ON SIMILAR POLYGONS. PROPOSITION IV. THEOREM. 279. Two triangles which are mutually equiangular are similar. A A ДД E H In the AABC and A'B'C' let & A, B, C be equal to Apply the ▲ A'B'C' to the ▲ ABC, so that A' shall coincide with A. Then the ▲ A'B'C' will take the position of ▲ A E H. Now ZAEH (same as ▲ B') = ▲ B. .. EH is || to BC, § 69 (when two straight lines, lying in the same plane, are cut by a third straight line, if the ext. int. & be equal the lines are parallel). = .. AB AE AC: AH, § 276 (one side of a A is to either part cut off by a line || to the base, as the other side is to the corresponding part). Substitute for A E and A H their equals A' B' and A' C'. 280. COR. 1. Two triangles are similar when two angles of the one are equal respectively to two angles of the other. 281. COR. 2. Two right triangles are similar when an acute angle of the one is equal to an acute angle of the other. In the triangles A B C and A'B'C' let H C We are to prove A, B, and C equal respectively to A', B', and C'. Take on A B, A E equal to A' B', Substitute in this equality, for A' B' and A' C' their equals (if a line divide two sides of a ▲ proportionally, it is || to the third side). Now in the AA BC and A EH = = = EH B'C', AE A' B', and AH A'C', . . ^ A E H = ▲ A' B' C', Hyp. Ax. 1 Cons. $108 (having three sides of the one equal respectively to three sides of the other). But AAEH is similar to AABC. .. A A'B'C' is similar to ▲ ABC. Q. E. D. 283. SCHOLIUM. The primary idea of similarity is likeness of form; and the two conditions necessary to similarity are: I. For every angle in one of the figures there must be an equal angle in the other, and II. the homologous sides must be in proportion. In the case of triangles either condition involves the other, but in the case of other polygons, it does not follow that if one condition exist the other does also. Thus in the quadrilaterals Q and Q', the homologous sides. are proportional, but the homologous angles are not equal and the figures are not similar. In the quadrilaterals R and R', the homologous angles are equal, but the sides are not proportional, and the figures are not similar. PROPOSITION VI. THEOREM. 284. Two triangles having an angle of the one equal to an angle of the other, and the including sides proportional, are similar. Apply the AA'B'C' to the AABC so that A' shall coincide with ▲ A. Then the point B' will fall somewhere upon A B, as at E, the point C will fall somewhere upon A C, as at H, and B'C' upon EH. Now A C = A B Hyp. Substitute for A' B' and A' C" their equals A E and A H. .. the line EH divides the sides A B and AC proportionally; .. EH is to BC, $ 277 (if a line divide two sides of a ▲ proportionally, it is || to the third side). .. the AA BC and A E H are mutually equiangular and similar. ..A A'B'C' is similar to ▲ ABC. Q. E. D. PROPOSITION VII. THEOREM. 285. Two triangles which have their sides respectively parallel are similar. A B' B A CI A C In the triangles ABC and A'B'C' let AB, AC, and BC be parallel respectively to A'B', A'C', and B'C'. We are to prove AABC and A' B' C' similar. The corresponding are either equal, $ 77 (two whose sides are 1, two and two, and lie in the same direction, or opposite directions, from their vertices are equal). or supplements of each other, $ 78 (if two have two sides || and lying in the same direction from their vertices, while the other two sides are || and lie in opposite directions, the are supplements of each other). Hence we may make three suppositions: Since the sum of the s of the two ▲ cannot exceed four right angles, the 3d supposition only is admissible. .. the two A A B C and A' B' C' are similar, (two mutually equiangular ▲ are similar). § 98 $279 Q. E. D. |