Section 4.4 Rectangles
¶Suppose that a rectangle exists in neutral geometry. What can we conclude?
The constructions below are described in general terms, and are not intended as rigorous proofs.
Claim 4.4.1.
If there is one rectangle, there are arbitrarily large rectangles.
Proof.
The given rectangle can be duplicated by extending two parallel sides. The new quadrilateral is again a rectangle, by SASAS congruence. Putting the two together makes a rectangle twice as large. Repeat (in either or both directions) as desired, as shown in Figure 4.4.2.
Claim 4.4.3.
If there are arbitrarily large rectangles, there are rectangles of any size.
Proof.
A rectangle can be divided into two arbitrary rectangular pieces using a line parallel to any side, as shown in Figure 4.4.4.
Claim 4.4.5.
If there are rectangles of arbitrary size, then all right triangles have angle sum \(180^\circ\text{.}\)
Proof.
A right triangle of any desired size can be obtained by dividing a suitable rectangle in half, as shown in Figure 4.4.6. The angles in each triangle clearly add up to \(180^\circ\text{.}\)
Claim 4.4.7.
If right triangles have angle sum \(180^\circ\text{,}\) so do all triangles.
Proof.
Any triangle can be divided into right triangles, as shown in Figure 4.4.8.
Claim 4.4.9.
If there exists a triangle with angle sum \(180^\circ\text{,}\) then there exists a rectangle.
Proof.
The rectangle can be constructed by dropping a perpendicular line segment from one vertex of the triangle to the opposite side, as shown in Figure 4.4.10.
We have therefore shown that rectangles can only exist in neutral geometry if all triangles have angle sum \(180^\circ\text{!}\) We now claim that either of these conditions can only occur in Euclidean geometry, that is, that the existance of rectangles in neutral geometry, or equivalently the existance of triangles with angle sum \(180^\circ\text{,}\) is enough to imply the Euclidean parallel postulate, and vice versa.
Claim 4.4.11.
If the Euclidean parallel postulate holds (in neutral geometry), then triangles have angle sum \(180^\circ\text{.}\)
Proof.
Suppose that lines \(m\) and \(n\) are parallel, as shown in Figure 4.4.12. Then the alternate interior angles \(\alpha\) and \(\beta\) must be equal, as otherwise we could construct additional parallel lines through the intersection points, thus violating the Euclidean parallel postulate. But now Figure 4.4.8 can be used to verify that the angle sum of any triangle must be \(180^\circ\text{.}\)
Claim 4.4.13.
If triangles have angle sum \(180^\circ\) (in neutral geometry), then the Euclidean parallel postulate holds.
Proof.
Suppose that lines \(m\) and \(n\) are both parallel to line \(l\text{,}\) as shown in Figure 4.4.14. If \(\alpha\) is the angle between \(m\) and \(n\text{,}\) we can choose a point \(P\) on \(l\) such that the angle \(\beta\) is smaller than \(\alpha\text{,}\) as shown. (Intuitively, choose \(P\) sufficiently far away.) But now \(\beta+\delta \lt \alpha+\delta \lt 90^\circ\text{,}\) so the triangle shown has angle sum less than \(180^\circ\text{.}\) Thus, if the geometry is not Euclidean, then not all triangles have angle sum \(180^\circ\text{.}\) Turning this around yields the desired claim.
We conclude that not only does the existence of one rectangle imply both that all triangles have angle sum \(180^\circ\) and that the Euclidean parallel postulate holds, but also that if the Euclidean parallel postulate does not hold, then the angle sum of every triangle must be less than \(180^\circ\text{,}\) and there can not be any rectangles.