Art. 97. ON SIMILAR TRIANGLES. The different cases, in which two triangles are similar, correspond to some extent to the cases in which two triangles are congruent. For this reason the cases in which two triangles are congruent will first be enumerated. Art. 98. Two triangles are congruent if (1) The three sides of one triangle are respectively equal to the three sides of the other triangle. (2 a)* Two sides and the included angle of one triangle are respectively equal to two sides and the included angle of the other triangle. (3a)† Two angles and the adjacent side of one triangle are respectively equal to two angles and the adjacent side in the other triangle. (3b)† One side, the opposite angle, and one other angle of one triangle are respectively equal to one side, the opposite angle, and one other angle in the other triangle. Besides the above cases there should be noted the following, in which three elements (sides or angles) of one triangle are respectively equal to the three corresponding elements of the other triangle, viz. those in which (26)* One angle, the opposite side, and one other side of one triangle are respectively equal to one angle, the opposite side and one other side of the other triangle. In this case the angles opposite the other pair of equal sides are either equal or supplementary, and in the former alternative the triangles are congruent. (This case is usually known as the Ambiguous Case.) (4) Three angles of one triangle are respectively equal to three angles of the other triangle. This last case is only mentioned in order to complete all the possible cases in which three elements of one triangle are respectively equal to the three corresponding elements of another triangle. In it the triangles are not generally congruent, but are always similar (see Prop. 26). Art. 99. To case (1) above corresponds in the case of similar triangles the proposition that if the sides of one triangle taken in order are proportional to the sides of another triangle taken in order, then the triangles are similar. * The numbers attached to the cases (2 a) and (2b) both contain the same number 2 because in each there are two sides and one angle given equal. The numbers attached to the cases (3 a) and (3b) both contain the same number 3 because in each there are two angles and one side given equal. To case (2 a) corresponds the proposition that if two sides of a triangle are proportional to two sides of another triangle, and the included angles are equal, then the triangles are similar. To case (2 b) corresponds the proposition that if two triangles have one angle of the one equal to one angle of the other, and the sides about one other angle proportional in such a manner that the sides opposite the equal angles correspond, then the triangles have their remaining angles either equal or supplementary, and in the former case the triangles are similar. To cases (3 a), (3b) and (4), in all of which the three angles of the one triangle are respectively equal to the three angles of the other triangle, corresponds the single proposition that if the angles of one triangle are respectively equal to the angles of another triangle, then the triangles are similar. Hence there are four cases of similar triangles to be dealt with. It should be noticed that the first and last amount to the proposition that, in the case of triangles, if either of the two sets of conditions for the similarity of rectilineal figures be satisfied, then the other set must also be satisfied. So that the two sets of conditions for the similarity of rectilineal figures are not independent when the rectilineal figures are triangles. Art. 100. In dealing with similar triangles the reader will find it useful to draw the similar triangles separately if they happen to overlap, and to mark equal angles with the same numbers, as in the figure. A 1 2 3 B D 1 2 3 E F Fig. 84. Â Then those sides which join the equal angles have the same numbers at their extremities, and it is therefore at once evident that they are corresponding sides. If in the triangles ABC, DEF, A=Î, B=Ê, and Ĉ-Ê, let A and D be marked 1, let B and E be marked 2, and let C and F be marked 3. Write down all the possible pairs of the numbers 1, 2, 3, viz.:-23, 31, 12. Now 2 and 3 are at the extremities of BC in one triangle, and at the extremities of EF in the other. Hence BC, EF are corresponding sides. In like manner the positions of the numbers 3 and 1 indicate that CA, DF are corresponding sides, and the positions of the numbers 1 and 2 indicate that AB and DE are corresponding sides. .. BC: EF – CA : FD = AB : DE. Art. 101. PROPOSITION XXVI. (Euc. VI. 4.) ENUNCIATION. If the three angles of one triangle are respectively equal to the three angles of another triangle, then the triangles are similar. Those sides correspond which join the vertices of equal angles. From A on AB measure off a length AG equal to B DE, and then draw GH parallel to BC cutting AC at H. KCE Fig. 85. It will first be shown that the triangles AGH, DEF are congruent. * Observe that if the triangles are similar, the vertex A of the triangle ABC corresponds to the vertex D of the triangle DEF; and the side AB of the triangle ABC corresponds to the side DE of the triangle DEF. Hence BA: DE - CA: DF = CB : EF, which, taking the letters in order, may be more conveniently written AB: DE = BC: EF = CA: FD. [Prop. 10. Now AB, DE join the vertices of equal angles, and are therefore corresponding sides. In like manner BC corresponds to EF, and CA to FD. Hence the two sets of conditions for the similarity of the triangles ABC, DEF are satisfied. Hence the triangles are similar. Art. 102. NOTE. It may be noticed that corresponding sides of the triangles are opposite to equal angles; e.g. AB corresponds to DE, and they are opposite to the equal angles and respectively. ↑ Ê Art. 103. COROLLARY TO PROP. 26. If a triangle be cut by a straight line parallel to one of the sides, the triangular portion cut off is similar to the whole triangle. For with the figure of Prop. 26, GH may be regarded as any straight line parallel to BC, the triangles AGH, ABC are equiangular, and therefore similar by Prop. 26. Art. 104. EXAMPLES. 32. Show how to draw a straight line across two of the sides of a triangle, but not parallel to the third side, which will cut off a triangle similar to the original triangle. When will it be impossible to do this? 33. If ABC be a triangle inscribed in a circle, and CD a diameter of the circle, and AE a perpendicular from A on the side BC, show that the triangles AEB, ACD are similar. H. E. 8 Art. 105. PROPOSITION XXVII.* (Euc. VI. 5.) ENUNCIATION. If the sides taken in order of one triangle are proportional to the sides taken in order of another triangle, prove that the triangles are similar, and that those angles are equal which are opposite to corresponding sides. In the triangles ABC, DEF let it be given that to prove that the triangles ABC, DEF are similar. AA Fig. 86. (I) From A the vertex of the triangle ABC which corresponds to D, measure off on AB, the side corresponding to DE, a length AG equal to DE. Draw GH parallel to BC, cutting AC at H. It will first be shown that the triangles AGH and DEF are congruent. The triangles AGH and ABC have the angles of the one respectively equal to the angles of the other. Hence each of the three ratios marked (I) is equal to each of the three ratios marked (II). .. BC: EF BC: GH, .. EF = GH. * See Note 8. [Prop. 21. |