35. Theorem. If two right triangles have equal hypotenuses and an unequal
acute angle, then the sides opposite the unequal angles are unequal, and the larger
side is opposite the larger angle.
Figure 34
Let the two triangles be ABC, A B C (Fig. 34), for which BC = B C ,
B B . We claim that AC A C .
To see this, we extend AC by its own length to get AD, and similarly extend
A C to get A D . We immediately have (29) BD = BC = B C = B D . More-
over, in isosceles triangle BDC median BA is also an angle bisector, so that angle
DBC is twice the original angle B. Likewise, angle D B C is twice the original
angle B , from which we have DBC D B C .
The two triangles DBC, D B C therefore have an unequal angle between equal
sides, from which it follows that DC D C , and therefore AC A C .
36. Theorem. The bisector of an angle is the locus of points in the interior
of the angle which are equidistant from the two sides.
Figure 35
As explained earlier (33), the proof consists of two parts:
Every point on the bisector is equidistant from the two sides.
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