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. 35
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