187. If three or more parallels intercept equal parts on one transversal, they intercept equal parts on every transversal. Let the parallels AH, BK, CM, DP intercept equal parts HK, KM, MP on the transversal HP. To prove that they intercept equal parts AB, BC, CD on the transversal AD. Proof. Suppose AE, BF, and CG drawn to HP. AEB, BFC, etc. = &HKE, KMF, etc., respectively. § 112 But HKE, KMF, etc. are equal. .. & AEB, BFC, etc. are equal. § 112 Ax. 1 Also BAE, CBF, etc. are equal. § 112 Now AE = HK, BF = KM, CG = MP, § 180 (parallels comprehended between parallels are equal). .. AE = BF = CG. Ax. 1 = (having two and the included side of each respectively equal). ..A ABE = ABCF ▲ CDG, § 139 .. AB = BC = CD. § 128 Q. E. D. 188. COR. 1. If a line is parallel to the base of a triangle and bisects one side, it bisects the other side also. B D E Let DE be to BC and bisect AB. Suppose a line is drawn through A || to BC. Then this line is I to DE, by § 106. The three parallels by hypothesis intercept equal parts on the transversal AB, and therefore, by § 187, they intercept equal parts on the transversal AC; that is, the line DE bisects AC. 189. COR. 2. The line which joins the middle points of two sides of a triangle is parallel to the third side, and is equal to half the third side. A line drawn through D, the middle point of AB, || to BC, passes through E, the middle point of AC, by § 188. Therefore, the line joining D and E coincides with this parallel and is to BC. Also, since EF drawn to AB bisects AC, it bisects BC, by § 188; that is, BF = FC = {BC. But BDEF is aby § 166, and therefore DE = BF = } BC. 190. COR. 3. The median of a trapezoid is parallel to the bases, and is equal to half the sum of the bases. = А E D AB. G B Draw the diagonal DB. In the ▲ ADB join E, the middle point of AD, to F, the middle point of DB. Then, by § 189, EF is to AB and In the A DBC join F to G, the middle point of BC. Then FG is || to DC and = DC. AB and FG, being || to DC, are || to each other. But only one line can be drawn through Fl to AB (§ 105). Therefore FG is the prolongation of EF. Hence, EFG is parallel to AB and DC, and equal to † (AB + DC). POLYGONS IN GENERAL. 191. A polygon is a portion of a plane bounded by straight lines. The bounding lines are the sides, and their sum, the perimeter of the polygon. The angles included by the adjacent sides are the angles of the polygon, and the vertices of these angles are the vertices of the polygon. The number of sides of a polygon is evidently equal to the number of its angles. 192. A diagonal of a polygon is a line joining the vertices of two angles not adjacent; as, AC (Fig. 1). 193. An equilateral polygon is a polygon which has all its sides equal. 194. An equiangular polygon is a polygon which has all its angles equal. 195. A convex polygon is a polygon of which no side, when produced, will enter the polygon. 196. A concave polygon is a polygon of which two or more sides, if produced, will enter the polygon. 197. Each angle of a convex polygon (Fig. 2) is called a salient angle, and is less than a straight angle. 198. The angle EDF of the concave polygon (Fig. 3) is called a re-entrant angle, and is greater than a straight angle. When the term polygon is used, a convex polygon is meant. 199. Two polygons are equal when they can be divided by diagonals into the same number of triangles, equal each to each, and similarly placed; for if the polygons are applied to each other, the corresponding triangles will coincide, and hence the polygons will coincide and be equal. 200. Two polygons are mutually equiangular, if the angles of the one are equal to the angles of the other, each to each, when taken in the same order. Figs. 1 and 2. 201. The equal angles in mutually equiangular polygons are called homologous angles; and the sides which are included by homologous angles are called homologous sides. 202. Two polygons are mutually equilateral, if the sides of the one are equal to the sides of the other, each to each, when taken in the same order. Figs. 1 and 2. FIG. 4. FIG. 5. FIG. 6. FIG. 7. 203. Two polygons may be mutually equiangular without being mutually equilateral; as, Figs. 4 and 5. And, except in the case of triangles, two polygons may be mutually equilateral without being mutually equiangular; as, Figs. 6 and 7. If two polygons are mutually equilateral and mutually equiangular, they are equal, for they can be made to coincide. 204. A polygon of three sides is called a triangle; one of four sides, a quadrilateral; one of five sides, a pentagon; one of six sides, a hexagon; one of seven sides, a heptagon; one of eight sides, an octagon; one of ten sides, a decagon; one of twelve sides, a dodecagon. PROPOSITION XXXIX. THEOREM. 205. The sum of the interior angles of a polygon is equal to two right angles, taken as many times less two as the figure has sides. Let the figure ABCDEF be a polygon, having n sides. To prove that A + Z B+ ZC, etc. = (n-2) 2 rt. 4. The sum of the s of the A is equal to the sum of the of the polygon. Now, there are (n − 2) ▲, and the sum of the s of each ▲ = 2 rt. . § 129 .. the sum of the s of the A, that is, the sum of the of the polygon is equal to (n-2)2 rt. . Q. E. D. 206. COR. The sum of the angles of a quadrilateral equals 4 right angles; and if the angles are all equal, each is a right angle. In general, each angle of an equiangular polygon of n sides is equal to 2(n-2) n right angles. Ex. 17. How many diagonals can be drawn in a polygon of n sides? |