Plane and solid geometry . drawlines to the vertices. 2. There will be formed n A. 3. The sum of the A of each A thus formed = 2 rt. A. 4. .*. the sum of the A of the n A thus formed = 2 nrt. A, 5. The sum of all the A about 0 = 4 rt. A, 6. .. the sum of all the A of the polygon = 2 ?i rt. A- 4 rt. A, Reasons 1. Straight line post I. § 54, 15. 2. Each side of the polygon will become the base ofa A. 3. The sum of the A of any A is 2 rt. A. § 204. 4. If equals are multiplied by equals, the productsare equal. § 54, 7 a. 5. The sum of all the A about a point = 4 rt. A. § 67. 6. The sum of a
Plane and solid geometry . drawlines to the vertices. 2. There will be formed n A. 3. The sum of the A of each A thus formed = 2 rt. A. 4. .*. the sum of the A of the n A thus formed = 2 nrt. A, 5. The sum of all the A about 0 = 4 rt. A, 6. .. the sum of all the A of the polygon = 2 ?i rt. A- 4 rt. A, Reasons 1. Straight line post I. § 54, 15. 2. Each side of the polygon will become the base ofa A. 3. The sum of the A of any A is 2 rt. A. § 204. 4. If equals are multiplied by equals, the productsare equal. § 54, 7 a. 5. The sum of all the A about a point = 4 rt. A. § 67. 6. The sum of all the A of the polygon = the sumof the A of the ii A —the sum of all the Aabout 0. 217. Cor. Each angle of an equiangular polygon of n sides is equal to 2(n - 2) n right angles. 82 PLAXE GEOMETRY Propositiox XXX. Theorem 218. // the sides of any polygon are prolonged in succes-sion one way, no two adjacent sides being prolongedthrough the same vertex, the sum of tl%e exterior anglesthus formed is four right \/D Given polygon P with Zl, Z2, Z3, Z4, ••• its successive ex-terior angles. To prove Zl + Z2 + ^3 + ^4 + ... = 4 rt. Z. Argument 1. Zl + ^6 = 2 rt. Z, Z2 + Z7 = 2 rt. Z, and so on; the sum of the and the ext. Z at onevertex = 2 rt. Z. 2. .*. the sum of the int. and ext. zs at the n vertices= 2 ?i rt. Z. 3. Denote the sum of all the interior Z by / and thesum of all the ext. Z byE\ then^4-i=2?irt. Z. 4. But /= 2 n rt. Z - 4 rt. Z. 6. .-. jF = 4 rt. Z; Zl+ Z2 + Z3-f Z4 -f ... = 4 rt. Z. Reasons 1. If one str. line meets an- other str. line, the sum ofthe two adj. Z is 2 rt. Z.§ 65. 2. If equals are multiplied by equals, the products areequal. § 54, 7 a. 3. Arg. 2. 4. The sum of all the Z of any polygon = 2 ?i rt. Z-4rt. Z. §216. 5. If equals are subtracted from equals, the remain-ders are equal. § 54, 3 BOOK I 83 219. Note. The formula 2 w rt. zl - 4 rt. ^i (§ 216) is sometimesmore useful iu the form (?i — 2) 2 rt. A. Ex. 24
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