates I and II).  Let $H$ be the intersection of the line $GK$ with the line $EF$.
Then $\angle AGH = \angle FHG$ and $\angle FHG = \angle DKG$ (Proposition 29).  Hence
also $\angle AGH = \angle DKG$ (Common Notion 1).  Therefore, $AB$ is parallel to $CD$
(Proposition 27).   

\begin{enumerate}
\item[(A)] There is a point here that is questionable in the sense that it does
not really follow from any of the Postulates or the previous results in Book I.  What
is this issue?
\item[(B)] To fix the problem you noted in part (A), prove the following statement:
\emph{If a line intersects one of a pair of parallel lines, then it intersects the
other one as well}.  Hint:  You should take ``line'' here to mean a line extended
as far as possible in both directions (Postulate 2).  
Argue by contradiction and note that two lines that do not intersect are parallel
by definition.  Your proof \emph{should} make use of results that depend on Postulate V.
\end{enumerate}
\vskip 10pt
\noindent  
II.  Give a proof of Proposition 35 in Book I of the \emph{Elements} in the cases that
the sides $CD$ and $EF$ of the two parallelograms overlap.  (See the slides on Book I.)
\vskip 10pt
\noindent
III.  Assume Postulates I - IV of the \emph{Elements} hold.  Consider the following statements:
\begin{enumerate}
\item[(A)]  Euclid's Postulate V.
\item[(B)]  A line perpendicular to one ray of an acute angle intersects the other
ray as well.
\item[(C)]  Through any point in the interior of an angle less than a straight angle 
($180^\circ$), there passes a line meeting each of the two rays at points other than 
the vertex.
\item[(D)]  The sum of the angles in any triangle is $180^\circ$.  
\end{enumerate}
Show that all these statements are equivalent by showing that (A) $\Rightarrow$ (B), 
(B) $\Rightarrow$ (C), (C) $\Rightarrow$ (D).  Hints: We know from class that 
(D) $\Rightarrow$ (A). So once you show those three implications, all four statements
are equivalent.  Also, see Problem 3.5 in McCleary, which gives an ``attempted
proof'' of Postulate V by A.-M. Legendre (1804).  You \emph{can} use this proof 
under the assumption that (C) is true.  Do you see why?  And do you see why Legendre's
proof fails if we don't know (C) is true? 
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