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

21, 22, 23] PARALLELS. HILBERT

43

Also it cannot be parallel to BC, because according to the
Axiom the two parallels are not to form one and the same
straight line.
Therefore the angles between a., a2, and AD must be acute.
We shall now show that they are equal.
If the angles are unequal, one of them must be the greater.
Let a1 make the greater angle with AD, and at A make
L DAP - DAK.
Then AP must cut BC when produced.
A
a2i / P\ a,
B R D C
FIG. 15.
Let it cut it at Q.
On the other side of D, from the line b cut off DR = DQ and
join AR.
Then the triangles DAQ and DAR are congruent, and AR
makes the same angle with AD as a2, so that AR and a2 must
coincide.
But a2 does not cut BC; therefore the angles which al, a2
make with AD are not unequal.
Thus we have shown that the perpendicular AD bisects the
angle between the parallels a1 and a2.
The angle which AD makes with either of these rays is called
the angle of parallelism for the distance AD, and is denoted,
after Lobatschewsky, by II(p), where AD =p.
The rays a1 and a2 are called the right-handed and lefthanded parallels from A to the line BC.
~ 23. In the above definition' of parallels, the starting point
A of the ray is material. We shall now show that
A straight line maintains its property of parallelism at all
its points.