Unformatted text preview: Math402 Problem Set 7 First Part This problem set is about equivalent statements of the Parallels (Euclidean) Axiom. In particular
you can use (only) axioms and theorems of neutral geometry. Problem 1. Show that the Converse of the Alternate Interior Angles Theorem implies Playfair
Axiom (the opposite direction was shown in class). Problem 2. Show that the Converse of the Alternate Interior Angles Theorem is equivalent to
the following statement: the opposite sides of any parallelogram are congruent. A parallelogram is
a quadrilateral such that opposite sides are parallel. Hints: Angles => Parallelogram: draw a diagonal of the parallelogram. Parallelogram => Angles: draw a line n such that indicated angles (a’s) are congruent. You
may assume that if n is close to l then it intersects m2 (this follows from the axioms of neutral
geometry, but you don’t have to prove it). a Problem 3. Show (by example) that if the line n from the above hint is far from I it could miss
m2 (use half plane model of hyperbolic geometry). ...
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 Spring '09
 malkin

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