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

28, 29, 30, 31] SUM OF ANGLES OF A TRIANGLE 53
Thus L D'AC =L DAC, and the triangles D'AC and DAC are
congruent.
Therefore L D'CA is a right angle, and DC, CD' form one
straight line.
D' C D
B1 A B
FIG. 30.
Applying the result of ~ 28 to the quadrilateral D'B'BD, it
follows that the angles at D' and D are equal and acute.
~ 31. The sum of the angles of every triangle is less
than two right angles.
Case I. Let the triangle ABC be any right-angled triangle
with C =90~.
At A make / BAD =L ABC.
A
0
C P B
FIG. 31.
From 0, the middle point of AB, draw the perpendiculars OP
and OQ to CB and AD respectively.
Then the triangles POB and AOQ are congruent, and it
follows that OP and OQ are in one and the same straight line.