CHAPTER IV

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

congruence.

First case of congruence. Two right triangles are congruent if they have

equal hypotenuses, and an equal acute angle.

Figure 33

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

CA.

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

follows.

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