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 AC D E F B, because AG is less than AC G. Thus ACDB ACHB ACGD E F B AC D E F B. QED Corollary. The perimeter of a convex polygon is less than the perimeter of any closed polygonal line surrounding it completely. Figure 29 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. Figure 30
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