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

86

NON-EUCLIDEAN GEOMETRY

[CH. III.

Therefore the quadrilateral BB'C'C and the triangle ABC are
equivalent.
Next, let A1B1C1 be another triangle with its side BiC1 equal
to BC, and the same defect as the triangle ABC.
For this triangle we get in the same way one of Saccheri's
Quadrilaterals, the acute angles at B1 and C1 being equal to
the acute angles at B and C, while the side B1C1 = the
side BC.
It is easy to see that these quadrilaterals must be congruent,
for if they were not, we should obtain a quadrilateral, in which
the sum of the angles would be four right angles, by a process
which amounts to placing the one quadrilateral upon the
other, so that the common sides coincide.
It follows that the triangles ABC and A1BiC1 are equivalent.
Thus we have shown that triangles with a side of the one equal
to a side of the other, and the same defect, are equivalent.
COROLLARY. The locus of the vertices of triangles on the
same base, with equal defects, is an Equidistant-Curve.
2. Any two triangles with the same defect and a side of the one
greater than a side of the other are equivalent.
_ ~ E
A
B C
FIG. 61.
Let ABC be the one triangle and A^lB1C the other, and let
the side AC1 (b1) be greater than the side AC(b).
Let E, F be the middle points of AC and AB.
From C draw CC' perpendicular to EF; CC' cannot be greater
than ~b.