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

81, 82, 83] METHOD OF Gf:RARD AND MANSION 141

~ 82. We turn now to the quadrilateral with three right
angles and one obtuse angle.
Let OABb be such a quadrilateral, the angles at 0, B, and b
being right angles.
Produce Ob, and cut off bc=ob and cd=bc.
Draw the perpendiculars to Ob produced at the points c and
d; and from A the perpendiculars to the lines just drawn.
D
0 b c d
FIG. 100.
We thus obtain three quadrilaterals OABb, OACc, OADd, of
this nature, standing on the bases ob, Oc, and Od.
It is easy to show that the obtuse angles of these quadrilaterals increase as the bases increase.
Let bB produced meet AC at H, AB produced meet Cc at I,
and AC produced meet Dd at J.
Then we have AB = BI, AB < AH, and Al > AC.
It follows that AB > AC- AB.
Also we have HC=CJ and AD < AJ.
Therefore AC - AH = AJ - AC, and finally AC - AB > AD - AC.
Thus AB > AC - AB > AD - AC.
~83. We return to the notation of ~ 80 and the figure
OMPM', in which the angles at 0, M, and M' are right angles,
and the sides OM, MP, PM', and OM' are denoted by x, y, x',
and y' respectively.
We shall now prove the following theorem:
In the quadrilateral with three right angles (x, y, x', y'), in