Since the polygons are similar, and ▲ ABC ~ ▲ A'B'C', Q. E D. In like manner, ▲ ADE and A'D'E' are similar. PROPOSITION XXV. THEOREM 314. Two polygons are similar if they are composed of the same number of triangles, similar each to each, and similarly placed. To prove polygon ABCDE~ polygon A'B'C'D'E'. HINT. The polygons are mutually equiangular by Axiom 2. The ratio of any pair of homologous sides is equal to the ratio of the next pair, for either ratio is equal to the ratio of the included homologous diagonals. N Ex. 875. If AF|| BE CD and BF CE, prove that ABEF is similar to BCDE. (See diagram.) C E Ex. 876. If AB|| A'B', BC B'C', and CD || C'D', then OABCD is similar to OA'B'C' D'. Ex. 877. Polygon ABCDE is similar to polygon A'B'C' D'E' if the sides of the first are respectively parallel to the sides of the second and if AC|| A'C', ADA'D'. B Ex. 878. If polygon ABCDEF is similar to polygon A'B'C'D'E'F', then ABCDE is similar to A'B'C' D'E'. [See practical problems, 61–63, p. 295.] 315. To construct a polygon similar to a given polygon upon a line homologous to a side of the given polygon. B Given polygon ABCDE, and line A'B'. B' HINT. Draw AD and AC and make corresponding & equal. Ex. 879. To construct a quadrilateral ABCD similar to a given quadrilateral at A'B'C'D' and having the diagonal equal to a given line. PROPOSITION XXVII. THEOREM 316. The perimeters of two similar polygons are to each other as any two homologous sides. Given P and P', the perimeters of the similar polygons ABCDE and A'B'C'D'E' respectively. = HINT. AB: A'B' = BC : B'C' = CD : C'D' — DE : D'E' = EA : E' A'. Apply (286). Ex. 880. The sides of a polygon are 4, 5, 6, 7, and 8 respectively. Find the perimeter of a similar polygon, if the side corresponding to 5 is 7. Ex. 881. The perimeters of two similar polygons are 20 and 25 in. respectively. If a side of the first polygon is 4 in., find the homologous side of the second polygon. Ex. 882. The perimeters of two similar polygons are to each other as any two homologous diagonals. Ex. 883. The perimeters of two similar triangles are to each other as any two homologous altitudes. Ex. 884. In the diagram for Prop. XXVII find the perimeter of ABCDE, if the perimeter of A'B'C' D'E' equals 20 inches, A'B' = 4 inches, B'C' = 3 inches, AC 10 inches, B'C': A'B' = A'B': A'C' and ABCDE~ A'B'C'D'E'. 317. METHOD XIX. = To prove the proportionality of four lines which do not form similar triangles, find a third ratio equal to each of the given ones. Thus, in the annexed figure, if ABCD is a parallelogram and AG is a straight line, then Each of these proportions is easily proved by means of the fundamental method (XVII). Ex. 885. In similar triangles the radii of the inscribed circles have the same ratio as any two homologous sides. Ex. 886. If triangle ABC is similar to triangle A'B'C', and AD and A'D' are angle bisectors, prove that AD: A'D' = BC: B'C'. Ex. 887. If in the similar triangles ABC and A'B'C', the points D and D' are taken respectively in BC and B'C' so that ▲ BAD=Z B'A'D', then BD: B'D': = BC: B'C'. Ex. 888. If D is the mid-point of AB, CF AB, and DG is a straight line, prove that E DE DG (Annexed diagram.) FE FG B PROPOSITION XXVIII. THEOREM 318. If two parallel lines are cut by three or more transversals passing through a common point, the corresponding segments are proportional. Given the transversals OA, OB, OC, and OD intersecting the parallel lines AD and A'D' in A, B, C, D, and A', B', C', D' respectively. HINT. Which method for demonstrating the proportionality of the first four lines must be applied? Why? Ex. 889. In the diagram for Prop. XXVIII, if AB = B'C', A'B' = 4, and BC= 9, what is the value of AB? Ex. 890. In quadrilateral ABCD, if EG || AC |