Download Quadrilaterals in Euclidean Geometry

Quadrilaterals in Euclidean Geometry
Rectangles exist
A rectangle is a quadrilateral with four right angles.
Recall the Saccheri Quadrilateral, with a triangle constructed as shown:
←−→ ←→
In Euclidean geometry, the midpoint connector theorem tells us right away that MN ||BC. So the
base and summit of this Saccheri quadrilateral are known to be parallel in Euclidean geometry. (We
don’t get that result from just absolute geometry.)
Remove all the extra bits, like the triangle. All we needed it for was to get MN ||BC, now P R||BC.
Extend the summit BC through U and V .
Why are ∠P CB and ∠RCB right angles?
In Euclidean geometry, the Saccheri quadrilateral is a rectangle. There is no way to construct a
Saccheri quadrilateral in Euclidean geometry and not end up with four right angles - congruent base
angles plus congruent sides forces parallel lines, which force right angles at the summit.
The fact that rectangles exist in Euclidean geometry does not alone tell us anything about the existence
or nonexistence of rectangles in hyperbolic geometry, but some other facts all taken together do. The
theorems of absolute geometry and the parallel postulate combine to give us a collection of statements
that are “if and only if”s - you can start by assuming any one of them as the axiom, and proving the
others.
So we also get the contrapositives:
Results in hyperbolic geometry
• The sum of the measures of the angles of a triangle is < 180.
• The sum of the measures of the angles of a convex quadrilateral is < 360.
• Rectangles don’t exist.
(Go into nonEuclid and try to draw one!)