SOAR Homework Two
These homework problems are meant to expand your understanding of what
goes on during class. Any you turn in will be graded and returned to
you. Answers may or may not be posted on the web, depending on demand.
Problems
 Prove that Euclid's Postulate 5 and Playfair's Postulate are
equivalent. That is, assuming ``absolute geometry'' (Euclid's first
four postulates), prove that...
 Postulate 5 implies Playfair's Postulate, and

Playfair's Postulate implies Euclid's Postulate 5.

Prove that the sum of angles in a quadrilateral is at most
360°. (Do not assume Euclid's Postulate 5  assume only
``absolute geometry.'')
 For this problem, consider the quadrilateral pictured in the pdf version in the context of
absolute geometry.
 If AC = BD, this is called a Saccheri quadrilateral.
Prove in this case that ∠C = ∠D.

If, on the other hand, AC < BD, show that ∠C > ∠ D.

Suppose now that ∠C > ∠D. Show that AC < BD.
(Hint: work by contradiction. That is, assume that AC ≥ BD
and show this implies that ∠C ≤ ∠D. Perhaps it's
easier to do two cases: AC = BD and AC > BD.)
Note that in parts (b) and (c) you've proved a statement and it's
converse. That is, you've shown that AC < BD if and only if
∠C > ∠D.)

For this problem, you may use only results from absolute geometry .
Consider a Lambert quadrilateral, which is the quadrilateral above
with ∠C also a right angle (so three of the four angles are
assumed to be right angles).
 Show that the fourth angle, ∠D, is never greater than
90°.

Show that if ∠D is also a right angle, then opposite sides
of Lambert's quadrilateral are congruent.

Show that if ∠D is acute (less than 90°), then
CD > AB and BD > AC.

Using the previous problem and the fact that the angle of
parallelism is acute (under 90°), show that, in hyperbolic
geometry, the distance between parallel lines is decreasing.

What can you say about the sum of the angles
of a quadrilateral in...
 in absolute geometry?
 in hyperbolic geometry?
These problems are also available as a pdf file.
Solutions
Please email Peter
if you are interested in answers or solutions for the web. Thanks.
SOAR Winter 2003 Course Homepage