28 II. ON TRIANGLES
The path ACDB is shorter than ACHB because they have a common portion
ACD, and the remainder BD of the ﬁrst is less than the remainder DHB of
the second. In turn, the path ACHB is shorter than AGD E F B because, after
removing the common parts AC, HB, the segment CH which is left is smaller
than the broken line CGD E F H. In the same way, AGD E F B is less than
D E F B, because AG is less than AC G. Thus
ACDB ACHB ACGD E F B AC D E F B.
Corollary. The perimeter of a convex polygon is less than the perimeter of
any closed polygonal line surrounding it completely.
Consider (Fig. 29) convex polygon ABCDE, and polygonal line
A B C D E F G A which surrounds it completely. We extend side AB in both
directions until it intersects the surrounding polygon at M , N. By the preceding
result, the length of the path AEDCB is less than AMB A G NB, and therefore
the perimeter of AEDCBA is less than the perimeter of the polygon NMB A G N.
This perimeter, in turn, is less than the surrounding line because the part MB A G N
is common to both, and MN MC D E F N.
28. Theorem. If two triangles have a pair of unequal angles contained by two
sides equal in pairs, and the third sides of the triangles are unequal, then the greater
side is opposite the greater angle.