The elements of non-Euclidean plane geometry and trigonometry, by H. S. Carslaw.

48

NON-EUCLIDEAN GEOMETRY

[CH. III.

Also, a straight line will be said to pass through this point
2, when it is parallel to these two lines in the same sense.
1. If a straight line passes through one of the angular points
A, B, or 2, and through a point inside this figure, it must cut
the opposite side. (Fig. 21.)
Let P be the point within the figure. Then AP must cut B2,
by the Axiom of Parallels. Let it cut B2 at Q. The line PQ
must cut one of the sides AB or BQ of the triangle ABQ, by
Pasch's Axiom. It cannot cut BQ, since it is parallel to BQ.
Therefore it must cut AB.
2. A straight line in the plane AB~2, not passing through an
angular point, which cuts one of the sides, also cuts one, and only
one, of the remaining sides of this figure.
Let the straight line pass through a point C on AB. Let CQ
be drawn through C parallel to A2 and BQ. If the given line
lies in the region bounded by AC and CQ, it must cut A2; and
if it lies in the region bounded by BCOand C2 it must cut B2.
Again, if the line passes through a point D on A2, and B, D
are joined, it is easy to show that it must cut either AB or B2.
We shall now prove some further properties of this figure.
3. The exterior angle at A or B is greater than the interior and
opposite angle.
C
A/-I _ M
~B~~FI. 22.
FIa. 22.
Consider the angle at A, and produce the line BA to C.
Make L CAM =L AB2. AM cannot intersect B2, since the
exterior angle of a triangle is greater than the interior and
opposite angle. Also it cannot coincide with A2, because then
the perpendicular to A2 from the middle point of AB would
also be perpendicular to B2. The angle of parallelism for
this common perpendicular would be a right angle, and this
is contrary to Hilbert's Axiom of Parallels.