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

43, 44] THE COMMON PARALLEL TO TWO LINES 75

It follows that

L.QA' = L QB;',

/ QAE = L,'BF = BF.
We shall now show that the' lines a' and b' neither intersect,
nor are parallel.
If possible, let them intersect at M.
The triangle AOB is isosceles, and L OAB=z_ OBA.
Therefore L BAM = LABM, and AM =BM.
0
A~ ~B
a bb
E F
U V
FIG. 50.
Through M draw the parallel MQ2 to Aig and B&2.
Then, since AM=BM and LMA= =L MB2, by ~26 (4), we
must have LAM L BM2,
which is absurd.
The lines AE and BF therefore do not intersect at an ordinary
point, and this proof applies also to the lines produced through
A and B.
Next, let us suppose that they are parallel.
Since the ray a' lies in the region BAt2, it must intersect B2.
Let it cut that line at D.
Then we have z 12AE = DBF, and z ADQ =/ BDE.
Also we are supposing DE and BF parallel, and we have
A2Q and D2 parallel.