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

35] THREE-RIGHT-ANGLED QUADRILATERAL 61

II. The Quadrilateral with Three Right Angles and an Acute
Angle.
Let PQRS be a quadrilateral in which the angles P, Q, R
are right angles. We denote the sides, for reasons that will
presently appear, by l1, a,, m1', and cl; the acute angle by
/31; and 11, c1 contain this angle 3,.
m~
C, R
p a, Q,
FIG. 41.
Produce c1 through R a distance ml, and draw the perpendicular at the end of that segment. Since H (ml) + II (n') = -,)
if the parallel through R-to PQ is supposed drawn, it follows
that this perpendicular is parallel to PQ (Fig. 41).
It follows that 1 + I (c +l) = 1,....................(I.)
and correspondingly y1 + II (7 + a',) = 1..................(I'.)
$-s
/AR
P
FIG. 42.
From RS cut off the segment mi; then it is obvious from
Fig. 42 that X, + = nI (cl - al),...................(II.)
and correspondingly y1 + f = II (11 - a/).................... (II'.)