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

77, 78] SUM OF THE ANGLES OF A TRIANGLE 133

Produce ED to F, so that ED= DF.
Join AF and PF.
Then the triangles ADF and DEB are congruent, and AF, FB
lie in one straight line.
C E pF
C E B P
FIG. 91.
But we know that L PAC > a right angle, since CP is greater
than V.
Therefore the sum of the angles at A and B in the rightangled triangle ACB is greater than a right angle in this case
as well as in the others.
Thus we have proved that
2. In any right-angled triangle the sum of the angles is greater
than two right angles.
Finally, let ABC be any triangle in A
which none of the angles are right angles. /
We need only consider the case when
two of the angles are acute.
Let LABC and LACB be acute.
From A draw AD perpendicular to
BC; D must lie on the segment BC.
Then, from (2), Y
B D C
L ABD +~ BAD > a right angle B D C
FIG. 92.
and LDAC + L ACD >a right angle.
It follows that the sum of the angles of the triangle ABC is
greater than two right angles.