Cases of Congruence for Right Triangles.
A Property of the Bisector of an Angle
34. Cases of congruence for right triangles. The general cases of congru-
ence for triangles apply, of course, to right triangles as well. For instance, two right
triangles are congruent if their legs are respectively equal (2nd case of congruence
for any triangles).
Aside from these general cases, right triangles present two special cases of
First case of congruence. Two right triangles are congruent if they have
equal hypotenuses, and an equal acute angle.
Consider two right triangles ABC, A B C (Fig. 33) in which BC = B C ,
B = B . We move the second triangle over the ﬁrst so that the angles B and B
coincide. Then B C will assume the direction of BC, and, since these two segments
are equal, C will fall on C. B A will assume the direction of BA, and therefore
C A will coincide with the perpendicular dropped from C on BA; that is, with
Second case of congruence. Two right triangles are congruent if they have
equal hypotenuses and one pair of corresponding legs equal.
Suppose the two triangles are ABC, A B C , and that BC = B C , AB = A B .
We move the second triangle onto the ﬁrst so that A B coincides with AB. The
side AC will assume the direction of A C . We then have two oblique lines from
point B to line AC; namely, BC and the new position of B C . These oblique
lines are equal by hypothesis and therefore (30), equidistant from the foot of the
perpendicular. This gives A C = AC, from which the congruence of the triangles