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

50

NON-EUCLIDEAN GEOMETRY

[CH. III

At A and B draw the rays which make with AB an angle
equal to the angle at A'.
These rays must intersect; let them meet at C.
From A'f' cut off A'C'=AC, and join B'C'.
The triangles ABC and A'B'C' are congruent, so that
L A'B'C' =L ABC =LA'B'2'
which is absurd.
Thus the angles at A and A' must be equal; and it follows
that the angles at A, B, A' and B' are equal to each other.
6. If the angles at A and A' are equal, and the angles at B and
B' are also equal, then the segment AB =the segment A'B'.
If AB is not equal to A'B', one of them must be the greater.
Let it be AB.
A A'
C/ / n
B B'
FIG. 25.
From AB cut off AC =A'B', and draw C02 parallel to Ai.
Then, by (4), L AC2 =zL A'B'2' =z ABf2.
But by (3), L AC2 > L ABt2.
Therefore AB cannot be greater than A'B', and the two
segments are equal.
~ 27. The Angle of Parallelism.
From ~ 26 (4), we can at once deduce that the angles of
parallelism corresponding to equal distances are equal.
1 p
P2
FIG. 26.
Combining this result with ~ 26 (3), we can assert that
If PI>p,) then (02) > TI(pl).