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

33, 34, 35] THE RIGHT-ANGLED TRIANGLE

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~35. The correspondence between a Right-Angled
Triangle and a Quadrilateral with Three Right
Angles and One Acute Angle.
The sides of a right-angled -triangle ABC, in which C is the
right angle, are denoted as usual by a, b, and c; the angles A
and B by X and u; and the distances corresponding to the
angles of parallelism X and,A are denoted by I and m.
Between these quantities a, b, c, 1, m, X, and Ax certain relations
hold.
Similarly the elements of a quadrilateral, in which three
angles areright angles, the remaining angle being necessarily
acute, are connected by certain relations.
We proceed to find the equations connecting these quantities, and to establish a very important correspondence between
the two figures.
a
C
b
A b C B a C
FIG. 37. FIG. 38.
I. The Right-Angled Triangle.
Let ABC be any right-angled triangle. Produce the hypothenuse through A a distance 1, and at the other end of the
segment I draw the parallel to the line CA. Also draw through
B the parallel to both these lines.
It follows from Fig. 38 that
+ II(c +Z)= (a) = c,..................(1)
and in the same way we have

+ n (c +n) = I(b)=3................(....(1')